WEBVTT 00:00:00.080 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.050 Your support will help MIT OpenCourseWare 00:00:06.050 --> 00:00:10.150 continue to offer high-quality educational resources for free. 00:00:10.150 --> 00:00:12.690 To make a donation or to view additional materials 00:00:12.690 --> 00:00:16.600 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.600 --> 00:00:17.305 at ocw.mit.edu. 00:00:25.715 --> 00:00:26.640 PROFESSOR: All right. 00:00:26.640 --> 00:00:27.910 Welcome back. 00:00:27.910 --> 00:00:30.680 Today is probably our last lecture about 3SAT, 00:00:30.680 --> 00:00:34.680 but this time it's about planar 3SAT, which we've 00:00:34.680 --> 00:00:38.060 sort of alluded to in the past. 00:00:38.060 --> 00:00:40.310 And we saw a version of what you might 00:00:40.310 --> 00:00:44.250 call planar 3SAT with planar CircuitSAT last class. 00:00:44.250 --> 00:00:49.400 But planar 3SAT is a very useful special case of 3SAT. 00:00:49.400 --> 00:00:54.850 It is just like 3SAT, but you also 00:00:54.850 --> 00:01:09.630 are told that the bipartite graph, let's say of variables 00:01:09.630 --> 00:01:14.320 versus clauses is a planar graph. 00:01:18.360 --> 00:01:27.410 So the idea is you have some vertices representing 00:01:27.410 --> 00:01:28.290 variables. 00:01:28.290 --> 00:01:30.350 v for variable here. 00:01:30.350 --> 00:01:32.615 And then you separately have some clauses. 00:01:36.848 --> 00:01:42.162 I guess I should really have more variables. 00:01:42.162 --> 00:01:49.490 You have an edge between a variable and a clause whenever 00:01:49.490 --> 00:01:51.140 that clause includes that variable, 00:01:51.140 --> 00:01:53.390 either in positive or negative form. 00:01:53.390 --> 00:01:54.890 So maybe some of these edges, I'll 00:01:54.890 --> 00:01:56.610 draw a red to indicate, actually, 00:01:56.610 --> 00:02:01.640 that's v bar that is included in this clause. 00:02:01.640 --> 00:02:10.110 So this clause is vi or vj bar or vk. 00:02:10.110 --> 00:02:10.620 And so on. 00:02:10.620 --> 00:02:13.465 And this graph should be planar, shouldn't have any crossings. 00:02:17.999 --> 00:02:23.611 Something like that would be a valid input to planar 3SAT. 00:02:23.611 --> 00:02:24.110 Question. 00:02:24.110 --> 00:02:28.110 AUDIENCE: So if you're given the fact 00:02:28.110 --> 00:02:33.826 that a particular bipartite graph is planar, 00:02:33.826 --> 00:02:38.456 is there a polynomial operation to find a planar arrangement? 00:02:38.456 --> 00:02:39.080 PROFESSOR: Yes. 00:02:39.080 --> 00:02:40.810 Deciding whether a graph is planar 00:02:40.810 --> 00:02:43.970 and finding a planar embedding when it is is linear time. 00:02:43.970 --> 00:02:46.697 So here I'm just saying it's given as a planar graph, 00:02:46.697 --> 00:02:47.780 but not with an embedding. 00:02:47.780 --> 00:02:49.660 So we're going to use the embeddings in our reduction. 00:02:49.660 --> 00:02:50.380 Yeah, good point. 00:02:50.380 --> 00:02:52.040 So we do actually need the ability 00:02:52.040 --> 00:02:54.580 to draw these things without crossings. 00:02:54.580 --> 00:02:56.240 Good. 00:02:56.240 --> 00:02:59.390 So first thing I'm going to do is prove that this is hard. 00:02:59.390 --> 00:03:01.670 And usually I don't prove base problems are hard, 00:03:01.670 --> 00:03:05.600 but this is a sufficiently sophisticated base problem. 00:03:05.600 --> 00:03:08.620 I mean, we do lots of reductions from planar 3SAT. 00:03:08.620 --> 00:03:10.340 But the proof is actually from 3SAT, 00:03:10.340 --> 00:03:12.340 and I think it's a representative example of how 00:03:12.340 --> 00:03:14.970 to build a crossover in this kind of setting, 00:03:14.970 --> 00:03:16.830 and what kind of extra properties 00:03:16.830 --> 00:03:18.430 you can get out of it. 00:03:18.430 --> 00:03:20.910 So here it is. 00:03:20.910 --> 00:03:25.200 This is the proof in a single diagram. 00:03:25.200 --> 00:03:27.820 So the notation here is that the little green circles 00:03:27.820 --> 00:03:28.970 are clauses. 00:03:28.970 --> 00:03:32.070 And the bigger circles are the variables. 00:03:32.070 --> 00:03:33.820 Purple circles are variables. 00:03:33.820 --> 00:03:36.945 So the idea is it in your initial instance-- 00:03:36.945 --> 00:03:38.430 so we're going to reduce from 3SAT. 00:03:38.430 --> 00:03:41.330 We're given an arbitrary 3SAT instance. 00:03:41.330 --> 00:03:44.340 So that has a corresponding bipartite graph. 00:03:44.340 --> 00:03:45.420 Variables versus clauses. 00:03:45.420 --> 00:03:46.600 Just not planar. 00:03:46.600 --> 00:03:47.960 So consider any drawing. 00:03:47.960 --> 00:03:50.311 Actually, we'll look at a specific drawing in a moment. 00:03:50.311 --> 00:03:52.310 But it's going to have some crossings like this. 00:03:52.310 --> 00:03:54.393 This is a clause-variable connection, another one. 00:03:54.393 --> 00:03:55.170 They cross here. 00:03:55.170 --> 00:03:58.580 We're going to replace that with this picture, which 00:03:58.580 --> 00:04:02.707 is technically not 3SAT, because there's a four-variable clause 00:04:02.707 --> 00:04:03.290 in the center. 00:04:03.290 --> 00:04:04.400 But we can fix that. 00:04:04.400 --> 00:04:05.880 That's not too hard. 00:04:05.880 --> 00:04:08.310 Let me first convince you-- and this'll 00:04:08.310 --> 00:04:11.930 take a little while-- that this is equivalent to this. 00:04:11.930 --> 00:04:14.160 So the ultimate claim-- so there's 00:04:14.160 --> 00:04:17.649 four Greek letters in the center as extra variables. 00:04:17.649 --> 00:04:22.180 The main claim is that a1 equals a2 equals a. 00:04:22.180 --> 00:04:24.980 And b1 equals b2 equals b. 00:04:24.980 --> 00:04:27.820 If that's true, then, of course, this connection 00:04:27.820 --> 00:04:30.010 simulates that connection. 00:04:30.010 --> 00:04:32.950 And that connection simulates this connection. 00:04:32.950 --> 00:04:34.750 And the claim is as long as you satisfy 00:04:34.750 --> 00:04:36.750 that, you can always set the stuff in the middle 00:04:36.750 --> 00:04:37.890 to be happy. 00:04:37.890 --> 00:04:40.940 So those are the only constraints, basically. 00:04:40.940 --> 00:04:47.060 So let's start with over here and down here. 00:04:47.060 --> 00:04:52.150 These little sort of length two paths-- sorry, so the coloring 00:04:52.150 --> 00:04:54.010 is similar to what I did on the board. 00:04:54.010 --> 00:04:55.700 The red are negative connections, 00:04:55.700 --> 00:04:59.080 meaning that clause has not a and this clause has not a2. 00:04:59.080 --> 00:05:01.990 And the blue connections are positive connections. 00:05:01.990 --> 00:05:06.420 So when you have this sort of alternating four-cycle here, 00:05:06.420 --> 00:05:09.380 you can read that as not a or a2, 00:05:09.380 --> 00:05:12.030 which is the same as saying a implies a2. 00:05:12.030 --> 00:05:13.535 And also, a2 implies a. 00:05:13.535 --> 00:05:15.700 That means they have the same value. 00:05:15.700 --> 00:05:17.040 a if and only if a2. 00:05:17.040 --> 00:05:19.970 Or I'll just write equality of their truth values. 00:05:19.970 --> 00:05:20.670 Same thing here. 00:05:20.670 --> 00:05:26.380 So that part is just to get warmed up. 00:05:26.380 --> 00:05:29.130 Now the fun part is on the inside. 00:05:29.130 --> 00:05:31.950 And I think the next step is to think 00:05:31.950 --> 00:05:36.520 about these sort of triangles in the corner. 00:05:36.520 --> 00:05:39.100 If you stare at them for a little while, 00:05:39.100 --> 00:05:40.690 you get these kinds of constraints. 00:05:40.690 --> 00:05:42.490 So let's work through this one. 00:05:42.490 --> 00:05:46.260 You'll see there's a certain symmetry among what the alpha, 00:05:46.260 --> 00:05:48.330 beta, gamma, deltas are. 00:05:48.330 --> 00:05:53.220 They're going to be all sort of all patterns of a2 00:05:53.220 --> 00:05:58.060 possibly complemented, anded with b2 possibly complemented. 00:05:58.060 --> 00:06:00.905 So these are supposed to be the four cases of a versus b. 00:06:00.905 --> 00:06:03.650 The trick is that some of them are 2's and some of them are 00:06:03.650 --> 00:06:04.550 1's. 00:06:04.550 --> 00:06:07.160 So that's what makes it a little bit annoying. 00:06:07.160 --> 00:06:09.250 It would be nice to make it perfectly symmetric, 00:06:09.250 --> 00:06:13.440 but that would, in terms of these labels-- but that 00:06:13.440 --> 00:06:14.940 would destroy planarity. 00:06:14.940 --> 00:06:17.940 So this is the planar version. 00:06:17.940 --> 00:06:20.300 So let's look at this. 00:06:20.300 --> 00:06:26.410 We have, for example, not alpha or b2. 00:06:26.410 --> 00:06:28.520 So alpha implies b2. 00:06:28.520 --> 00:06:30.740 Also, alpha implies a2. 00:06:30.740 --> 00:06:34.730 So if alpha is true, then they must both be true. 00:06:34.730 --> 00:06:39.070 And conversely, I want a2 and b2 to imply alpha. 00:06:39.070 --> 00:06:39.570 Right. 00:06:39.570 --> 00:06:41.740 So if these are both true, then this guy's 00:06:41.740 --> 00:06:43.530 unsatisfied from those two red clauses. 00:06:43.530 --> 00:06:44.450 So alpha must be true. 00:06:44.450 --> 00:06:44.950 Good. 00:06:44.950 --> 00:06:46.280 So it works in both directions. 00:06:46.280 --> 00:06:47.230 You get equality. 00:06:47.230 --> 00:06:49.396 It's exactly the same in each of these four corners, 00:06:49.396 --> 00:06:52.250 just the coloring is different. 00:06:52.250 --> 00:06:53.840 And you're connected to a different b. 00:06:53.840 --> 00:06:56.960 So here if you reflect, you're connected to b1 instead of b2. 00:06:56.960 --> 00:06:58.990 And there's a complement here basically 00:06:58.990 --> 00:07:03.904 because these two colors switched from here. 00:07:03.904 --> 00:07:05.320 So it's exactly the same argument, 00:07:05.320 --> 00:07:08.060 but you just imagine flipping the value of b1. 00:07:08.060 --> 00:07:10.120 And then reflect over here. 00:07:10.120 --> 00:07:12.060 Switch the value of the a's. 00:07:12.060 --> 00:07:15.030 Reflect over here, switch the coloring of the b's. 00:07:15.030 --> 00:07:18.550 So that's what's happening in the corners. 00:07:18.550 --> 00:07:22.930 And at this point, I had to draw a table of all the cases. 00:07:22.930 --> 00:07:23.830 Oh, sorry. 00:07:23.830 --> 00:07:26.595 This is a little bit about what happens in the center. 00:07:26.595 --> 00:07:29.220 So of course, the center clause says that at least one of these 00:07:29.220 --> 00:07:29.720 is true. 00:07:29.720 --> 00:07:32.680 We want exactly one to be true, because exactly one 00:07:32.680 --> 00:07:34.280 of these cases should happen. 00:07:34.280 --> 00:07:36.350 But that's a weak constraint. 00:07:36.350 --> 00:07:38.770 And then we also have some constraints 00:07:38.770 --> 00:07:47.010 like if alpha is true, then this is not satisfied from alpha. 00:07:47.010 --> 00:07:48.000 And similarly, this. 00:07:48.000 --> 00:07:50.580 And therefore, beta must be false. 00:07:50.580 --> 00:07:53.430 So if alpha is true, then beta and delta are false. 00:07:53.430 --> 00:07:55.080 I think that's actually all you need. 00:07:55.080 --> 00:07:56.720 I think these might not be necessary. 00:07:56.720 --> 00:07:59.500 But they make for a nice symmetric diagram. 00:07:59.500 --> 00:08:01.290 And they don't hurt. 00:08:01.290 --> 00:08:06.326 So now I drew a table-- there may be a more direct argument-- 00:08:06.326 --> 00:08:09.710 of all the possible settings a2 and b2, 00:08:09.710 --> 00:08:12.110 and what you can derive from that. 00:08:12.110 --> 00:08:15.890 And everything except this part is really easy to derive, 00:08:15.890 --> 00:08:18.920 assuming I can remember what I did last night. 00:08:18.920 --> 00:08:22.420 So let's say a2 and b2 are false. 00:08:22.420 --> 00:08:25.990 I claim alpha is false, because we have this equation. 00:08:25.990 --> 00:08:26.850 That's easy. 00:08:26.850 --> 00:08:29.100 And in general, you can compute alpha directly, 00:08:29.100 --> 00:08:30.960 because it's just a function of a2 and b2. 00:08:30.960 --> 00:08:32.350 It's the end function. 00:08:32.350 --> 00:08:33.659 So that's good. 00:08:33.659 --> 00:08:34.409 What about beta? 00:08:34.409 --> 00:08:42.429 I claim if alpha is false here, then in this particular case-- 00:08:42.429 --> 00:08:43.059 well, OK. 00:08:43.059 --> 00:08:44.400 Here's one thing we wrote. 00:08:44.400 --> 00:08:46.570 Alpha implies not beta and not gamma. 00:08:46.570 --> 00:08:49.370 So in this last row, if alpha is true, 00:08:49.370 --> 00:08:52.510 we know that beta and gamma are false. 00:08:52.510 --> 00:08:53.120 That's good. 00:08:53.120 --> 00:08:57.000 We hope also that beta and delta are false. 00:08:57.000 --> 00:08:58.430 We hope also that gamma is false. 00:08:58.430 --> 00:09:03.710 But I don't think we need that, because if you 00:09:03.710 --> 00:09:07.920 look at this equation on delta, which we know 00:09:07.920 --> 00:09:14.000 is 0, that must mean that one of these two things is 0. 00:09:14.000 --> 00:09:15.910 But b2 is true. 00:09:15.910 --> 00:09:16.410 Sorry. 00:09:16.410 --> 00:09:18.243 I'm going to use true and 1 interchangeably, 00:09:18.243 --> 00:09:21.630 and 0 and false for whatever reason today. 00:09:21.630 --> 00:09:25.920 So this needs to be 0, but this is 1. 00:09:25.920 --> 00:09:29.690 So this must be 0, which means that a1 is 1. 00:09:29.690 --> 00:09:33.100 And symmetrically on the top, this is a symmetric between a 00:09:33.100 --> 00:09:33.600 and b. 00:09:33.600 --> 00:09:39.040 So up here, you know that a2 is 1. 00:09:39.040 --> 00:09:42.990 And therefore, b1 bar must be 0. 00:09:42.990 --> 00:09:44.480 So b1 must be 1. 00:09:44.480 --> 00:09:46.644 So we transfer the information. 00:09:46.644 --> 00:09:48.810 And the only other thing to check in all these cases 00:09:48.810 --> 00:09:50.518 is that actually everything is satisfied. 00:09:50.518 --> 00:09:52.246 But that's sort of boring. 00:09:52.246 --> 00:09:53.870 I want to do the other direction, which 00:09:53.870 --> 00:09:56.150 is that you are forced to communicate 00:09:56.150 --> 00:09:58.259 the copies of a and the copies of b. 00:09:58.259 --> 00:09:59.300 So that was the last row. 00:09:59.300 --> 00:10:00.674 I think the first row should also 00:10:00.674 --> 00:10:03.970 be easy for a slightly different reason. 00:10:03.970 --> 00:10:07.575 So a2 and b2 are false. 00:10:10.830 --> 00:10:11.910 Alpha is false. 00:10:11.910 --> 00:10:12.830 Beta is false. 00:10:12.830 --> 00:10:16.810 That's as far as we've gotten so far. 00:10:16.810 --> 00:10:18.350 Let's see. 00:10:18.350 --> 00:10:24.440 Delta is a1 bar and b2, right? 00:10:27.781 --> 00:10:28.280 Right. 00:10:28.280 --> 00:10:31.835 Because b2 is false, we know that delta is false. 00:10:31.835 --> 00:10:33.820 So it doesn't matter what a1 is. 00:10:33.820 --> 00:10:35.650 So it's sort of the reverse direction. 00:10:35.650 --> 00:10:39.490 And similarly, we should be able to-- oh, 00:10:39.490 --> 00:10:42.221 because from this, we know at least one of them is true, 00:10:42.221 --> 00:10:43.970 at least one of the Greek letters is true, 00:10:43.970 --> 00:10:45.410 then we've got three down. 00:10:45.410 --> 00:10:47.000 So it must be gamma that is true. 00:10:47.000 --> 00:10:48.500 And once you know gamma is true, you 00:10:48.500 --> 00:10:50.230 know exactly what a1 and b1 are. 00:10:50.230 --> 00:10:52.510 So that's another good case. 00:10:52.510 --> 00:10:53.940 These are slightly more annoying, 00:10:53.940 --> 00:10:55.880 at least as far as I could see. 00:10:55.880 --> 00:10:58.090 Maybe there's a cleaner argument. 00:10:58.090 --> 00:10:59.820 So let's say they're symmetric, though. 00:10:59.820 --> 00:11:01.640 So let's do the second row. 00:11:01.640 --> 00:11:04.000 So a2 is 0. 00:11:04.000 --> 00:11:06.220 b2 is 0. 00:11:06.220 --> 00:11:08.710 I think just like this argument, we 00:11:08.710 --> 00:11:13.470 get that beta is 0, because beta involves a2. 00:11:13.470 --> 00:11:18.000 We know a2 is 0, so beta equals 0, just like this case, 00:11:18.000 --> 00:11:19.010 actually. 00:11:19.010 --> 00:11:19.510 OK. 00:11:19.510 --> 00:11:21.480 So that's good for that. 00:11:21.480 --> 00:11:23.590 I think it's harder to figure out delta. 00:11:26.700 --> 00:11:28.480 Or here's one way to argue it. 00:11:28.480 --> 00:11:30.170 Suppose that gamma were 1. 00:11:30.170 --> 00:11:34.700 It should be 0, because gamma's supposed to be only 1 00:11:34.700 --> 00:11:36.480 in this situation. 00:11:36.480 --> 00:11:43.290 If it's 1, then I claim that alpha 1 and beta 1 must be 0. 00:11:43.290 --> 00:11:45.570 That's from this equation. 00:11:45.570 --> 00:11:48.150 And then we should get a contradiction. 00:11:48.150 --> 00:11:49.530 Help me find the contradiction. 00:11:49.530 --> 00:11:51.500 AUDIENCE: Delta becomes 1. 00:11:51.500 --> 00:11:53.060 PROFESSOR: Delta becomes 1. 00:11:53.060 --> 00:11:55.920 AUDIENCE: And then the clause between gamma and delta is-- 00:11:55.920 --> 00:11:57.530 PROFESSOR: And then this is unhappy. 00:11:57.530 --> 00:11:58.030 Good. 00:11:58.030 --> 00:11:58.530 OK. 00:11:58.530 --> 00:12:01.960 So we do need a couple other constraints like this one. 00:12:01.960 --> 00:12:02.460 Cool. 00:12:02.460 --> 00:12:03.620 So that's the idea. 00:12:03.620 --> 00:12:05.710 Therefore, gamma is 0. 00:12:05.710 --> 00:12:08.690 And then by that, delta must be 1. 00:12:08.690 --> 00:12:10.406 And then we're happy again. 00:12:10.406 --> 00:12:11.040 OK? 00:12:11.040 --> 00:12:14.490 So that's the idea in a nutshell. 00:12:14.490 --> 00:12:17.010 There's maybe other ways to see it. 00:12:17.010 --> 00:12:20.930 And this is an old proof by Lichtenstein in 1982. 00:12:20.930 --> 00:12:21.741 It's a cool paper. 00:12:21.741 --> 00:12:22.990 It has a lot of results in it. 00:12:22.990 --> 00:12:25.140 We'll be covering a few different results. 00:12:25.140 --> 00:12:28.320 But at this point, we have shown this. 00:12:28.320 --> 00:12:30.350 You just apply this to each crossing. 00:12:30.350 --> 00:12:34.100 Eventually you get a planar bipartite graph 00:12:34.100 --> 00:12:37.280 that's satisfiable if and only if the original is a planar 00:12:37.280 --> 00:12:39.150 3SAT is NP-hard. 00:12:39.150 --> 00:12:39.830 Question? 00:12:39.830 --> 00:12:41.452 So many questions. 00:12:41.452 --> 00:12:43.160 AUDIENCE: I guess this is probably silly. 00:12:43.160 --> 00:12:45.544 But [INAUDIBLE] 3SAT, every clause 00:12:45.544 --> 00:12:46.710 had to have three variables. 00:12:46.710 --> 00:12:47.060 PROFESSOR: Yes. 00:12:47.060 --> 00:12:47.430 OK. 00:12:47.430 --> 00:12:47.964 Right. 00:12:47.964 --> 00:12:48.880 AUDIENCE: Two or four. 00:12:48.880 --> 00:12:51.630 PROFESSOR: I cheated a little bit here. 00:12:51.630 --> 00:12:52.130 OK. 00:12:52.130 --> 00:12:53.020 Yeah. 00:12:53.020 --> 00:12:53.695 Some have two. 00:12:56.510 --> 00:13:00.790 So for that, I need to be able to construct the value false. 00:13:00.790 --> 00:13:01.960 And some have four. 00:13:01.960 --> 00:13:06.440 The four, I believe the claim is you 00:13:06.440 --> 00:13:09.460 can take a little clause like this 00:13:09.460 --> 00:13:17.310 and convert it into a clause with another variable. 00:13:17.310 --> 00:13:19.727 AUDIENCE: And one of the two edges connecting 00:13:19.727 --> 00:13:20.810 the variables [INAUDIBLE]. 00:13:20.810 --> 00:13:21.518 PROFESSOR: Right. 00:13:24.130 --> 00:13:24.710 Like this? 00:13:30.720 --> 00:13:34.010 So the claim is this four-way clause to the 4SAT clause 00:13:34.010 --> 00:13:36.570 is equivalent to these two 3SAT clauses. 00:13:36.570 --> 00:13:38.210 Let's call this new variable x. 00:13:45.500 --> 00:13:47.270 So the idea is x is going to represent 00:13:47.270 --> 00:13:50.640 whether the left side satisfies the clause or the right side 00:13:50.640 --> 00:13:53.480 satisfies the clause. 00:13:53.480 --> 00:13:56.055 If x is 1, then it satisfies this for free. 00:13:56.055 --> 00:13:58.430 But it doesn't satisfy this one, so at least one of these 00:13:58.430 --> 00:14:00.470 must be satisfied. 00:14:00.470 --> 00:14:05.470 On the other hand, if x is 0, then it satisfies this one. 00:14:05.470 --> 00:14:07.890 And so at least one of these two should be satisfied. 00:14:07.890 --> 00:14:08.390 OK? 00:14:08.390 --> 00:14:14.360 So that's a reduction from 4SAT to 3SAT, so to speak. 00:14:14.360 --> 00:14:18.580 Now, I didn't create any degree-two clauses, 00:14:18.580 --> 00:14:21.050 but I have tons of degree-two clauses over there. 00:14:21.050 --> 00:14:22.782 AUDIENCE: Just use parallel edges. 00:14:22.782 --> 00:14:24.240 PROFESSOR: Just use parallel edges. 00:14:24.240 --> 00:14:30.000 So you could just repeat the same variable in the clause 00:14:30.000 --> 00:14:31.330 multiple times. 00:14:31.330 --> 00:14:32.490 That sounds good. 00:14:32.490 --> 00:14:35.340 I don't know if there's an easy way to avoid multiple edges. 00:14:35.340 --> 00:14:36.560 I think it's a little tricky. 00:14:36.560 --> 00:14:39.360 You can't construct false in the CircuitSAT sense, 00:14:39.360 --> 00:14:42.542 because everything you build should be satisfiable. 00:14:42.542 --> 00:14:44.000 So you have to be a little careful. 00:14:44.000 --> 00:14:46.950 But it might be possible to pair them up, 00:14:46.950 --> 00:14:53.772 because there should be an even number of those guys. 00:14:53.772 --> 00:14:56.230 And so maybe you pair them up and connect them in some way. 00:14:56.230 --> 00:14:58.260 But let's just allow multiple edges here. 00:14:58.260 --> 00:14:59.715 Yeah? 00:14:59.715 --> 00:15:01.655 AUDIENCE: In the original instance of 3SAT, 00:15:01.655 --> 00:15:06.990 does it allow two clauses with two variables? 00:15:06.990 --> 00:15:08.024 For getting planar? 00:15:08.024 --> 00:15:08.690 PROFESSOR: Yeah. 00:15:08.690 --> 00:15:12.580 I mean, in general, this would work for any kind of SAT, 00:15:12.580 --> 00:15:14.330 any kind of CNF-SAT. 00:15:14.330 --> 00:15:18.199 We're not relying on three-ness at all in this picture. 00:15:18.199 --> 00:15:20.240 But it's not going to split things up, of course. 00:15:20.240 --> 00:15:21.740 AUDIENCE: But the definition of 3SAT 00:15:21.740 --> 00:15:24.010 has three distinct variables in each clause. 00:15:24.010 --> 00:15:25.760 PROFESSOR: There are a couple definitions. 00:15:25.760 --> 00:15:28.910 One has exactly three, but we never, I 00:15:28.910 --> 00:15:31.132 think, specify that they are distinct. 00:15:31.132 --> 00:15:33.090 I think that is hard when they're all distinct. 00:15:33.090 --> 00:15:36.740 But I don't know the reduction offhand. 00:15:36.740 --> 00:15:38.990 I think the way we define it is there's exactly three, 00:15:38.990 --> 00:15:41.750 but we didn't say any uniqueness. 00:15:41.750 --> 00:15:42.640 So let's say that. 00:15:42.640 --> 00:15:44.670 And then we're still happy here, because we 00:15:44.670 --> 00:15:47.488 can duplicate variables. 00:15:47.488 --> 00:15:48.300 Yeah? 00:15:48.300 --> 00:15:53.900 AUDIENCE: Why did we include the a equals a2, and b equals b2? 00:15:53.900 --> 00:15:56.160 PROFESSOR: I think it's for the next slide. 00:15:56.160 --> 00:15:58.860 [LAUGHTER] 00:15:58.860 --> 00:16:04.760 The next claim is that not only is that bipartite graph planar, 00:16:04.760 --> 00:16:08.350 but it also remains planar if we connect all the variables 00:16:08.350 --> 00:16:21.930 together in a cycle. 00:16:34.740 --> 00:16:36.950 So let's say we number the variables 1 through n. 00:16:36.950 --> 00:16:39.240 It doesn't matter how you number them. 00:16:39.240 --> 00:16:41.120 Well, in a certain sense. 00:16:41.120 --> 00:16:45.210 You can restrict to instances of 3SAT 00:16:45.210 --> 00:16:51.940 that this cycle plus the variable clause adjacency graph 00:16:51.940 --> 00:16:55.350 is together planar. 00:16:55.350 --> 00:16:56.310 OK? 00:16:56.310 --> 00:17:02.150 So that's less clear here. 00:17:02.150 --> 00:17:03.680 But let me prove it. 00:17:03.680 --> 00:17:07.000 This is proved in the same paper. 00:17:07.000 --> 00:17:12.430 So here's a top-level diagram of what we're trying to do. 00:17:12.430 --> 00:17:14.160 So we have clauses on one hand. 00:17:14.160 --> 00:17:15.839 Let's put them all on the y-axis. 00:17:15.839 --> 00:17:17.750 We have variables on the other hand. 00:17:17.750 --> 00:17:21.369 These are the original variables in the original 3SAT instance. 00:17:21.369 --> 00:17:23.060 Put them on the x-axis. 00:17:23.060 --> 00:17:25.579 Then draw all the connections in the bipartite graph. 00:17:25.579 --> 00:17:30.434 And furthermore, draw this cycle on the vertices down here. 00:17:30.434 --> 00:17:31.350 OK? 00:17:31.350 --> 00:17:33.240 And obviously, we put them in the order 00:17:33.240 --> 00:17:34.490 that they appear in the cycle. 00:17:34.490 --> 00:17:37.980 Otherwise we're making our lives harder. 00:17:37.980 --> 00:17:39.820 And here is that crossover gadget. 00:17:39.820 --> 00:17:42.480 This is how it's originally drawn. 00:17:42.480 --> 00:17:45.580 In this case, it's already been expanded in the center 00:17:45.580 --> 00:17:46.830 into that picture over there. 00:17:46.830 --> 00:17:50.460 This is where I got that reduction from. 00:17:50.460 --> 00:17:52.410 And then there's no color here, but you'll 00:17:52.410 --> 00:17:57.800 notice the curvy line is an attempt to connect together 00:17:57.800 --> 00:18:00.210 all of the variables that appear in the crossover 00:18:00.210 --> 00:18:06.420 gadget in linear order without introducing any crossings. 00:18:06.420 --> 00:18:11.790 So that diagram is still planar when we add all of those edges. 00:18:11.790 --> 00:18:15.300 And so what this says is that if we-- we're 00:18:15.300 --> 00:18:17.590 trying to build a global cycle that 00:18:17.590 --> 00:18:18.952 connects all the variables. 00:18:18.952 --> 00:18:20.910 So obviously, we have this for these variables, 00:18:20.910 --> 00:18:23.410 but we need to add into this cycle 00:18:23.410 --> 00:18:26.170 all the variables that appear in all these crossings. 00:18:26.170 --> 00:18:29.020 And so what this says is if we are coming in either here 00:18:29.020 --> 00:18:32.070 or here into a1, we can follow this path 00:18:32.070 --> 00:18:35.900 and then leave either here or here on a2. 00:18:35.900 --> 00:18:37.599 OK? 00:18:37.599 --> 00:18:39.390 And because the diagram is symmetric, also, 00:18:39.390 --> 00:18:43.620 if you're coming into b1, you can end up in b2. 00:18:43.620 --> 00:18:48.070 And I'm guessing this is why we separate b2. 00:18:48.070 --> 00:18:51.890 The reason we separated b2 from b in general 00:18:51.890 --> 00:18:54.180 is to basically add this connection, 00:18:54.180 --> 00:18:57.970 so we can separate this gadget from this gadget. 00:18:57.970 --> 00:19:00.860 So just showing that we can copy and extend things. 00:19:00.860 --> 00:19:02.660 It may not be necessary, but it definitely 00:19:02.660 --> 00:19:04.330 matches this diagram. 00:19:04.330 --> 00:19:04.830 OK. 00:19:04.830 --> 00:19:07.160 So this diagram is this diagram, but where 00:19:07.160 --> 00:19:11.210 every intersection plus a few extra places, like over here, 00:19:11.210 --> 00:19:12.560 we add these little diamonds. 00:19:12.560 --> 00:19:15.230 The diamonds indicate that gadget. 00:19:15.230 --> 00:19:19.240 Then sorry there's no color, but in curved lines 00:19:19.240 --> 00:19:21.950 is the attempt to connect all the variables together 00:19:21.950 --> 00:19:23.260 in one single path. 00:19:23.260 --> 00:19:23.935 Question. 00:19:23.935 --> 00:19:26.110 AUDIENCE: So what are the extra crossover gadgets 00:19:26.110 --> 00:19:27.570 that don't cross anything over? 00:19:27.570 --> 00:19:30.290 PROFESSOR: They're just to do this path thing. 00:19:30.290 --> 00:19:32.280 So what are they? 00:19:32.280 --> 00:19:33.860 I mean, they are exactly this gadget, 00:19:33.860 --> 00:19:35.484 just there's no connection on the left. 00:19:35.484 --> 00:19:38.320 AUDIENCE: So they're a crossover of the path on the variables. 00:19:38.320 --> 00:19:39.300 Is that right? 00:19:39.300 --> 00:19:39.570 PROFESSOR: Yeah. 00:19:39.570 --> 00:19:41.200 So instead of going straight here, 00:19:41.200 --> 00:19:43.680 you just put a crossover gadget in the middle. 00:19:43.680 --> 00:19:44.940 You can still go through. 00:19:44.940 --> 00:19:47.260 You're copying some arbitrary information here 00:19:47.260 --> 00:19:49.030 to some arbitrary information here. 00:19:49.030 --> 00:19:51.470 So you don't preserve the number of solutions, let's say. 00:19:51.470 --> 00:19:53.980 But who cares? 00:19:53.980 --> 00:19:57.500 The reason for that is so that we can take this path, 00:19:57.500 --> 00:19:59.700 connect to this vertex, and then get out over here. 00:19:59.700 --> 00:20:02.470 AUDIENCE: Yeah, that's what I meant by crossing. 00:20:02.470 --> 00:20:07.090 PROFESSOR: So that we can do this kind of loop thing. 00:20:07.090 --> 00:20:08.530 So how do we follow the path? 00:20:08.530 --> 00:20:10.940 I mean, we basically do it in scan-line order. 00:20:10.940 --> 00:20:13.717 It's like some printers used to do 00:20:13.717 --> 00:20:15.300 this, where they print back and forth, 00:20:15.300 --> 00:20:16.790 left and right, line printers. 00:20:16.790 --> 00:20:19.010 So we come in here. 00:20:19.010 --> 00:20:21.950 I don't know quite why they draw it coming in the middle here. 00:20:21.950 --> 00:20:24.410 But it's essentially coming to this vertex. 00:20:24.410 --> 00:20:27.070 Then ending on this vertex on the left side. 00:20:27.070 --> 00:20:31.524 Then going into this vertex, which brings us to here. 00:20:31.524 --> 00:20:32.690 Let's say coming on the top. 00:20:32.690 --> 00:20:33.670 Doesn't matter. 00:20:33.670 --> 00:20:36.120 And then we go to the right till we get to the end. 00:20:36.120 --> 00:20:39.140 Here we add an extra crossover, so we can basically cross over 00:20:39.140 --> 00:20:40.960 this line with our path. 00:20:40.960 --> 00:20:42.870 And add another one. 00:20:42.870 --> 00:20:45.600 And whenever we visit an intersection, we can just go. 00:20:45.600 --> 00:20:47.516 And then we go to the next line and so on. 00:20:47.516 --> 00:20:48.600 We clear? 00:20:48.600 --> 00:20:51.560 Then at the very end, we're either on the right or left, 00:20:51.560 --> 00:20:53.450 depending on parity. 00:20:53.450 --> 00:20:55.420 Probably we want to be on the right. 00:20:55.420 --> 00:20:58.000 So if we're on the left, we'll just add extra crossovers here 00:20:58.000 --> 00:20:59.370 to end up on the right. 00:20:59.370 --> 00:21:02.340 Then we can go through all these variables in reverse order 00:21:02.340 --> 00:21:04.050 and then come back up here. 00:21:04.050 --> 00:21:08.700 So that will be one non-crossing path in this planar embedding. 00:21:08.700 --> 00:21:12.180 Therefore, we've constructed a new formula 00:21:12.180 --> 00:21:14.520 where the bipartite graph plus the path through all 00:21:14.520 --> 00:21:16.220 the variables is planar. 00:21:16.220 --> 00:21:19.230 And so this new version of planar 3SAT is hard. 00:21:19.230 --> 00:21:22.046 This version is usually called planar 3SAT, but if you 00:21:22.046 --> 00:21:23.420 want to distinguish it, you could 00:21:23.420 --> 00:21:27.893 call it planar 3SAT with a variable cycle or something. 00:21:27.893 --> 00:21:29.160 Yeah? 00:21:29.160 --> 00:21:31.305 AUDIENCE: Why is it that we care about this? 00:21:31.305 --> 00:21:32.430 PROFESSOR: Why do you care? 00:21:32.430 --> 00:21:33.550 Aha. 00:21:33.550 --> 00:21:35.680 We'll get to that a little bit later. 00:21:35.680 --> 00:21:38.060 Well, we've seen a lot of proofs, actually, 00:21:38.060 --> 00:21:41.500 where you visit a variable and make a choice, 00:21:41.500 --> 00:21:44.860 and then visit the next variable and make a choice, and so on. 00:21:44.860 --> 00:21:47.790 And when you make a choice, you go visit the clauses and things 00:21:47.790 --> 00:21:48.540 like that. 00:21:48.540 --> 00:21:51.170 So when we're reducing from planar 3SAT, 00:21:51.170 --> 00:21:54.080 it's often handy to have this traversable path. 00:21:54.080 --> 00:21:58.500 If we have a robot or somebody moving around, 00:21:58.500 --> 00:22:02.620 then we can guarantee planarity of that motion still. 00:22:02.620 --> 00:22:04.490 And the connections between the variables 00:22:04.490 --> 00:22:05.790 don't have to cross anything. 00:22:05.790 --> 00:22:09.050 In general, our goal here is to avoid crossover gadgets 00:22:09.050 --> 00:22:12.260 when we're reducing from 3SAT to something else. 00:22:12.260 --> 00:22:15.080 So this gives you kind of a generic crossover, 00:22:15.080 --> 00:22:17.600 so that as long as you can do variables and clauses more 00:22:17.600 --> 00:22:20.841 or less independently and you can build connections, 00:22:20.841 --> 00:22:23.340 then you don't have to worry about the connections crossing. 00:22:23.340 --> 00:22:23.909 Yeah? 00:22:23.909 --> 00:22:25.950 AUDIENCE: When you're given this bipartite graph, 00:22:25.950 --> 00:22:30.500 how do you actually tell whether an edge is negated or not? 00:22:33.950 --> 00:22:36.980 PROFESSOR: You're still given a formula, let's say. 00:22:36.980 --> 00:22:39.704 And then you're just told this extra fact 00:22:39.704 --> 00:22:42.120 that the bipartite graph plus this path and the variables, 00:22:42.120 --> 00:22:44.880 in that order, is planar. 00:22:44.880 --> 00:22:46.894 AUDIENCE: The variable nodes, you 00:22:46.894 --> 00:22:49.301 have one for every variable and its first negation? 00:22:49.301 --> 00:22:50.070 Or do you just-- 00:22:50.070 --> 00:22:50.710 PROFESSOR: No. 00:22:50.710 --> 00:22:51.640 Sorry. 00:22:51.640 --> 00:22:55.050 There's only vertices for variables and for clauses, 00:22:55.050 --> 00:22:57.130 not for literals. 00:22:57.130 --> 00:22:59.730 But the next version is actually about that case. 00:23:03.890 --> 00:23:12.629 The next one is-- so the idea, if you were just 00:23:12.629 --> 00:23:14.170 given the bipartite graph, you'd have 00:23:14.170 --> 00:23:16.850 to be told the coloring of the edges here 00:23:16.850 --> 00:23:18.690 to know which literal you're talking about. 00:23:21.240 --> 00:23:24.590 But also, the problem remains planar 00:23:24.590 --> 00:23:41.720 if we use literals instead of variables. 00:23:41.720 --> 00:23:44.990 So if you were wondering about that version, 00:23:44.990 --> 00:23:47.540 that will also be planar. 00:23:47.540 --> 00:23:49.400 It doesn't follow immediately. 00:23:49.400 --> 00:23:52.525 But the idea is suppose we have vi. 00:23:52.525 --> 00:23:54.930 So in the regular bipartite graph, 00:23:54.930 --> 00:23:58.870 we have some positive edges and we have some negative edges. 00:23:58.870 --> 00:24:07.990 Instead, I want to represent that as vi and vi bar. 00:24:07.990 --> 00:24:10.176 And the positive connections are over here. 00:24:10.176 --> 00:24:11.800 The negative connections are over here. 00:24:11.800 --> 00:24:12.970 Now I don't need the two coloring, 00:24:12.970 --> 00:24:15.260 because I have the labels to tell me which is which. 00:24:15.260 --> 00:24:17.740 And furthermore, I can add this edge connecting them. 00:24:17.740 --> 00:24:18.870 That will still be planar. 00:24:18.870 --> 00:24:21.990 So just split vi into two parts. 00:24:21.990 --> 00:24:25.600 What this tells me is that the positive connections to vi 00:24:25.600 --> 00:24:27.780 appear as a contiguous chunk. 00:24:27.780 --> 00:24:30.550 And the negative connections appear as a contiguous chunk. 00:24:30.550 --> 00:24:32.920 So it's not like alternating white, red, white, red. 00:24:32.920 --> 00:24:36.690 So far, it is if you look at the reduction. 00:24:36.690 --> 00:24:39.200 But we can fix that. 00:24:39.200 --> 00:24:39.700 OK. 00:24:42.640 --> 00:24:50.460 So I just want to add, with an edge between the xi and xi bar. 00:24:50.460 --> 00:24:50.960 Yeah? 00:24:50.960 --> 00:24:52.956 AUDIENCE: When you say you add an edge between them, 00:24:52.956 --> 00:24:53.902 what does that mean? 00:24:53.902 --> 00:24:55.318 Because all the edges are supposed 00:24:55.318 --> 00:24:57.220 to be in between literals and clauses. 00:24:57.220 --> 00:24:59.720 PROFESSOR: Well, I mean, the graph already wasn't bipartite. 00:25:02.430 --> 00:25:05.520 We start with a bipartite graph between variables and literals. 00:25:05.520 --> 00:25:07.680 Then we also add the edges between variables 00:25:07.680 --> 00:25:08.780 and their complements. 00:25:08.780 --> 00:25:14.310 And then we also add a cycle among all the variables, 00:25:14.310 --> 00:25:18.020 among all the literals, let's say. 00:25:18.020 --> 00:25:20.830 So I guess this could be part of the cycle, for example. 00:25:20.830 --> 00:25:23.984 And that graph, which is not bipartite, must be planar. 00:25:23.984 --> 00:25:24.900 That's the constraint. 00:25:24.900 --> 00:25:28.090 The claim is that all such problems are NP-hard. 00:25:28.090 --> 00:25:31.690 In general, we want to make as small a problem 00:25:31.690 --> 00:25:34.330 as we can, as special a case as we can NP-hard, 00:25:34.330 --> 00:25:35.790 because then we're reducing. 00:25:35.790 --> 00:25:38.500 We have more structure that we get to exploit. 00:25:38.500 --> 00:25:42.310 This is a special case of this is a special case of this. 00:25:42.310 --> 00:25:45.660 So the more we can confine it, the better. 00:25:45.660 --> 00:25:46.560 Cool. 00:25:46.560 --> 00:25:48.900 So this is proved also in the Lichtenstein paper. 00:25:48.900 --> 00:25:51.340 Although, we'll see a cleaner proof in a moment, 00:25:51.340 --> 00:25:53.310 so I won't cover this in too much detail. 00:25:53.310 --> 00:25:57.650 But the idea is this. 00:25:57.650 --> 00:26:01.660 So we're taking each variable, we're 00:26:01.660 --> 00:26:04.199 replacing it with a big cycle like this. 00:26:04.199 --> 00:26:06.240 I think it's the same kind of trick as over here. 00:26:06.240 --> 00:26:10.260 It doesn't matter. 00:26:10.260 --> 00:26:13.714 All these a's should be the same or sort of alternating. 00:26:13.714 --> 00:26:15.380 And we're connecting the negative side-- 00:26:15.380 --> 00:26:17.920 so this is really two vertices connected by an edge. 00:26:17.920 --> 00:26:19.680 But I draw it like that to make clear 00:26:19.680 --> 00:26:24.470 that that represents the same variable to literals of it. 00:26:24.470 --> 00:26:26.900 And do this thing. 00:26:26.900 --> 00:26:29.680 And then this is the path. 00:26:29.680 --> 00:26:33.290 So suppose the path used to go through this way. 00:26:33.290 --> 00:26:35.840 Then the path now bisects. 00:26:35.840 --> 00:26:38.490 And you see that all of the positive copies 00:26:38.490 --> 00:26:42.632 are on one side, and negative copies are on the other side. 00:26:42.632 --> 00:26:44.857 That's the idea. 00:26:44.857 --> 00:26:46.940 We will see another proof of a stronger statement, 00:26:46.940 --> 00:26:49.250 so I think let's not worry about this 00:26:49.250 --> 00:26:51.865 too much unless there are questions. 00:26:57.260 --> 00:26:58.530 Here's the stronger statement. 00:27:42.970 --> 00:27:50.000 So in general, the variable cycle-- 00:27:50.000 --> 00:27:52.840 so I'm going to not subdivide into literals for the moment. 00:27:52.840 --> 00:27:56.190 But we could put that back. 00:27:56.190 --> 00:27:59.130 So here's the variable cycle. 00:27:59.130 --> 00:28:01.370 This decomposes the plane at two regions-- 00:28:01.370 --> 00:28:03.015 the interior and the exterior. 00:28:03.015 --> 00:28:04.390 Or course, I could flip the edge, 00:28:04.390 --> 00:28:06.220 and it doesn't matter which one's interior and exterior. 00:28:06.220 --> 00:28:07.780 But there are two regions. 00:28:07.780 --> 00:28:11.610 And what I'm going to require is that every clause 00:28:11.610 --> 00:28:16.730 that I draw here should have entirely positive connections 00:28:16.730 --> 00:28:18.120 on this side. 00:28:18.120 --> 00:28:22.320 And every clause that I draw out here 00:28:22.320 --> 00:28:25.235 should have entirely negative connections. 00:28:28.150 --> 00:28:28.650 OK? 00:28:28.650 --> 00:28:32.240 This is monotone 3SAT again. 00:28:32.240 --> 00:28:34.450 So this implies monotone. 00:28:38.110 --> 00:28:39.397 We know monotone 3SAT is hard. 00:28:39.397 --> 00:28:41.230 Now we're claiming that planar monotone 3SAT 00:28:41.230 --> 00:28:45.670 is hard when all the positive clauses are 00:28:45.670 --> 00:28:47.210 on one side of the variable cycle, 00:28:47.210 --> 00:28:49.335 and all the negative clauses are on the other side. 00:28:49.335 --> 00:28:52.454 This is really helpful. 00:28:52.454 --> 00:28:54.120 I haven't yet given this problem a name, 00:28:54.120 --> 00:28:56.250 because it has a name that uses another word, which 00:28:56.250 --> 00:28:57.970 I should first introduce. 00:28:57.970 --> 00:29:00.965 So that's the next page. 00:29:11.539 --> 00:29:16.230 That word is "rectilinear." so this 00:29:16.230 --> 00:29:18.520 is a relatively simple modification to what we 00:29:18.520 --> 00:29:21.040 have already. 00:29:21.040 --> 00:29:24.690 I want the variables to live on the x-axis. 00:29:24.690 --> 00:29:27.590 You can think of them as horizontal segments 00:29:27.590 --> 00:29:28.810 on the x-axis. 00:29:28.810 --> 00:29:31.430 I prefer this drawing. 00:29:31.430 --> 00:29:34.450 So imagine each variable is like a little horizontal segment, 00:29:34.450 --> 00:29:36.500 or in this case, a box. 00:29:36.500 --> 00:29:38.990 And the clauses are horizontal segments. 00:29:38.990 --> 00:29:41.540 And then they're connected by vertical segments 00:29:41.540 --> 00:29:43.970 between the clause and the variable. 00:29:43.970 --> 00:29:47.690 And all of the clauses that are above the x-axis 00:29:47.690 --> 00:29:49.200 should be all positive. 00:29:49.200 --> 00:29:51.020 All the clauses that are below the x-axis 00:29:51.020 --> 00:29:52.520 should be all negative. 00:29:52.520 --> 00:29:54.740 That would be planar monotone rectilinear 3SAT, 00:29:54.740 --> 00:29:55.920 and I jumped the gun. 00:29:55.920 --> 00:29:57.670 Planar rectilinear has no such constraint. 00:29:57.670 --> 00:30:00.170 Each of these could be positive or negative. 00:30:00.170 --> 00:30:01.000 OK. 00:30:01.000 --> 00:30:06.150 So variables are segments. 00:30:06.150 --> 00:30:06.650 All right. 00:30:06.650 --> 00:30:10.705 So each variable is a segment on the x-axis. 00:30:13.630 --> 00:30:24.180 And each clause is another horizontal segment, 00:30:24.180 --> 00:30:31.570 plus vertical connections to the three variables 00:30:31.570 --> 00:30:34.410 that it includes in positive or negative form. 00:30:39.580 --> 00:30:40.280 OK. 00:30:40.280 --> 00:30:54.220 And then planar monotone rectilinear 3SAT 00:30:54.220 --> 00:30:57.030 is a special case of monotone 3SAT, 00:30:57.030 --> 00:31:09.390 where all positive clauses are above the line 00:31:09.390 --> 00:31:14.970 and conversely, above x-axis. 00:31:14.970 --> 00:31:18.530 Every clause above the x-axis should be all positive. 00:31:18.530 --> 00:31:23.810 And every clause below the x-axis should be all negative. 00:31:33.050 --> 00:31:33.770 OK. 00:31:33.770 --> 00:31:36.810 So I guess this one you could call monotone planar 00:31:36.810 --> 00:31:38.340 3SAT without rectilinear. 00:31:38.340 --> 00:31:42.670 But of course, rectilinear doesn't change much. 00:31:42.670 --> 00:31:46.770 I didn't say why, but if you have a planar drawing 00:31:46.770 --> 00:31:50.550 like we had before, this is just a particularly nice way 00:31:50.550 --> 00:31:52.390 to make that drawing. 00:31:52.390 --> 00:31:55.900 To go here, I just used the fact that there are no crossings. 00:31:55.900 --> 00:31:58.580 And I stretched out the variables to decent lengths, 00:31:58.580 --> 00:32:01.380 so that clauses can just go straight down. 00:32:01.380 --> 00:32:06.920 You can prove any planar graph can be drawn in this way. 00:32:06.920 --> 00:32:11.370 Any planar graph that has the vi cycle and has some degree-- 00:32:11.370 --> 00:32:13.690 three vertices that connect to variables. 00:32:13.690 --> 00:32:15.580 They have to nest in this way, because if you 00:32:15.580 --> 00:32:17.150 think of a clause, they could either 00:32:17.150 --> 00:32:22.540 be in this pocket or outside of it, or down in this region. 00:32:22.540 --> 00:32:24.765 And so you can just figure out how big 00:32:24.765 --> 00:32:27.390 this thing needs to be, and then make the next one a little bit 00:32:27.390 --> 00:32:28.200 bigger. 00:32:28.200 --> 00:32:29.800 And so you can represent the nesting 00:32:29.800 --> 00:32:32.320 in this nice orthogonal, or rectilinear structure. 00:32:32.320 --> 00:32:35.910 Rectilinear just means horizontal and vertical lines. 00:32:35.910 --> 00:32:36.660 Cool? 00:32:36.660 --> 00:32:40.840 So that's planar rectilinear 3SAT first observed 00:32:40.840 --> 00:32:44.270 by Knuth and Raghunathan. 00:32:44.270 --> 00:32:47.840 Now we want to-- let's prove that planar monotone 00:32:47.840 --> 00:32:49.890 rectilinear 3SAT is also hard. 00:32:49.890 --> 00:32:50.390 Question. 00:32:50.390 --> 00:32:51.015 AUDIENCE: Wait. 00:32:51.015 --> 00:32:54.760 If rectilinearity just is a nice way to draw the graph, 00:32:54.760 --> 00:32:57.380 can we use the fact that it's rectilinear in our reductions? 00:32:57.380 --> 00:32:59.130 I mean, is that going to help us any more? 00:32:59.130 --> 00:33:00.879 PROFESSOR: The rectilinear helps us mainly 00:33:00.879 --> 00:33:04.600 when we're reducing to a problem that lives on a grid. 00:33:04.600 --> 00:33:06.660 It's just a convenient way of thinking about it. 00:33:06.660 --> 00:33:08.670 It doesn't directly help us. 00:33:08.670 --> 00:33:10.890 I mean, it's a linear time reduction. 00:33:10.890 --> 00:33:12.390 I mean, all of these are reductions, 00:33:12.390 --> 00:33:15.200 so of course, none of them help us in theory. 00:33:15.200 --> 00:33:17.900 But the cleaner you can make the problem-- I mean, 00:33:17.900 --> 00:33:18.950 this looks pretty. 00:33:18.950 --> 00:33:20.650 And we'll see, I think at the end 00:33:20.650 --> 00:33:23.550 of class, a proof that directly mimics this structure. 00:33:23.550 --> 00:33:24.809 So that's the only reason. 00:33:24.809 --> 00:33:26.600 You could, of course, as a first step, say, 00:33:26.600 --> 00:33:27.900 hey, draw it rectilinearly. 00:33:27.900 --> 00:33:32.130 But this saves you that step. 00:33:32.130 --> 00:33:33.800 Cool. 00:33:33.800 --> 00:33:34.440 All right. 00:33:34.440 --> 00:33:38.010 So planar monotone rectilinear 3SAT. 00:33:41.740 --> 00:33:43.365 Here's an explicit example. 00:33:43.365 --> 00:33:44.990 Although it's not going to be modified, 00:33:44.990 --> 00:33:48.480 it's just another example of planar rectilinear 3SAT 00:33:48.480 --> 00:33:49.450 without the monotone. 00:33:49.450 --> 00:33:52.920 You see there's negations above, lack of negations below. 00:33:52.920 --> 00:33:55.285 And here is the trick to fix things. 00:33:58.130 --> 00:34:01.250 So suppose you have a variable xi on the x-axis, 00:34:01.250 --> 00:34:03.970 you have a clause above it that uses that variable 00:34:03.970 --> 00:34:05.550 in the wrong orientation. 00:34:05.550 --> 00:34:07.825 So of course, or the reflective picture. 00:34:07.825 --> 00:34:08.929 But let's say it's above. 00:34:08.929 --> 00:34:10.610 It's U is negated. 00:34:10.610 --> 00:34:12.790 So in general, we have a bunch of connections up. 00:34:12.790 --> 00:34:14.610 We have a bunch of connections down. 00:34:14.610 --> 00:34:17.270 This one is somewhere in the middle. 00:34:17.270 --> 00:34:19.000 These bold lines are just to distinguish 00:34:19.000 --> 00:34:21.666 these are the ones to the right, these are the ones to the left. 00:34:21.666 --> 00:34:23.650 They don't mean negation, necessarily. 00:34:23.650 --> 00:34:25.489 Because this one should be negated. 00:34:25.489 --> 00:34:25.989 OK. 00:34:25.989 --> 00:34:29.060 So we're just going to replace it like this. 00:34:29.060 --> 00:34:32.049 This is the trick that we saw in the right and bottom 00:34:32.049 --> 00:34:33.340 corner of the crossover gadget. 00:34:33.340 --> 00:34:35.909 We're going to duplicate xi, way except we'll 00:34:35.909 --> 00:34:37.260 negate it at the same time. 00:34:37.260 --> 00:34:40.909 So the other one had a negation here and one negation here, 00:34:40.909 --> 00:34:42.080 not here. 00:34:42.080 --> 00:34:47.022 This xi or a, together with xi bar or a bar-- again, 00:34:47.022 --> 00:34:47.980 these are 2SAT clauses. 00:34:47.980 --> 00:34:52.520 You have to duplicate an edge to make it 3SAT. 00:34:52.520 --> 00:34:56.239 That forces xi to be not a, I think. 00:34:56.239 --> 00:34:58.753 Maybe I even have that as a fate. 00:34:58.753 --> 00:35:00.710 OK, xi is not equal to a. 00:35:00.710 --> 00:35:04.370 Similarly, a does not equal b, which means b equals xi. 00:35:04.370 --> 00:35:04.870 OK? 00:35:04.870 --> 00:35:08.920 So we just duplicated xi to be used in all the old scenarios. 00:35:08.920 --> 00:35:10.720 And maybe these guys still all go down. 00:35:10.720 --> 00:35:13.250 We could move some of them over here if we wanted to. 00:35:13.250 --> 00:35:15.000 The ones to the left have to be over here. 00:35:15.000 --> 00:35:17.120 The ones to the right have to be over here. 00:35:17.120 --> 00:35:20.280 This one gets to use a negated copy of xi. 00:35:20.280 --> 00:35:23.870 So boom, one negation down. 00:35:23.870 --> 00:35:25.760 One negation that was on the wrong side. 00:35:25.760 --> 00:35:29.816 Repeat this n times. 00:35:29.816 --> 00:35:31.815 Where you have a negation you're not supposed to 00:35:31.815 --> 00:35:33.981 and everywhere you don't have a negation that you're 00:35:33.981 --> 00:35:36.006 supposed to, apply this trick. 00:35:36.006 --> 00:35:37.630 You'll increase the number of variables 00:35:37.630 --> 00:35:40.360 by some constant factor. 00:35:40.360 --> 00:35:42.090 And now you're in the monotone case. 00:35:42.090 --> 00:35:43.810 So that's actually really easy. 00:35:43.810 --> 00:35:45.920 It's our first proof that monotone 3SAT is hard. 00:35:45.920 --> 00:35:47.650 But it's a pretty easy one. 00:35:47.650 --> 00:35:48.689 Yeah? 00:35:48.689 --> 00:35:50.480 AUDIENCE: Can we use a similar construction 00:35:50.480 --> 00:35:53.210 to recover splitting the variables into literals? 00:35:56.210 --> 00:35:58.030 PROFESSOR: Oh, to do this? 00:35:58.030 --> 00:35:59.230 AUDIENCE: Yeah. 00:35:59.230 --> 00:36:03.002 Because we dropped that when we moved over here, right? 00:36:03.002 --> 00:36:04.960 AUDIENCE: But you said it was just [INAUDIBLE]. 00:36:04.960 --> 00:36:06.567 PROFESSOR: Well, we sort of did. 00:36:06.567 --> 00:36:09.150 No, actually, this is supposed to be a generalization of this, 00:36:09.150 --> 00:36:11.910 because you could just split apart-- I mean, you 00:36:11.910 --> 00:36:14.700 can carve vi in half to the positive side 00:36:14.700 --> 00:36:16.400 and the negative side. 00:36:16.400 --> 00:36:18.200 And all the negative sides are below, 00:36:18.200 --> 00:36:20.310 and all the positive sides are above. 00:36:20.310 --> 00:36:23.170 So yeah, we could already do that. 00:36:26.450 --> 00:36:26.950 Right. 00:36:26.950 --> 00:36:29.870 Because this was not about having them-- 00:36:29.870 --> 00:36:34.445 it was about having the vertices labeled as literals instead of 00:36:34.445 --> 00:36:35.020 as variables. 00:36:35.020 --> 00:36:37.660 So we don't need to explicitly say that this one does not 00:36:37.660 --> 00:36:38.430 equal that one. 00:36:38.430 --> 00:36:40.410 That's already in the problem. 00:36:40.410 --> 00:36:43.000 So we can combine all of the things I've said so far. 00:36:43.000 --> 00:36:46.810 That may not always be the case, but so far, we can combine all. 00:36:46.810 --> 00:36:47.310 Yeah? 00:36:47.310 --> 00:36:48.351 AUDIENCE: Silly question. 00:36:48.351 --> 00:36:50.910 If we're carving them into positive side of the variable 00:36:50.910 --> 00:36:52.170 on one side and negative on the other side, 00:36:52.170 --> 00:36:53.770 wouldn't that make all the connections 00:36:53.770 --> 00:36:55.620 below that positive connections instead 00:36:55.620 --> 00:36:57.270 of negative connections? 00:36:57.270 --> 00:36:58.240 PROFESSOR: Sorry. 00:36:58.240 --> 00:37:00.830 I don't mean split it in the way of that diagram. 00:37:00.830 --> 00:37:03.480 I mean that we could, in this sense, 00:37:03.480 --> 00:37:06.910 we could have vertices represent a positive vi 00:37:06.910 --> 00:37:09.077 and a negative vi, and have the connections. 00:37:09.077 --> 00:37:10.660 This was saying that we could separate 00:37:10.660 --> 00:37:12.850 the positive connections from the negative ones. 00:37:12.850 --> 00:37:15.300 And I'm just saying we already have that here. 00:37:15.300 --> 00:37:17.730 This is a stronger property to say that they're also 00:37:17.730 --> 00:37:19.086 all in the same direction. 00:37:19.086 --> 00:37:20.710 If you just had this property, then you 00:37:20.710 --> 00:37:23.520 could have some white here and some white here, 00:37:23.520 --> 00:37:24.962 and red here and red here. 00:37:24.962 --> 00:37:26.920 But this is saying you can get them all aligned 00:37:26.920 --> 00:37:29.831 in the right direction. 00:37:29.831 --> 00:37:30.330 Yeah. 00:37:30.330 --> 00:37:38.992 In fact, if we look at this diagram, 00:37:38.992 --> 00:37:40.700 maybe you didn't understand this diagram, 00:37:40.700 --> 00:37:44.360 but it has the same issue. 00:37:44.360 --> 00:37:46.050 Or this is exactly that kind of picture. 00:37:46.050 --> 00:37:47.470 So here we have a5. 00:37:47.470 --> 00:37:50.330 The positive is over here, the negative is over here. 00:37:50.330 --> 00:37:51.200 Here we have a4. 00:37:51.200 --> 00:37:53.640 The negative is there, and the positive is there. 00:37:53.640 --> 00:37:56.284 So this was getting the weaker property. 00:37:56.284 --> 00:37:57.950 Again, you don't have to worry about it, 00:37:57.950 --> 00:38:00.980 because we got the stronger one. 00:38:00.980 --> 00:38:02.650 But that's just to show that there's 00:38:02.650 --> 00:38:05.880 a difference between the two. 00:38:05.880 --> 00:38:07.860 Good. 00:38:07.860 --> 00:38:13.840 So that was planar monotone rectilinear 3SAT. 00:38:13.840 --> 00:38:14.470 OK. 00:38:14.470 --> 00:38:20.475 I have a governor general's warning. 00:38:26.070 --> 00:38:28.490 So here's another problem. 00:38:28.490 --> 00:38:29.955 Suppose we have the variable cycle. 00:38:35.290 --> 00:38:39.370 And I say, well, every clause is inside the cycle. 00:38:39.370 --> 00:38:42.990 And you are free to have positive and negative 00:38:42.990 --> 00:38:43.490 connections. 00:38:49.260 --> 00:38:50.670 It's going to get hard. 00:38:50.670 --> 00:38:51.370 OK. 00:38:51.370 --> 00:38:53.900 Something like that. 00:38:53.900 --> 00:38:56.000 And it's planar. 00:38:56.000 --> 00:38:59.520 That problem is polynomially solvable. 00:38:59.520 --> 00:39:02.440 So I'll rephrase this. 00:39:02.440 --> 00:39:22.140 If all the clauses are on one side, of the variable cycle, 00:39:22.140 --> 00:39:25.630 then that version of 3SAT is polynomially solvable. 00:39:25.630 --> 00:39:29.800 And the intuition is that this kind of nesting structure 00:39:29.800 --> 00:39:32.700 is essentially a tree. 00:39:32.700 --> 00:39:35.335 If you look at what nests inside of what, 00:39:35.335 --> 00:39:38.550 what clauses fall underneath other clauses, 00:39:38.550 --> 00:39:42.460 that is a nesting relation, which corresponds to a tree. 00:39:42.460 --> 00:39:45.530 And then you just do dynamic programming over the tree. 00:39:45.530 --> 00:39:46.860 That's easy. 00:39:46.860 --> 00:39:48.930 The sort of surprising thing is that just 00:39:48.930 --> 00:39:51.640 having two of these trees, one on the inside of the cycle, one 00:39:51.640 --> 00:39:54.215 on the outside, is enough to make the problem hard. 00:39:54.215 --> 00:39:55.590 That's a typical thing, actually, 00:39:55.590 --> 00:39:58.340 because the trees interdigitate. 00:39:58.340 --> 00:40:00.240 They have no real relation to each other, 00:40:00.240 --> 00:40:02.240 and there's no way to keep track of both of them 00:40:02.240 --> 00:40:05.500 simultaneously with the dynamic program unless P equals mp. 00:40:10.080 --> 00:40:10.580 OK. 00:40:10.580 --> 00:40:12.319 So that's one gotcha to be careful of. 00:40:12.319 --> 00:40:13.860 So when you're doing your reductions, 00:40:13.860 --> 00:40:16.380 you have to worry about things above the line and things 00:40:16.380 --> 00:40:17.050 below the line. 00:40:20.850 --> 00:40:23.640 Also, here's a special case of this. 00:40:41.950 --> 00:40:44.670 Suppose I phrase the following problem. 00:40:44.670 --> 00:40:47.690 I want the bipartite graph on variables versus clauses, 00:40:47.690 --> 00:40:52.110 plus the cycle v1 through vn back to v1, 00:40:52.110 --> 00:40:56.460 plus another cycle on the clause's c1 to cn and back 00:40:56.460 --> 00:40:56.990 to c1. 00:40:56.990 --> 00:41:01.010 I want that whole graph to be planar. 00:41:01.010 --> 00:41:03.070 OK? 00:41:03.070 --> 00:41:04.820 If you think about that for a second, that 00:41:04.820 --> 00:41:06.990 would imply all the clauses are on one side 00:41:06.990 --> 00:41:08.442 of the variable cycle. 00:41:08.442 --> 00:41:10.150 And so we're in this case, which is easy. 00:41:10.150 --> 00:41:12.225 So this is a special case of that case. 00:41:12.225 --> 00:41:15.650 Now, this one I mentioned, because if this were hard, 00:41:15.650 --> 00:41:16.910 life would be so easy. 00:41:16.910 --> 00:41:18.230 It's not so life isn't easy. 00:41:18.230 --> 00:41:21.140 But you already knew that. 00:41:21.140 --> 00:41:25.270 Recall, say, proving push-1 is hard in 2D. 00:41:25.270 --> 00:41:28.389 Well, first we did it in 3D and then we had a crossover gadget. 00:41:28.389 --> 00:41:30.430 And most of the work was in the crossover gadget. 00:41:30.430 --> 00:41:32.530 The base proof was trivial or very easy. 00:41:36.180 --> 00:41:38.850 So we can get all of these connections between variables 00:41:38.850 --> 00:41:40.560 and clauses to be planar. 00:41:40.560 --> 00:41:42.870 We can get all of these connections, the variables 00:41:42.870 --> 00:41:45.510 in a line, to be planar together, 00:41:45.510 --> 00:41:46.980 all of those things together. 00:41:46.980 --> 00:41:51.040 But we cannot also have a path connecting all the clauses 00:41:51.040 --> 00:41:51.660 together. 00:41:51.660 --> 00:41:52.420 Even a path. 00:41:52.420 --> 00:41:58.840 So I said a path here, which is what we want. 00:41:58.840 --> 00:42:00.847 So that's annoying, because if we 00:42:00.847 --> 00:42:02.430 could make all of these things planar, 00:42:02.430 --> 00:42:04.840 we wouldn't have to worry about these crossovers. 00:42:04.840 --> 00:42:07.960 Same thing with all the Nintendo proofs, like Super Mario 00:42:07.960 --> 00:42:09.570 Brothers again. 00:42:09.570 --> 00:42:13.170 The player is going from one variable to the next, 00:42:13.170 --> 00:42:16.320 so you need those edges connecting all the variables 00:42:16.320 --> 00:42:17.142 in a path. 00:42:17.142 --> 00:42:18.850 You need the edges connecting the clauses 00:42:18.850 --> 00:42:20.740 in a path for the final check. 00:42:20.740 --> 00:42:22.620 So planar 3SAT basically doesn't buy 00:42:22.620 --> 00:42:24.550 you anything in these proofs. 00:42:24.550 --> 00:42:28.330 Now, if we could somehow avoid adding these connections, 00:42:28.330 --> 00:42:30.000 everything else could be planar. 00:42:30.000 --> 00:42:31.940 That's sort of the point of planar 3SAT. 00:42:31.940 --> 00:42:33.690 So if you can avoid these things-- 00:42:33.690 --> 00:42:35.898 and there are a lot of games where that would happen. 00:42:35.898 --> 00:42:37.896 Just you have to get a point here, 00:42:37.896 --> 00:42:39.520 and it's just added to your total score 00:42:39.520 --> 00:42:42.440 in some Never Never Land not in the plane, 00:42:42.440 --> 00:42:46.950 then life would be much easier, because you wouldn't need 00:42:46.950 --> 00:42:49.570 a crossover gadget, because you can just 00:42:49.570 --> 00:42:52.170 reduce from planar 3SAT. 00:42:52.170 --> 00:42:53.702 OK? 00:42:53.702 --> 00:42:55.160 That should give you some intuition 00:42:55.160 --> 00:42:57.460 why we care about planar 3SAT, but also 00:42:57.460 --> 00:42:59.825 why we didn't use it for those problems. 00:42:59.825 --> 00:43:01.200 As far as we know, there's no way 00:43:01.200 --> 00:43:04.015 to use planar 3SAT and make a simpler Super Mario Brothers 00:43:04.015 --> 00:43:04.765 proof or whatever. 00:43:08.660 --> 00:43:10.990 OK. 00:43:10.990 --> 00:43:14.030 So next topic. 00:43:25.000 --> 00:43:30.350 We will see some planar 3SAT proofs, 00:43:30.350 --> 00:43:31.579 reductions from planar 3SAT. 00:43:31.579 --> 00:43:33.870 But let me first tell you about other versions of 3SAT. 00:43:33.870 --> 00:43:36.078 Remember there are three main versions-- planar 3SAT, 00:43:36.078 --> 00:43:38.460 planar 1-in-3SAT, and-- sorry. 00:43:38.460 --> 00:43:41.550 There are three versions of 3SAT-- 3SAT, 1-in-3SAT, 00:43:41.550 --> 00:43:43.180 and Not All Equal SAT. 00:43:43.180 --> 00:43:47.760 Turns out 1-in-3SAT is hard when its graphs are planar. 00:43:47.760 --> 00:43:50.680 Not All Equal SAT is easy when graphs are planar. 00:43:50.680 --> 00:43:53.131 So I will show you both. 00:43:53.131 --> 00:43:54.630 AUDIENCE: For this easy case, do you 00:43:54.630 --> 00:43:56.463 have to have both the variables in the cycle 00:43:56.463 --> 00:43:58.280 and a clauses in a cycle? 00:43:58.280 --> 00:44:01.380 Or is it just like seeing the clauses in a path or cycle 00:44:01.380 --> 00:44:04.410 immediately triggers this easy condition? 00:44:04.410 --> 00:44:07.640 PROFESSOR: So I don't know. 00:44:07.640 --> 00:44:09.570 I think if the clauses are in a path 00:44:09.570 --> 00:44:12.290 and the variables are not connected at all, 00:44:12.290 --> 00:44:13.730 it should still be hard. 00:44:13.730 --> 00:44:15.360 But I couldn't find a proof of that. 00:44:15.360 --> 00:44:17.580 It's just a vague recollection. 00:44:17.580 --> 00:44:20.020 So I'm pretty sure if variables are in a path 00:44:20.020 --> 00:44:22.890 and clauses are in a path, then that's enough to be toast. 00:44:22.890 --> 00:44:25.132 But all I proved here was cycle. 00:44:25.132 --> 00:44:26.340 But I think a path is enough. 00:44:26.340 --> 00:44:29.466 It will still behave like a tree. 00:44:29.466 --> 00:44:29.965 Mm. 00:44:29.965 --> 00:44:30.600 Maybe not. 00:44:30.600 --> 00:44:33.691 I'm not sure. 00:44:33.691 --> 00:44:34.190 Yeah. 00:44:34.190 --> 00:44:36.030 So it's a good question. 00:44:36.030 --> 00:44:36.730 All right. 00:44:36.730 --> 00:44:38.710 Let's do planar 1-in-3SAT. 00:44:38.710 --> 00:44:41.440 So it's pretty much the same thing. 00:44:41.440 --> 00:44:43.280 I won't duplicate all of the text here. 00:44:43.280 --> 00:44:48.520 We take the bipartite graph of variables versus clauses. 00:44:48.520 --> 00:44:53.210 And we also add a path, or cycle, 00:44:53.210 --> 00:44:55.850 connecting all the variables. 00:44:55.850 --> 00:44:59.150 And that graph should be planar. 00:44:59.150 --> 00:45:03.580 And otherwise, it's 1-in-3SAT, exactly 1-in-3SAT. 00:45:03.580 --> 00:45:05.790 So here's the proof. 00:45:05.790 --> 00:45:08.902 It's one of the easiest proofs we'll cover. 00:45:08.902 --> 00:45:09.985 Suppose you have a clause. 00:45:14.470 --> 00:45:20.260 It's a reduction from planar 3SAT to planar 1-in-3SAT. 00:45:20.260 --> 00:45:21.470 OK? 00:45:21.470 --> 00:45:25.200 So this is also our first proof of 1-in-3SAT being hard. 00:45:25.200 --> 00:45:28.580 We are going to take a regular 3SAT clause 00:45:28.580 --> 00:45:32.520 and turn it into three 1-in-3SAT clauses. 00:45:32.520 --> 00:45:35.929 And that transformation preserves planarity, so done. 00:45:35.929 --> 00:45:37.720 So all of our proofs are going to just keep 00:45:37.720 --> 00:45:38.720 building up in this way. 00:45:38.720 --> 00:45:40.440 We've already proved this problem's hard 00:45:40.440 --> 00:45:42.100 when the graph is planar. 00:45:42.100 --> 00:45:44.576 So let's do this. 00:45:44.576 --> 00:45:45.950 At least one of them should be 1. 00:45:45.950 --> 00:45:47.241 We don't want them all to be 0. 00:45:47.241 --> 00:45:50.200 So let's try when they're all 0, see if something should break. 00:45:50.200 --> 00:45:52.730 Let's say this variable is 0, easier to reach. 00:45:52.730 --> 00:45:58.380 So that makes this guy satisfied, 00:45:58.380 --> 00:46:00.250 which means both of these must be false. 00:46:00.250 --> 00:46:01.730 This variable's just hanging out. 00:46:01.730 --> 00:46:04.180 It's just to make everything degree three. 00:46:04.180 --> 00:46:09.680 So if this is 0, this is 1, so this must be 0. 00:46:09.680 --> 00:46:10.970 So this is 0. 00:46:10.970 --> 00:46:12.290 This is 0. 00:46:12.290 --> 00:46:14.720 That means exactly one of these should be 1. 00:46:14.720 --> 00:46:16.870 If cp is 1, obviously, we're happy. 00:46:16.870 --> 00:46:19.065 But this will end up forcing this guy to be 0. 00:46:19.065 --> 00:46:19.680 Or sorry. 00:46:19.680 --> 00:46:22.660 Actually, it'll leave this guy free to do whatever. 00:46:22.660 --> 00:46:28.650 So if this is 0, 0, 0, then this must be 1. 00:46:28.650 --> 00:46:29.910 This is 1. 00:46:29.910 --> 00:46:30.480 This is 1. 00:46:30.480 --> 00:46:31.920 Therefore, both of these are 0. 00:46:31.920 --> 00:46:35.780 If zr bar is 0, that must mean this one is 1. 00:46:35.780 --> 00:46:39.520 So if both of these were 0, it forced this guy to be a 1. 00:46:39.520 --> 00:46:43.175 If this was 0 and this is 1, for example, this is 1. 00:46:45.750 --> 00:46:49.020 Then this must be 0/ OK. 00:46:49.020 --> 00:46:50.220 This is 0. 00:46:50.220 --> 00:46:52.320 But we're free to set either one of these to 1. 00:46:52.320 --> 00:46:55.420 So zr is free, which is what we want to do. 00:46:55.420 --> 00:46:56.700 We're simulating regular 3SAT. 00:46:56.700 --> 00:46:58.684 If this is 1, then both of these guys 00:46:58.684 --> 00:47:00.350 should be able to do whatever they want. 00:47:00.350 --> 00:47:03.150 And that's what these guys let you do. 00:47:03.150 --> 00:47:04.840 They're free floating. 00:47:04.840 --> 00:47:07.170 This guy can toggle if it's not constrained 00:47:07.170 --> 00:47:08.940 to be the only guy that's 1. 00:47:08.940 --> 00:47:09.850 Yeah? 00:47:09.850 --> 00:47:13.450 AUDIENCE: What's the dashed line represent? 00:47:13.450 --> 00:47:16.102 PROFESSOR: Dashed line is the outline of the gadget. 00:47:16.102 --> 00:47:20.750 It's just saying you replace that blob with this blob. 00:47:20.750 --> 00:47:22.945 I mean, slightly more formally, I 00:47:22.945 --> 00:47:24.320 think it's saying something like, 00:47:24.320 --> 00:47:26.952 if you contract this to a point or if you contract this 00:47:26.952 --> 00:47:28.910 to a point, then it's exactly the same diagram. 00:47:28.910 --> 00:47:30.451 So it's one way to argue you preserve 00:47:30.451 --> 00:47:32.170 planarity, something like that. 00:47:35.150 --> 00:47:37.630 So ignore it, basically. 00:47:37.630 --> 00:47:40.310 Now, I guess you would have to argue 00:47:40.310 --> 00:47:44.482 that you can still have a path that visits all the variables. 00:47:44.482 --> 00:47:45.190 Another question? 00:47:45.190 --> 00:47:46.174 AUDIENCE: Yeah. 00:47:46.174 --> 00:47:50.632 What's preventing you from setting all three to false? 00:47:50.632 --> 00:47:52.590 PROFESSOR: That is one of the cases I just did. 00:47:52.590 --> 00:47:57.550 If these two are false, then these have to be true. 00:47:57.550 --> 00:47:58.520 No, sorry. 00:47:58.520 --> 00:47:59.480 These have to be 0. 00:47:59.480 --> 00:48:00.920 So these are both 0. 00:48:00.920 --> 00:48:04.030 And then this has to be a 1 or satisfy this. 00:48:06.650 --> 00:48:08.820 It's easy to get confused, because that one is 3SAT, 00:48:08.820 --> 00:48:10.130 this one is 1-in-3SAT. 00:48:10.130 --> 00:48:13.490 So these all have exactly 1 constraints. 00:48:13.490 --> 00:48:14.740 OK. 00:48:14.740 --> 00:48:16.960 Now, this proof has negations. 00:48:16.960 --> 00:48:20.380 And you may recall that monotone-- 00:48:20.380 --> 00:48:23.150 I'm going to switch to saying positive 1-in-3SAT where you 00:48:23.150 --> 00:48:25.720 have no negations is also hard. 00:48:25.720 --> 00:48:32.980 So we can also define planar positive 1-in-3SAT. 00:48:40.170 --> 00:48:45.110 This is also hard by a slightly more complicated proof, 00:48:45.110 --> 00:48:47.000 more recent proof also. 00:48:47.000 --> 00:48:51.380 Surprisingly, this planar monotone rectilinear 3SAT, 00:48:51.380 --> 00:48:54.290 or just planar monotone 3SAT, is also very recent. 00:48:54.290 --> 00:48:55.960 I think 2010. 00:48:55.960 --> 00:48:58.560 So a lot of these just fell recently 00:48:58.560 --> 00:49:02.270 but are a natural culmination of all these simplifications. 00:49:02.270 --> 00:49:04.450 So this is a version that-- well, these 00:49:04.450 --> 00:49:06.090 are some gadgets, actually. 00:49:06.090 --> 00:49:09.010 This is one way to force the two variables to have 00:49:09.010 --> 00:49:11.770 opposite values and force the two variables to have 00:49:11.770 --> 00:49:16.340 equal values, using-- I guess this is really 00:49:16.340 --> 00:49:21.730 planar positive rectilinear 1-in-3SAT. 00:49:21.730 --> 00:49:23.810 It's the 1-in-3SAT version of this problem. 00:49:23.810 --> 00:49:25.640 So again, variables are on the x-axis. 00:49:25.640 --> 00:49:27.570 Everything above is implicitly all positive. 00:49:27.570 --> 00:49:29.835 Everything below is implicitly all negative. 00:49:29.835 --> 00:49:31.360 Oh, sorry. 00:49:31.360 --> 00:49:33.056 Here, everything is all positive. 00:49:33.056 --> 00:49:33.986 Yeah? 00:49:33.986 --> 00:49:35.486 AUDIENCE: Another nice thing to note 00:49:35.486 --> 00:49:36.944 about that particular construction 00:49:36.944 --> 00:49:39.860 is that each of the variables in the clauses are unique, 00:49:39.860 --> 00:49:40.832 so it's exactly 3. 00:49:40.832 --> 00:49:44.234 And they're all unique, which someone 00:49:44.234 --> 00:49:45.514 was asking about before. 00:49:45.514 --> 00:49:46.180 PROFESSOR: Good. 00:49:46.180 --> 00:49:51.660 So in this construction, 1-in-3SAT, all of the clauses 00:49:51.660 --> 00:49:53.900 do not repeat any variables. 00:49:53.900 --> 00:49:56.000 Exactly three distinct guys, yeah. 00:49:56.000 --> 00:50:00.250 AUDIENCE: So to be clear, this problem is hard, 00:50:00.250 --> 00:50:06.780 but in this terminology, planar positive rectilinear 3SAT, 00:50:06.780 --> 00:50:10.440 not 1-in-3SAT, is easy. 00:50:10.440 --> 00:50:13.420 Because if everything was on one side of the line-- 00:50:13.420 --> 00:50:16.550 PROFESSOR: I mean, positive 3SAT is easy. 00:50:16.550 --> 00:50:20.372 You set all the variables to true. 00:50:20.372 --> 00:50:21.330 AUDIENCE: I'm confused. 00:50:21.330 --> 00:50:23.004 Positive means that-- 00:50:23.004 --> 00:50:25.170 PROFESSOR: Positive is sort of the 1-in-3SAT version 00:50:25.170 --> 00:50:25.910 of monotone. 00:50:25.910 --> 00:50:29.260 So 3SAT monotone means every clause is 00:50:29.260 --> 00:50:30.770 all positive or all negative. 00:50:30.770 --> 00:50:34.070 In 1-in-3SAT, you just need everything positive. 00:50:34.070 --> 00:50:35.051 And it's still hard. 00:50:35.051 --> 00:50:36.925 But with 3SAT, that doesn't work, because you 00:50:36.925 --> 00:50:38.800 can set everything to true. 00:50:38.800 --> 00:50:43.090 1-in-3SAT requires some false things. 00:50:43.090 --> 00:50:45.320 So it's sort of the analog. 00:50:45.320 --> 00:50:49.650 This is sometimes called planar monotone rectilinear 1-in-3SAT. 00:50:49.650 --> 00:50:53.130 But to avoid confusion, because they are somewhat different, 00:50:53.130 --> 00:50:54.580 I'm calling it positive. 00:50:54.580 --> 00:50:56.550 I think this paper actually calls it positive, 00:50:56.550 --> 00:50:58.250 which makes me happy. 00:50:58.250 --> 00:51:01.560 Anyway, you can check these gadgets that they force. 00:51:01.560 --> 00:51:04.790 In particular here, you're basically forcing a to b 0 00:51:04.790 --> 00:51:06.880 by this little construction. 00:51:06.880 --> 00:51:08.930 You can see as 1. 00:51:08.930 --> 00:51:12.520 Therefore, this clause forces exactly one of these 00:51:12.520 --> 00:51:13.936 to be true, the other to be false. 00:51:13.936 --> 00:51:15.352 So it forces them to be different. 00:51:15.352 --> 00:51:17.140 You repeat that twice, you get equality. 00:51:17.140 --> 00:51:18.250 Then you do this. 00:51:21.840 --> 00:51:24.670 First, you get rid of all negations, 00:51:24.670 --> 00:51:26.100 because you have this way to force 00:51:26.100 --> 00:51:28.270 two things to be different. 00:51:28.270 --> 00:51:30.110 These Xed out ones are the red lines. 00:51:30.110 --> 00:51:32.950 Those are connected to negated copies. 00:51:32.950 --> 00:51:35.850 You just duplicate the variable in negated form 00:51:35.850 --> 00:51:37.860 and then use a wire there. 00:51:37.860 --> 00:51:39.670 So this wire becomes that one. x2 00:51:39.670 --> 00:51:42.150 is different from x1. x3 is different from x2. 00:51:42.150 --> 00:51:43.190 So it's equal to x1. 00:51:43.190 --> 00:51:45.050 Same trick we saw before. 00:51:45.050 --> 00:51:46.320 Now you have no negations. 00:51:46.320 --> 00:51:50.340 So now this type of clause is a 3SAT clause. 00:51:50.340 --> 00:51:51.460 It's a little weird. 00:51:51.460 --> 00:51:54.187 Right of the arrows, these are 1-in-3SAT clauses. 00:51:54.187 --> 00:51:56.020 Left of the arrows are regular 3SAT clauses. 00:51:56.020 --> 00:51:58.430 So we replace this with this construction. 00:52:02.834 --> 00:52:04.750 It's sort of like the old one but a little bit 00:52:04.750 --> 00:52:06.242 spread out, with the duplications, 00:52:06.242 --> 00:52:08.885 with these equal gadgets. 00:52:08.885 --> 00:52:11.260 We get a copy of x over here. 00:52:11.260 --> 00:52:13.820 That's so that if we have more connections below, 00:52:13.820 --> 00:52:17.570 we can easily access x still. 00:52:17.570 --> 00:52:21.030 And yeah. 00:52:21.030 --> 00:52:25.717 These 1-in-3 gadgets constrain this. 00:52:25.717 --> 00:52:27.050 Yeah, there is an equal gadget-- 00:52:27.050 --> 00:52:27.470 AUDIENCE: The equals and not equals 00:52:27.470 --> 00:52:29.159 are not part of the problem, they're 00:52:29.159 --> 00:52:30.700 just referring to the earlier gadget. 00:52:30.700 --> 00:52:30.946 PROFESSOR: No. 00:52:30.946 --> 00:52:31.445 Right. 00:52:31.445 --> 00:52:33.580 That's shorthand for these gadgets. 00:52:33.580 --> 00:52:35.750 So you would plug in each of those things 00:52:35.750 --> 00:52:37.640 into those little pictures. 00:52:37.640 --> 00:52:38.140 Yeah. 00:52:40.460 --> 00:52:40.960 OK. 00:52:40.960 --> 00:52:43.070 So let's skip those details. 00:52:43.070 --> 00:52:48.840 But you get that planar positive rectilinear 1-in-3SAT is hard. 00:52:48.840 --> 00:52:52.361 And that was 2008. 00:52:52.361 --> 00:52:52.860 Cool. 00:52:56.640 --> 00:53:06.590 Let's do planar Not All Equal 3SAT. 00:53:09.670 --> 00:53:10.650 This is polynomial. 00:53:10.650 --> 00:53:13.020 Important thing to remember-- in the planar world, 00:53:13.020 --> 00:53:16.820 you have to be careful with all sorts of things. 00:53:16.820 --> 00:53:19.400 It would be nice to have a Schaefer-type dichotomy 00:53:19.400 --> 00:53:23.610 theorem for planar graphs, but I don't know of any such theorem. 00:53:23.610 --> 00:53:26.465 So meanwhile, this is the sort of main characterization. 00:53:26.465 --> 00:53:28.820 You have those three standard problems. 00:53:28.820 --> 00:53:30.920 This one falls in the planar case. 00:53:30.920 --> 00:53:31.980 Same setup. 00:53:31.980 --> 00:53:34.680 We have Not All Equal clauses and the connections 00:53:34.680 --> 00:53:38.720 between those and the variables negated or not. 00:53:38.720 --> 00:53:42.690 Even when you allow negations, this problem is polynomial. 00:53:42.690 --> 00:53:46.889 The fun thing is the proof of this theorem is a reduction, 00:53:46.889 --> 00:53:48.680 and it looks exactly like a hardness proof. 00:53:48.680 --> 00:53:51.170 I mean-- [LAUGHS] It just happens 00:53:51.170 --> 00:53:55.330 to be from this problem to a known polynomial time problem, 00:53:55.330 --> 00:53:58.010 namely, planar max cut. 00:53:58.010 --> 00:54:00.810 So we're going to reduce to-- this 00:54:00.810 --> 00:54:06.370 is one of the few times we will reduce to something-- planar 00:54:06.370 --> 00:54:08.370 max cut. 00:54:08.370 --> 00:54:10.620 So max cut is I give you a graph, 00:54:10.620 --> 00:54:13.340 and I want to color the vertices two different colors. 00:54:13.340 --> 00:54:15.640 Think of them as the two sides of a cut. 00:54:15.640 --> 00:54:16.866 Say red and blue. 00:54:16.866 --> 00:54:18.240 And I want to maximize the number 00:54:18.240 --> 00:54:21.510 of edges that are red, blue. 00:54:21.510 --> 00:54:27.280 So I want to maximize the number of, here, white-black edges, 00:54:27.280 --> 00:54:29.080 bichromatic edges. 00:54:29.080 --> 00:54:29.887 That's max cut. 00:54:29.887 --> 00:54:32.220 In general graphs, that's NP-hard, but in planar graphs, 00:54:32.220 --> 00:54:35.460 it's easy, because if you look at the dual graph 00:54:35.460 --> 00:54:38.210 and then solve the Chinese postman tour problem, which 00:54:38.210 --> 00:54:40.670 is the shortest path that visits all of the edges at least 00:54:40.670 --> 00:54:42.520 once, which you can do in polynomial time 00:54:42.520 --> 00:54:46.930 by perfect matching and a bipartite null in a cleek. 00:54:46.930 --> 00:54:49.350 Anyway, you do that. 00:54:49.350 --> 00:54:52.740 And then all the edges that you double are not in the cut, 00:54:52.740 --> 00:54:53.240 I think. 00:54:53.240 --> 00:54:55.710 And all the edges that you don't double in that tour 00:54:55.710 --> 00:55:00.090 are in the max cut. 00:55:00.090 --> 00:55:01.040 Fun fact. 00:55:01.040 --> 00:55:04.490 So this is a known fact. 00:55:04.490 --> 00:55:05.970 Planar max cut is easy. 00:55:05.970 --> 00:55:09.660 So if we reduce from planar Not Equal 3SAT to max cut, 00:55:09.660 --> 00:55:10.730 the problem is easy. 00:55:10.730 --> 00:55:11.850 And here's the proof. 00:55:11.850 --> 00:55:16.720 It's got a variable gadget and a Not All Equal clause gadget. 00:55:16.720 --> 00:55:20.370 So we want to represent Not All Equal. 00:55:20.370 --> 00:55:23.080 And so the idea is we're going to represent a variable 00:55:23.080 --> 00:55:25.550 with this alternating chain. 00:55:25.550 --> 00:55:27.970 And if you want to maximize the number of red-blue edges, 00:55:27.970 --> 00:55:30.137 you should alternate, because this have even length. 00:55:30.137 --> 00:55:31.761 If you don't alternate anywhere, you're 00:55:31.761 --> 00:55:33.160 going to not get as many edges. 00:55:33.160 --> 00:55:36.382 In general, we'll say, well, the target cut 00:55:36.382 --> 00:55:38.340 size you're trying to get, the decision problem 00:55:38.340 --> 00:55:40.640 is, is there a cut of size at least something? 00:55:40.640 --> 00:55:42.880 The something is going to be, in particular, it's 00:55:42.880 --> 00:55:44.921 going to be a sum of things, but for this gadget, 00:55:44.921 --> 00:55:46.710 it is two times the number of occurrences 00:55:46.710 --> 00:55:50.510 that the variable would say, or basically the length 00:55:50.510 --> 00:55:52.549 of this cycle. 00:55:52.549 --> 00:55:54.090 We'll just make the cycle big enough, 00:55:54.090 --> 00:55:57.930 so we get lots of copies of xi, lots of copies of xi bar. 00:55:57.930 --> 00:56:00.300 So this is one possible setting. 00:56:00.300 --> 00:56:04.370 If blue means true, this is like saying that we set xi to true. 00:56:04.370 --> 00:56:05.400 We could do the reverse. 00:56:05.400 --> 00:56:09.910 We could set xi to false and set xi bar to true. 00:56:09.910 --> 00:56:11.260 But they have to be opposites. 00:56:11.260 --> 00:56:14.090 So now we have-- this is basically a split gadget. 00:56:14.090 --> 00:56:19.060 We have several copies of xi and xi bar. 00:56:19.060 --> 00:56:21.860 Then we're going to connect them with this Not All Equal clause, 00:56:21.860 --> 00:56:24.330 which is just a triangle. 00:56:24.330 --> 00:56:28.380 And the idea is if these are-- so this is really 00:56:28.380 --> 00:56:31.260 a negated version of xi, but it doesn't matter. 00:56:31.260 --> 00:56:34.420 If you negate all the variables, they'll still be not all equal. 00:56:34.420 --> 00:56:36.090 And exactly when they're not all equal, 00:56:36.090 --> 00:56:39.040 we get two points out of this gadget. 00:56:39.040 --> 00:56:40.890 Well, we also have to get the three points 00:56:40.890 --> 00:56:42.440 from these connections. 00:56:42.440 --> 00:56:43.750 That forces this alternation. 00:56:43.750 --> 00:56:45.650 So we say the target in this gadget 00:56:45.650 --> 00:56:47.580 is to get a cut of size 5. 00:56:47.580 --> 00:56:49.570 We want five red-blue edges. 00:56:49.570 --> 00:56:52.210 That's the most you could hope for because of an odd cycle 00:56:52.210 --> 00:56:53.030 here. 00:56:53.030 --> 00:56:57.770 And if you get the sum of all these cut sizes, in total, 00:56:57.770 --> 00:57:01.320 you must have Not All Equal in every clause. 00:57:01.320 --> 00:57:05.000 So that's simulating Not All Equal-- planar 00:57:05.000 --> 00:57:06.040 Not All Equal 3SAT. 00:57:06.040 --> 00:57:07.630 This also preserves planarity. 00:57:07.630 --> 00:57:09.192 And we just contract the variable 00:57:09.192 --> 00:57:10.650 to a point, the clauses to a point. 00:57:10.650 --> 00:57:14.900 And it is the bipartite graph of the Not All Equal picture. 00:57:14.900 --> 00:57:17.390 And we've simulated that planar Not All Equal 3SAT instance 00:57:17.390 --> 00:57:21.284 with planar max cut, which here, we actually 00:57:21.284 --> 00:57:22.450 have a polynomial algorithm. 00:57:22.450 --> 00:57:24.074 So that gives us a polynomial algorithm 00:57:24.074 --> 00:57:26.250 for planar Not All Equal 3SAT. 00:57:26.250 --> 00:57:28.280 So that's kind of fun. 00:57:28.280 --> 00:57:28.970 Similar style. 00:57:32.680 --> 00:57:34.200 Yeah. 00:57:34.200 --> 00:57:34.700 Good. 00:57:38.330 --> 00:57:39.580 So many problems. 00:58:03.600 --> 00:58:09.190 So I guess now we're going to do some reductions 00:58:09.190 --> 00:58:13.280 from various planar 3SAT things to problems you might 00:58:13.280 --> 00:58:15.690 care a little bit more about. 00:58:15.690 --> 00:58:17.230 I'm going to start-- I mean, it's 00:58:17.230 --> 00:58:19.300 sort of a vague transition, because these, 00:58:19.300 --> 00:58:22.120 you could think of just more problems of the same type. 00:58:22.120 --> 00:58:25.740 X3C is something we talked about in the context of 3PARTITION. 00:58:25.740 --> 00:58:30.040 This was exact cover with sets of size 3. 00:58:30.040 --> 00:58:33.620 I think Exact Cover with sets of size 3. 00:58:33.620 --> 00:58:36.574 So this was-- you could think of it as a hyper-graph. 00:58:36.574 --> 00:58:38.490 Or you could think of it as a bipartite graph. 00:58:38.490 --> 00:58:43.290 You have sets of size 3 which cover variables. 00:58:43.290 --> 00:58:45.890 You want every variable to be covered exactly once. 00:58:49.720 --> 00:58:52.496 But you can only choose sets of size 3. 00:58:52.496 --> 00:58:54.210 Yeah. 00:58:54.210 --> 00:59:01.250 So this is closely related, I guess, to planar 1-in-3SAT. 00:59:01.250 --> 00:59:04.420 Here, each of these things could have 00:59:04.420 --> 00:59:06.480 arbitrary degree-- each element can 00:59:06.480 --> 00:59:09.420 appear in many different sets. 00:59:09.420 --> 00:59:13.410 Whereas over-- and you're covering these with these. 00:59:13.410 --> 00:59:21.110 I think this is the reverse setup, where every-- let's see. 00:59:23.620 --> 00:59:26.230 The clauses, again, have to be exactly covered 00:59:26.230 --> 00:59:29.300 by these things. 00:59:29.300 --> 00:59:34.266 And these things have arbitrary size. 00:59:34.266 --> 00:59:36.140 So these are the things that you're choosing. 00:59:36.140 --> 00:59:38.056 I'm choosing to make this variable true, which 00:59:38.056 --> 00:59:39.389 covers this guy. 00:59:39.389 --> 00:59:40.680 These have arbitrary size here. 00:59:40.680 --> 00:59:43.590 The thing that I'm choosing has size 3. 00:59:43.590 --> 00:59:44.090 OK? 00:59:44.090 --> 00:59:45.447 So it's sort of the dual of-- 00:59:45.447 --> 00:59:46.780 AUDIENCE: What are you choosing? 00:59:46.780 --> 00:59:49.400 PROFESSOR: --of planar 1-in-3SAT. 00:59:49.400 --> 00:59:55.060 So the goal-- maybe I should write down the problem again. 00:59:55.060 --> 00:59:56.850 You're given three sets. 00:59:56.850 --> 00:59:58.680 These are sets of size 3. 00:59:58.680 --> 01:00:03.820 And you want to choose, let's say, k of them. 01:00:08.150 --> 01:00:12.765 Actually, it would be choose n/3 of them that are disjoint. 01:00:15.450 --> 01:00:18.510 And therefore, every element-- they're n elements-- 01:00:18.510 --> 01:00:23.570 every element is covered exactly once by exactly one 3SAT. 01:00:23.570 --> 01:00:25.340 So it's, I think, sort of complementary 01:00:25.340 --> 01:00:26.910 to planar 1-in-3SAT. 01:00:26.910 --> 01:00:31.350 The planar version is that this bipartite graph is planar. 01:00:31.350 --> 01:00:36.010 And Dyer and Freeze, this is same people 01:00:36.010 --> 01:00:40.240 that approved one of these many problems is hard. 01:00:40.240 --> 01:00:44.396 I think the-- I'll look back. 01:00:47.890 --> 01:00:50.070 Planar 1-in-3SAT. 01:00:50.070 --> 01:00:51.540 So no surprise. 01:00:51.540 --> 01:00:54.864 So we're going to reduce from planar 1-in-3SAT 01:00:54.864 --> 01:00:57.840 to this problem, prove that this is hard. 01:00:57.840 --> 01:00:59.600 And here's a very simple proof. 01:00:59.600 --> 01:01:00.940 They make it more complicated. 01:01:00.940 --> 01:01:05.110 But for starters, let's make a variable 01:01:05.110 --> 01:01:07.740 by this kind of even cycle trick. 01:01:07.740 --> 01:01:11.750 And the picture here is that the big circles are 3SATs, 01:01:11.750 --> 01:01:13.970 the dots are the elements you're trying to cover. 01:01:13.970 --> 01:01:15.761 And every element should be covered exactly 01:01:15.761 --> 01:01:17.700 by exactly one set. 01:01:17.700 --> 01:01:20.380 So this looks good, because there are exactly two ways 01:01:20.380 --> 01:01:21.970 to solve this thing. 01:01:21.970 --> 01:01:25.490 You could choose these guys and cover those points exactly 01:01:25.490 --> 01:01:28.987 once, in which case, this is covered, this is covered, 01:01:28.987 --> 01:01:29.820 and this is covered. 01:01:29.820 --> 01:01:31.840 But these other guys are uncovered. 01:01:31.840 --> 01:01:34.630 Those are going to attach to other gadgets. 01:01:34.630 --> 01:01:36.510 Or you could do the reverse. 01:01:36.510 --> 01:01:39.590 So that's going to correspond to a true or false setting. 01:01:39.590 --> 01:01:41.430 And I think-- this is not in their paper, 01:01:41.430 --> 01:01:46.040 but I think this would be a clause for exactly one 3SAT. 01:01:46.040 --> 01:01:51.870 Just connect-- you could have even negated 01:01:51.870 --> 01:01:53.700 versions of your variables. 01:01:53.700 --> 01:01:57.430 But let's say we have all positive planar 1-in-3SAT. 01:01:57.430 --> 01:01:59.700 3, So I'll just take a positive copy 01:01:59.700 --> 01:02:02.800 of xi, a positive copy of xj from somewhere, 01:02:02.800 --> 01:02:06.365 and then just bring them together at a common dot. 01:02:06.365 --> 01:02:07.740 Then that point should be covered 01:02:07.740 --> 01:02:10.073 by exactly one of them, which means exactly one of those 01:02:10.073 --> 01:02:11.339 is true. 01:02:11.339 --> 01:02:11.838 Done. 01:02:11.838 --> 01:02:13.320 Very easy. 01:02:13.320 --> 01:02:15.310 Now, they want to prove more things, 01:02:15.310 --> 01:02:20.150 so they end up using-- I think I will just show you for fun. 01:02:20.150 --> 01:02:24.520 They end up using a more complicated clause and a more 01:02:24.520 --> 01:02:29.400 complicated way to connect these things into the clause, 01:02:29.400 --> 01:02:33.400 because they want to prove another problem hard, 01:02:33.400 --> 01:02:39.597 which is planar three-dimensional matching. 01:02:39.597 --> 01:02:41.555 Three-dimensional matching was a generalization 01:02:41.555 --> 01:02:45.780 of numerical three-dimensional matching, which was 01:02:45.780 --> 01:02:48.300 closely related to 3PARTITION. 01:02:48.300 --> 01:02:52.170 In general, three-dimensional matching, 01:02:52.170 --> 01:02:53.480 it's like this problem. 01:02:53.480 --> 01:02:55.390 But you also have the extra information 01:02:55.390 --> 01:02:59.280 that for every set of size 3, one of these is red, 01:02:59.280 --> 01:03:03.430 another one is blue, another one is yellow. 01:03:03.430 --> 01:03:06.260 This paper uses the additive primary colors. 01:03:06.260 --> 01:03:07.516 I don't know why. 01:03:07.516 --> 01:03:10.780 It could be green if you prefer. 01:03:10.780 --> 01:03:12.840 So there are three types of elements. 01:03:12.840 --> 01:03:15.730 And you're told that every set has one of each type. 01:03:15.730 --> 01:03:19.790 So that's extra information that's useful in some proofs. 01:03:19.790 --> 01:03:22.610 So this is a better construction, 01:03:22.610 --> 01:03:26.400 because it's going to end up being three-colorable. 01:03:26.400 --> 01:03:28.834 Yeah. 01:03:28.834 --> 01:03:31.000 I don't know how much you care about this reduction. 01:03:31.000 --> 01:03:34.100 I think I will skip the details, although I spent 01:03:34.100 --> 01:03:37.270 a lot of time understanding it. 01:03:37.270 --> 01:03:40.207 The rough idea is that these connectors-- there's 01:03:40.207 --> 01:03:42.290 a positive and a negative version, because they're 01:03:42.290 --> 01:03:43.289 worried about negations. 01:03:43.289 --> 01:03:46.190 But these connections will attach 01:03:46.190 --> 01:03:48.090 to these three terminals for one variable, 01:03:48.090 --> 01:03:49.506 these three terminals for another, 01:03:49.506 --> 01:03:50.720 and these three for another. 01:03:50.720 --> 01:03:53.530 And just by a counting argument in here, 01:03:53.530 --> 01:04:00.290 there is 1, 2, 3, 4, 5, 6, 7, 8, 9 of these guys. 01:04:00.290 --> 01:04:03.455 And there are however many-- I guess I should really 01:04:03.455 --> 01:04:04.580 be counting the black dots. 01:04:04.580 --> 01:04:11.360 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12 points here. 01:04:11.360 --> 01:04:17.210 So the best you can do is to choose three of the sets 01:04:17.210 --> 01:04:20.680 inside that will cover all of the black dots 01:04:20.680 --> 01:04:22.230 except for three of them. 01:04:22.230 --> 01:04:24.490 And so from that, you get exactly one 1SAT. 01:04:24.490 --> 01:04:28.196 Exactly one of these should be covered from somewhere else. 01:04:28.196 --> 01:04:29.820 That would correspond to this situation 01:04:29.820 --> 01:04:31.954 where you cover these three points as 01:04:31.954 --> 01:04:34.370 opposed to this situation, where none of them get covered. 01:04:34.370 --> 01:04:35.844 That's like these guys. 01:04:35.844 --> 01:04:37.510 And then you can always satisfy the rest 01:04:37.510 --> 01:04:39.130 by adding three in there. 01:04:39.130 --> 01:04:42.870 So that's roughly how that clause works. 01:04:42.870 --> 01:04:45.860 Then that picture is three-colorable. 01:04:45.860 --> 01:04:50.449 And this way, you can color all the dots with three colors. 01:04:50.449 --> 01:04:51.490 And it pretty much works. 01:04:51.490 --> 01:04:54.210 The variable alternates red, yellow. 01:04:54.210 --> 01:04:56.570 So we always get blue connections, which is good. 01:04:56.570 --> 01:04:59.250 So we can attach to anything, except that the way 01:04:59.250 --> 01:05:00.730 that we attach is like this. 01:05:00.730 --> 01:05:05.400 And these are three colors which match here. 01:05:05.400 --> 01:05:08.552 And in this case, these three colors match here. 01:05:08.552 --> 01:05:10.510 But you might want to attach here, for example. 01:05:10.510 --> 01:05:15.530 So you need another connector, which is slightly different. 01:05:15.530 --> 01:05:17.370 This does exactly the same thing, 01:05:17.370 --> 01:05:20.350 but now this color pattern matches 01:05:20.350 --> 01:05:22.020 here, but in negated form. 01:05:22.020 --> 01:05:28.510 And if you switch these two colors, it matches here, 01:05:28.510 --> 01:05:29.780 I think when it's upside-down. 01:05:29.780 --> 01:05:32.238 So there's this version, and there's the reflected version. 01:05:32.238 --> 01:05:37.130 And then there's also this coloring of the same gadget. 01:05:37.130 --> 01:05:38.462 And so you do all these things. 01:05:38.462 --> 01:05:40.170 You know what you're supposed to connect, 01:05:40.170 --> 01:05:42.253 and so you just choose one of these three gadgets, 01:05:42.253 --> 01:05:44.989 and it connects to one of the three appropriate places. 01:05:44.989 --> 01:05:45.780 That's their proof. 01:05:45.780 --> 01:05:47.480 It was fun to read. 01:05:47.480 --> 01:05:49.730 And that proves that planar three-dimensional matching 01:05:49.730 --> 01:05:51.270 is hard. 01:05:51.270 --> 01:05:51.870 OK. 01:05:51.870 --> 01:05:54.426 Plus, in their diagrams, oh, they 01:05:54.426 --> 01:05:56.300 have dashed lines through everything in order 01:05:56.300 --> 01:05:58.550 to illustrate that you can have one path that 01:05:58.550 --> 01:06:01.495 visits all of the elements, just like in planar 3SAT. 01:06:01.495 --> 01:06:04.630 So also, both planar three-dimensional matching and 01:06:04.630 --> 01:06:11.512 planar exact cover by 3SATs, you can have one cycle 01:06:11.512 --> 01:06:12.595 that visits every element. 01:06:16.240 --> 01:06:18.120 Yeah. 01:06:18.120 --> 01:06:19.300 Cool. 01:06:19.300 --> 01:06:22.385 Here's another relatively simple proof. 01:06:22.385 --> 01:06:24.260 This is in the original Lichtenstein paper, 01:06:24.260 --> 01:06:27.540 so this is one of their motivations for planar 3SAT. 01:06:27.540 --> 01:06:29.040 Planar vertex cover. 01:06:29.040 --> 01:06:32.190 So what's vertex cover? 01:06:32.190 --> 01:06:33.430 You're given a graph. 01:06:33.430 --> 01:06:36.440 You want to choose a set of vertices 01:06:36.440 --> 01:06:37.810 that cover all of the edges. 01:06:37.810 --> 01:06:39.950 So for example, if you have a triangle, 01:06:39.950 --> 01:06:42.085 and I choose this vertex, it covers this edge, 01:06:42.085 --> 01:06:43.470 and it covers this edge. 01:06:43.470 --> 01:06:44.660 I have one edge uncovered. 01:06:44.660 --> 01:06:46.790 So I'm going to add this one and cover that. 01:06:46.790 --> 01:06:49.440 So I don't require exactly one cover. 01:06:49.440 --> 01:06:52.150 Just every edge has to be covered by at least one 01:06:52.150 --> 01:06:53.810 of its endpoints. 01:06:53.810 --> 01:06:57.970 So you can think of it as an oar of its two endpoints. 01:06:57.970 --> 01:06:58.470 OK. 01:06:58.470 --> 01:07:01.750 But this problem is NP-hard even for planar graphs. 01:07:01.750 --> 01:07:03.710 And here's one way to do. 01:07:03.710 --> 01:07:05.910 This is like the whole proof in one little picture. 01:07:05.910 --> 01:07:08.450 You have a variable gadget, which, 01:07:08.450 --> 01:07:12.040 again, is just a even cycle. 01:07:12.040 --> 01:07:16.610 And in an even cycle, the vertex cover-- 01:07:16.610 --> 01:07:18.350 let's say this has size k. 01:07:18.350 --> 01:07:21.570 And I give you the budget of only k/2 for your vertex cover, 01:07:21.570 --> 01:07:22.924 then you have to alternate. 01:07:22.924 --> 01:07:25.090 So either these three guys are in your vertex cover, 01:07:25.090 --> 01:07:26.794 or these three guys are. 01:07:26.794 --> 01:07:28.210 This proof actually looks the same 01:07:28.210 --> 01:07:30.120 as the last one we saw, right? 01:07:30.120 --> 01:07:32.620 Then those things are just connected to triangles. 01:07:32.620 --> 01:07:33.970 Same deal here. 01:07:33.970 --> 01:07:35.900 I'm going to give you a budget of two 01:07:35.900 --> 01:07:38.700 for this clause, because to cover all three of these edges, 01:07:38.700 --> 01:07:42.480 you need at least two vertices, like we did over there. 01:07:42.480 --> 01:07:45.752 And if you're going to get away with only two-- so 01:07:45.752 --> 01:07:47.460 if I choose, for example, these two, that 01:07:47.460 --> 01:07:50.020 also covers this edge and this edge for free. 01:07:50.020 --> 01:07:51.450 But it does not cover this edge. 01:07:51.450 --> 01:07:53.640 This would be the one edge not covered by the clause 01:07:53.640 --> 01:07:55.640 alone if you only get a budget of two. 01:07:55.640 --> 01:07:57.106 And that's the one that better be 01:07:57.106 --> 01:07:58.480 covered by this side, which means 01:07:58.480 --> 01:08:00.117 this variable is set to true. 01:08:00.117 --> 01:08:02.200 And in general, at least one of these three things 01:08:02.200 --> 01:08:02.960 better be covered. 01:08:02.960 --> 01:08:05.110 Otherwise you won't have enough budget 01:08:05.110 --> 01:08:07.620 to finish that triangle clause. 01:08:07.620 --> 01:08:08.420 Question? 01:08:08.420 --> 01:08:11.620 AUDIENCE: So the hexagon versus square here 01:08:11.620 --> 01:08:13.370 is just to give you more connection points 01:08:13.370 --> 01:08:13.810 into other clauses? 01:08:13.810 --> 01:08:14.030 PROFESSOR: Yeah. 01:08:14.030 --> 01:08:15.405 Just make these cycles big enough 01:08:15.405 --> 01:08:18.120 to connect up to all the things you need it in. 01:08:18.120 --> 01:08:20.340 So if you have ni occurrences, you're 01:08:20.340 --> 01:08:22.754 going to do, like, 2 ni or something. 01:08:25.970 --> 01:08:26.470 Cool. 01:08:26.470 --> 01:08:28.330 So that's planar vertex cover. 01:08:28.330 --> 01:08:31.939 And because this reduction preserved planarity, 01:08:31.939 --> 01:08:35.180 we get planar vertex cover's hard, 01:08:35.180 --> 01:08:36.390 because planar 3SAT is hard. 01:08:36.390 --> 01:08:37.848 Here we didn't need any connections 01:08:37.848 --> 01:08:39.186 between the variables. 01:08:39.186 --> 01:08:40.810 Here's one where we do need connections 01:08:40.810 --> 01:08:42.330 between the variables. 01:08:42.330 --> 01:08:46.970 So this is planar-directed Hamiltonian cycle. 01:08:46.970 --> 01:08:49.810 So I think you all know what Hamiltonian cycle is. 01:08:49.810 --> 01:08:52.090 And you're given a directed graph here. 01:08:52.090 --> 01:08:54.560 You want to find one path that visits every vertex exactly 01:08:54.560 --> 01:08:55.490 once. 01:08:55.490 --> 01:08:57.149 Don't care about edges. 01:08:57.149 --> 01:08:59.980 So we can do a similar kind of thing. 01:08:59.980 --> 01:09:02.615 The proof is a little bit-- but checking this works is 01:09:02.615 --> 01:09:03.740 a little bit more annoying. 01:09:03.740 --> 01:09:08.262 But here's Lichtenstein's idea for variable. 01:09:08.262 --> 01:09:09.095 Essentially, a wire. 01:09:11.720 --> 01:09:13.069 The graph is directed here. 01:09:13.069 --> 01:09:15.724 So we get to say there's an incoming edge here. 01:09:15.724 --> 01:09:16.640 Then you get a choice. 01:09:16.640 --> 01:09:18.340 So which way it'll go. 01:09:18.340 --> 01:09:20.414 But then you basically have to alternate. 01:09:20.414 --> 01:09:22.080 Because of these vertices in the middle, 01:09:22.080 --> 01:09:23.540 you've got to visit all of them. 01:09:23.540 --> 01:09:25.649 And the only way to do that is to alternate. 01:09:25.649 --> 01:09:27.520 So you get this alternating pattern, 01:09:27.520 --> 01:09:31.330 which means if you look at these edges, a1 and a1 bar, 01:09:31.330 --> 01:09:33.430 exactly one of these is in. 01:09:33.430 --> 01:09:35.899 And then the opposite of these are in. 01:09:35.899 --> 01:09:38.170 The variable here is called a. 01:09:38.170 --> 01:09:40.529 So there's many instances of a. 01:09:40.529 --> 01:09:44.439 And these are going to be in-- they're going 01:09:44.439 --> 01:09:45.630 to be both in or both out. 01:09:45.630 --> 01:09:48.700 And then these are going to be both out or both in and so on. 01:09:48.700 --> 01:09:51.569 So because I have to zigzag, I made 01:09:51.569 --> 01:09:56.090 a bunch of copies, ni copies of the variable a. 01:09:56.090 --> 01:09:58.460 But you're free to choose one setting or the other. 01:09:58.460 --> 01:10:01.420 And then the clause is just going to be a single vertex. 01:10:01.420 --> 01:10:07.617 And the idea is if we want b bar to satisfy this clause, 01:10:07.617 --> 01:10:09.200 we're going to add in those two edges. 01:10:09.200 --> 01:10:12.140 So instead of going straight here, you could have done this 01:10:12.140 --> 01:10:13.620 and grabbed the clause for free. 01:10:13.620 --> 01:10:15.180 You don't have to. 01:10:15.180 --> 01:10:17.650 But this is going to get grabbed if and only 01:10:17.650 --> 01:10:20.692 if exactly-- at least one of these 01:10:20.692 --> 01:10:21.900 chooses the appropriate edge. 01:10:21.900 --> 01:10:23.384 If you're using this edge, there's 01:10:23.384 --> 01:10:24.550 no way to cover this clause. 01:10:24.550 --> 01:10:27.010 But if using this edge, you can do that. 01:10:27.010 --> 01:10:29.715 You have to check that you can't let go from here to there 01:10:29.715 --> 01:10:30.880 to over here. 01:10:30.880 --> 01:10:32.470 That's what this figure is about. 01:10:32.470 --> 01:10:35.080 So basically, if you're alternating here, 01:10:35.080 --> 01:10:38.300 you switch sides, and then you're alternating here, 01:10:38.300 --> 01:10:40.420 this guy is uncovered. 01:10:40.420 --> 01:10:42.010 And apparently, this proof works even 01:10:42.010 --> 01:10:44.590 when the graph is undirected, but that's even less clear. 01:10:44.590 --> 01:10:46.954 We'll see other Hamiltonian cycle proofs 01:10:46.954 --> 01:10:48.120 that are stronger than this. 01:10:48.120 --> 01:10:49.650 But it's a nice illustration. 01:10:49.650 --> 01:10:52.910 Here we're using that we can connect all the vertices 01:10:52.910 --> 01:10:53.480 together. 01:10:53.480 --> 01:10:56.980 That's what these connections are in the cycle. 01:10:56.980 --> 01:11:00.342 So there's this big vertex loop on the outside 01:11:00.342 --> 01:11:02.300 taking these gadgets, and pasting them together 01:11:02.300 --> 01:11:06.110 in a big loop, because we want an overall Hamiltonian cycle. 01:11:06.110 --> 01:11:10.340 Because we know that preserves linearity, life is good. 01:11:10.340 --> 01:11:10.840 OK. 01:11:10.840 --> 01:11:12.550 So there'll be some clauses inside the cycle, 01:11:12.550 --> 01:11:13.841 some clauses outside the cycle. 01:11:13.841 --> 01:11:16.500 But that doesn't matter. 01:11:16.500 --> 01:11:17.000 OK. 01:11:17.000 --> 01:11:19.345 That's planar-directed Hamiltonian cycle. 01:11:22.310 --> 01:11:24.930 Time for a Nikoli game. 01:11:24.930 --> 01:11:28.060 I bumped into this slide that I made a while ago. 01:11:28.060 --> 01:11:30.610 So for fun, these are all the Nikoli games 01:11:30.610 --> 01:11:32.850 I know that have been proved hard except the one I'm 01:11:32.850 --> 01:11:34.180 going to talk about. 01:11:34.180 --> 01:11:37.300 These are the references. 01:11:37.300 --> 01:11:40.120 We covered one of them, right, the Light Up. 01:11:40.120 --> 01:11:42.800 But there's a lot. 01:11:42.800 --> 01:11:44.440 And lots of papers proving them. 01:11:44.440 --> 01:11:46.580 So here's a relatively new game. 01:11:46.580 --> 01:11:48.970 It came out a couple years ago in Nikoli land, 01:11:48.970 --> 01:11:50.520 although it was invented before that. 01:11:50.520 --> 01:11:54.800 Shakashaka, which is like a shaking sound. 01:11:54.800 --> 01:12:00.359 So we have a square grid, blank squares, and obstacle squares. 01:12:00.359 --> 01:12:02.900 The obstacle squares, some of them have numbers, some of them 01:12:02.900 --> 01:12:04.970 are wild cards. 01:12:04.970 --> 01:12:10.720 And what you're allowed to do for a blank square-- or some 01:12:10.720 --> 01:12:14.670 of the blank squares, you can fill in one half. 01:12:14.670 --> 01:12:16.540 So you have like four different halves 01:12:16.540 --> 01:12:18.340 of the square you can choose. 01:12:18.340 --> 01:12:21.040 There they are-- this one, this one, this one, and this one. 01:12:21.040 --> 01:12:23.730 And you can fill it in half black, half white. 01:12:23.730 --> 01:12:26.170 So you cut along a diagonal, and then you 01:12:26.170 --> 01:12:28.705 choose one of the two halves to fill in black. 01:12:28.705 --> 01:12:29.455 You don't have to. 01:12:29.455 --> 01:12:32.280 You could just leave it white. 01:12:32.280 --> 01:12:35.020 And when you have a numbered square, 01:12:35.020 --> 01:12:37.950 then the number of filled things must be exactly that number. 01:12:37.950 --> 01:12:42.150 So here, there's one adjacent to it, and nothing down here. 01:12:42.150 --> 01:12:46.120 This guy has exactly two half-filled squares next to it. 01:12:46.120 --> 01:12:47.380 This one has exactly one. 01:12:47.380 --> 01:12:50.900 This one has exactly zero half-filled squares. 01:12:50.900 --> 01:12:56.530 Think of blank squares as zeros and these as counting as one. 01:12:56.530 --> 01:12:59.440 Plus, the goal is that the regions you make-- this 01:12:59.440 --> 01:13:02.296 would all be easy so far, I think. 01:13:02.296 --> 01:13:04.670 But the extra constraint is that all the regions you make 01:13:04.670 --> 01:13:07.220 must be rectangles. 01:13:07.220 --> 01:13:07.720 OK? 01:13:07.720 --> 01:13:11.120 This is a 45-degree-rotated rectangle. 01:13:11.120 --> 01:13:12.460 This is a regular rectangle. 01:13:12.460 --> 01:13:13.730 So you can use either one. 01:13:13.730 --> 01:13:14.855 So it's a funny constraint. 01:13:14.855 --> 01:13:17.540 It's actually quite a fun game to play. 01:13:17.540 --> 01:13:19.970 You're not allowed to fill in a square 100%. 01:13:19.970 --> 01:13:22.250 That one was already filled. 01:13:22.250 --> 01:13:24.190 That's the rules. 01:13:24.190 --> 01:13:26.840 Here's a reduction from planar 3SAT. 01:13:26.840 --> 01:13:28.230 Pretty simple. 01:13:28.230 --> 01:13:30.830 Looks a lot like Minesweeper, in some sense, 01:13:30.830 --> 01:13:32.640 a little bit thickened. 01:13:32.640 --> 01:13:36.600 But this is a wire, because the 1 01:13:36.600 --> 01:13:38.610 says that one of the two sides is half-filled. 01:13:38.610 --> 01:13:41.068 Once that's half-filled, you have to make this a rectangle. 01:13:41.068 --> 01:13:43.300 You can't just leave it as a pentagon. 01:13:43.300 --> 01:13:45.830 So you've got to fill that in to a rotated square. 01:13:45.830 --> 01:13:48.030 But because of the 1, this must be an empty square. 01:13:48.030 --> 01:13:49.330 So it alternates. 01:13:49.330 --> 01:13:51.130 You can do this or this. 01:13:51.130 --> 01:13:51.630 OK. 01:13:51.630 --> 01:13:54.170 You can easily split just by connecting those things 01:13:54.170 --> 01:13:55.740 in the obvious way. 01:13:55.740 --> 01:13:57.210 You can do 90-degree turns. 01:13:57.210 --> 01:13:57.710 No problem. 01:14:00.520 --> 01:14:04.790 Here's a slightly more sophisticated-- slightly more 01:14:04.790 --> 01:14:06.350 sophisticated gadget. 01:14:06.350 --> 01:14:09.120 You can think of this as a terminator. 01:14:09.120 --> 01:14:12.072 It would be hard to just stop this somewhere. 01:14:12.072 --> 01:14:15.466 Well, maybe you could just go all black. 01:14:15.466 --> 01:14:17.560 I think that's OK as well. 01:14:17.560 --> 01:14:19.080 This could also be a terminator. 01:14:19.080 --> 01:14:23.510 You can also think of this as a negator. 01:14:23.510 --> 01:14:25.680 Of course, also, the splitter's like a negator. 01:14:25.680 --> 01:14:28.770 This guide's just probably not so necessary. 01:14:28.770 --> 01:14:31.340 Let's get to Clause. 01:14:31.340 --> 01:14:33.890 So Clause is bringing three things together 01:14:33.890 --> 01:14:34.970 with one of these blocks. 01:14:34.970 --> 01:14:36.480 But we do it in a funny way. 01:14:36.480 --> 01:14:38.200 There's no 1's here or here. 01:14:38.200 --> 01:14:39.960 So we end up with this L blank shape. 01:14:39.960 --> 01:14:42.520 Everything else is now drawn as black. 01:14:42.520 --> 01:14:43.450 That's just obstacles. 01:14:43.450 --> 01:14:46.050 So this is not happy the way it is. 01:14:46.050 --> 01:14:48.280 There's lots of ways to resolve it. 01:14:48.280 --> 01:14:50.270 The one case where we can't resolve 01:14:50.270 --> 01:14:57.360 it is when all of these wires are 0's, because there's 01:14:57.360 --> 01:14:59.870 already stuff here, and stuff here, and stuff here, 01:14:59.870 --> 01:15:01.480 so we can't put anything here. 01:15:01.480 --> 01:15:03.360 Can't put anything here, or here, or here, 01:15:03.360 --> 01:15:04.440 because of those 1's. 01:15:04.440 --> 01:15:07.800 Then you're forced to have the L shape, and you're toast. 01:15:07.800 --> 01:15:10.430 In every other case, we can decompose 01:15:10.430 --> 01:15:13.710 into sometimes a big rectangle, sometimes little squares, 01:15:13.710 --> 01:15:14.950 sometimes rotated squares. 01:15:14.950 --> 01:15:15.810 But that's OK. 01:15:15.810 --> 01:15:19.040 So this is a regular 3SAT clause. 01:15:19.040 --> 01:15:22.680 The one thing that's missing at this point 01:15:22.680 --> 01:15:27.290 is a parity shift gadget, because a lot of these gadgets 01:15:27.290 --> 01:15:30.880 have very specific lengths, modular 3 or whatever. 01:15:30.880 --> 01:15:35.360 So it turns out this is a way to shift things slightly. 01:15:35.360 --> 01:15:37.750 It can be filled in two ways. 01:15:37.750 --> 01:15:42.630 You can fill in all of these guys like that 01:15:42.630 --> 01:15:44.880 and make this giant rotated square. 01:15:44.880 --> 01:15:48.370 Or if these are not allowed because these are here, 01:15:48.370 --> 01:15:50.512 then you can do this weird filling. 01:15:50.512 --> 01:15:52.970 You end up with a rectangle there, and two rotated squares, 01:15:52.970 --> 01:15:55.659 another rectangle. 01:15:55.659 --> 01:15:56.950 And you count the total length. 01:15:56.950 --> 01:16:02.387 This is length 4, whereas most of the things are length 2? 01:16:02.387 --> 01:16:03.580 AUDIENCE: 3 [INAUDIBLE]. 01:16:03.580 --> 01:16:04.180 PROFESSOR: 3. 01:16:04.180 --> 01:16:04.680 OK. 01:16:04.680 --> 01:16:08.032 So that makes me happy. 01:16:08.032 --> 01:16:08.740 Shift the parity. 01:16:08.740 --> 01:16:11.899 And that is Shakashaka. 01:16:11.899 --> 01:16:13.690 Here's an overall picture where we actually 01:16:13.690 --> 01:16:15.500 plugged some gadgets together. 01:16:15.500 --> 01:16:16.420 That was fun. 01:16:19.370 --> 01:16:26.450 So I think I have one more proof to just sketch. 01:16:26.450 --> 01:16:30.220 So this is a problem of so-called fixed-angle chains. 01:16:30.220 --> 01:16:32.320 So you have this forced 90-degree angle. 01:16:32.320 --> 01:16:34.560 We had a similar thing with the Hamiltonian path 01:16:34.560 --> 01:16:37.240 with cubes and elastics between them. 01:16:37.240 --> 01:16:40.380 You could twist one edge around the other, 01:16:40.380 --> 01:16:43.710 one block around the other, but you couldn't adjust this angle. 01:16:43.710 --> 01:16:44.210 OK? 01:16:44.210 --> 01:16:46.492 So the problem is I give you such a chain, which 01:16:46.492 --> 01:16:48.575 is specified by lengths and then 90-degree angles. 01:16:48.575 --> 01:16:51.280 So it can be-- you don't know whether the 90-degree angle 01:16:51.280 --> 01:16:52.870 goes up or down. 01:16:52.870 --> 01:16:56.850 And you want to just put it into the plane without crossings. 01:16:56.850 --> 01:16:59.030 And this is an old proof from 2000. 01:16:59.030 --> 01:17:01.150 The problem is weakly NP-hard. 01:17:01.150 --> 01:17:02.690 It's basically a partition proof, 01:17:02.690 --> 01:17:04.490 so it is a partition proof. 01:17:04.490 --> 01:17:07.150 You get to choose, for each of these things, 01:17:07.150 --> 01:17:08.550 whether it goes left or right. 01:17:08.550 --> 01:17:10.260 And you need to line up this with that. 01:17:10.260 --> 01:17:12.342 Otherwise you get collisions. 01:17:12.342 --> 01:17:13.550 You would just need to argue. 01:17:13.550 --> 01:17:16.260 This outer structure has a unique embedding, roughly. 01:17:16.260 --> 01:17:17.370 Unique enough. 01:17:17.370 --> 01:17:19.150 It always looks like that. 01:17:19.150 --> 01:17:24.180 So we proved a few years ago that it is strongly NP-hard. 01:17:24.180 --> 01:17:25.790 And I will just sketch the proof, 01:17:25.790 --> 01:17:28.440 because it's fun and cool. 01:17:28.440 --> 01:17:30.060 So here's the rough idea. 01:17:30.060 --> 01:17:32.910 Imagine this is like a piece of wire. 01:17:32.910 --> 01:17:35.610 And you can flip it up or down. 01:17:35.610 --> 01:17:37.700 All the angles stay 90 degrees. 01:17:37.700 --> 01:17:38.820 These are the down-case. 01:17:38.820 --> 01:17:40.290 Here are some example up-cases. 01:17:40.290 --> 01:17:42.740 One of them's going to represent false, the other true. 01:17:42.740 --> 01:17:43.240 Question? 01:17:43.240 --> 01:17:45.823 AUDIENCE: What's the difference between this and the Carpenter 01:17:45.823 --> 01:17:46.685 Ruler problem? 01:17:46.685 --> 01:17:47.810 PROFESSOR: Which Carpenter? 01:17:47.810 --> 01:17:50.140 AUDIENCE: Like you have to fold the ruler to fit. 01:17:50.140 --> 01:17:51.973 PROFESSOR: Oh, to fit inside a given length. 01:17:54.954 --> 01:17:56.120 Well, the goal is different. 01:17:56.120 --> 01:17:59.490 Here, the goal is to draw it in the plane without crossings. 01:17:59.490 --> 01:18:03.600 Before, the goal was to draw it in the plane with minimum span. 01:18:06.220 --> 01:18:07.330 To fit inside box. 01:18:07.330 --> 01:18:07.830 Yeah. 01:18:07.830 --> 01:18:09.790 I mean, this proof is pretty much the same. 01:18:09.790 --> 01:18:11.496 This proof will be totally different. 01:18:11.496 --> 01:18:13.120 That problem was also only weakly hard. 01:18:13.120 --> 01:18:15.060 There was a pseudo poly-algorithm. 01:18:15.060 --> 01:18:16.620 This problem is strongly NP-hard. 01:18:16.620 --> 01:18:17.994 There is no pseudo poly-algorithm 01:18:17.994 --> 01:18:19.140 unless P equals np. 01:18:19.140 --> 01:18:21.314 So just a slightly different goal. 01:18:21.314 --> 01:18:22.730 That one was more one-dimensional. 01:18:22.730 --> 01:18:24.896 This is going to be much more two-dimensional That's 01:18:24.896 --> 01:18:25.860 the other difference. 01:18:25.860 --> 01:18:26.360 OK. 01:18:26.360 --> 01:18:30.610 So this is our some kind of variable gadget. 01:18:30.610 --> 01:18:32.645 This is the variables gadget. 01:18:32.645 --> 01:18:34.570 All n variables are represented here. 01:18:34.570 --> 01:18:36.022 This represents one variable. 01:18:36.022 --> 01:18:38.230 At this point, I only have one copy of each variable, 01:18:38.230 --> 01:18:40.600 because there's no constraint between these. 01:18:40.600 --> 01:18:43.060 Each of them can independently flip up or down. 01:18:43.060 --> 01:18:44.120 Next, we have a clause. 01:18:44.120 --> 01:18:46.510 This is where things get fun. 01:18:46.510 --> 01:18:51.240 So there's some infrastructure, but the main action is here. 01:18:51.240 --> 01:18:53.250 Each of these is sort of independently 01:18:53.250 --> 01:18:55.270 pop in or outable. 01:18:55.270 --> 01:18:57.810 It's like some kind of pleated form. 01:18:57.810 --> 01:19:00.300 It can either pleat back and forth, like it's doing here, 01:19:00.300 --> 01:19:02.780 where it pleats in and out and out and in. 01:19:02.780 --> 01:19:06.120 And that gets this to this position. 01:19:06.120 --> 01:19:08.030 With a slightly different pleating, 01:19:08.030 --> 01:19:09.495 it ends up in this position. 01:19:09.495 --> 01:19:10.870 Or a slightly different pleating, 01:19:10.870 --> 01:19:13.700 where they're all out, you end up in this position. 01:19:13.700 --> 01:19:17.100 And notice this structure. 01:19:17.100 --> 01:19:18.930 So I'll go through it again. 01:19:18.930 --> 01:19:20.610 Here's the left one. 01:19:20.610 --> 01:19:24.410 And then the middle one. 01:19:24.410 --> 01:19:26.540 And then the last one. 01:19:26.540 --> 01:19:29.510 What ends up happening is that these pegs, in order 01:19:29.510 --> 01:19:36.700 to avoid hitting that peg, I occupy either P1 or P2 or P3. 01:19:36.700 --> 01:19:38.640 So I'll go through it one more time. 01:19:38.640 --> 01:19:40.900 Here it's hitting P1, because this basically 01:19:40.900 --> 01:19:43.576 pushes this thing down. 01:19:43.576 --> 01:19:45.200 Whereas the other ones can be up-- this 01:19:45.200 --> 01:19:48.440 is up and up and up and up. 01:19:48.440 --> 01:19:50.390 Or if we shift it into the middle, 01:19:50.390 --> 01:19:52.220 then this one has to be down. 01:19:52.220 --> 01:19:56.040 But this guy can be up and avoid P1 but hit P2. 01:19:56.040 --> 01:20:00.530 Or this one's down, these are up, and we hit P3. 01:20:00.530 --> 01:20:02.740 We want to avoid collisions, so we 01:20:02.740 --> 01:20:06.440 have to collide with exactly one of these things, 01:20:06.440 --> 01:20:08.790 or at least one of them, I guess. 01:20:08.790 --> 01:20:11.890 So that's the sort of picture. 01:20:11.890 --> 01:20:14.710 This is a bunch of those clauses. 01:20:14.710 --> 01:20:21.427 There's one here, there's one here, and there's one here. 01:20:21.427 --> 01:20:22.010 Look familiar? 01:20:22.010 --> 01:20:28.080 This is planar monotone rectilinear 3SAT, I think. 01:20:31.160 --> 01:20:32.910 So far so good, except this chain 01:20:32.910 --> 01:20:34.380 is separate from this one, so we're 01:20:34.380 --> 01:20:35.671 going to connect them together. 01:20:37.980 --> 01:20:39.550 Is a little more, I think. 01:20:39.550 --> 01:20:40.400 Yeah. 01:20:40.400 --> 01:20:41.790 Connect them together. 01:20:41.790 --> 01:20:43.990 So now it is one big chain. 01:20:43.990 --> 01:20:46.340 What I ended up doing, there's still this big thing 01:20:46.340 --> 01:20:48.180 to represent the entire variable. 01:20:48.180 --> 01:20:50.770 But I basically made a whole bunch of copies of it, 01:20:50.770 --> 01:20:54.070 so that I could attach things here. 01:20:54.070 --> 01:20:57.120 But for example, when this is up, all of these guys 01:20:57.120 --> 01:20:58.630 have to simultaneously be up. 01:20:58.630 --> 01:21:01.980 So it forces all the copies of the variable 01:21:01.980 --> 01:21:03.430 here to be one way or the other. 01:21:03.430 --> 01:21:05.669 This is one variable, this is another variable, 01:21:05.669 --> 01:21:07.960 this is another variable, and this is another variable, 01:21:07.960 --> 01:21:09.370 just like before. 01:21:09.370 --> 01:21:11.780 So same thing, but now I can connect them all together. 01:21:11.780 --> 01:21:16.830 Up here, these are all the negative, I think, clauses. 01:21:16.830 --> 01:21:17.510 Doesn't matter. 01:21:17.510 --> 01:21:20.070 And then down here, you put all the positive clauses. 01:21:20.070 --> 01:21:21.128 And that's the proof. 01:21:21.128 --> 01:21:21.627 Question. 01:21:21.627 --> 01:21:23.263 AUDIENCE: --all those three dots in a row? 01:21:23.263 --> 01:21:25.679 Like, it seems to me that if you have two dots [INAUDIBLE] 01:21:25.679 --> 01:21:28.940 straight line, they might as well just merge into one dot. 01:21:28.940 --> 01:21:31.070 PROFESSOR: I think the idea is, yeah, this 01:21:31.070 --> 01:21:32.470 could be a single line. 01:21:32.470 --> 01:21:34.970 We subdivide it just to point out that you can attach things 01:21:34.970 --> 01:21:35.970 to it. 01:21:35.970 --> 01:21:36.810 Yeah. 01:21:36.810 --> 01:21:41.260 There's extra dots, but just for consistency's sake, I guess. 01:21:41.260 --> 01:21:45.242 So now everywhere here, you can attach one of the clauses. 01:21:45.242 --> 01:21:46.700 And that way, they all kind of hold 01:21:46.700 --> 01:21:48.238 to the same infrastructure. 01:21:48.238 --> 01:21:51.230 Yeah. 01:21:51.230 --> 01:21:51.830 Clear? 01:21:51.830 --> 01:21:52.929 [LAUGHS] 01:21:52.929 --> 01:21:54.970 So obviously, you get pretty complicated pictures 01:21:54.970 --> 01:21:55.470 in the end. 01:21:55.470 --> 01:21:57.980 But this is a proof that clearly benefited 01:21:57.980 --> 01:22:01.400 from the rectilinear aspect and the monotone aspect 01:22:01.400 --> 01:22:04.920 that all the true thing-- we could not get some true above 01:22:04.920 --> 01:22:06.070 and some true below. 01:22:06.070 --> 01:22:07.832 They all wanted to go the same way. 01:22:07.832 --> 01:22:09.290 And the rectilinear was nice, so we 01:22:09.290 --> 01:22:11.570 could draw this all on a horizontal line 01:22:11.570 --> 01:22:13.660 and only have to worry about vertical connections 01:22:13.660 --> 01:22:14.990 between the top and the bottom. 01:22:14.990 --> 01:22:16.490 So that gets to your question about, 01:22:16.490 --> 01:22:19.570 why do we need rectilinear? 01:22:19.570 --> 01:22:20.360 But there you go. 01:22:20.360 --> 01:22:25.310 So that was just a few examples of planar 3SAT. 01:22:25.310 --> 01:22:28.240 But in general, the idea is that a lot of 3SAT-- 01:22:28.240 --> 01:22:31.320 or many 3SAT proofs get easier when you use 01:22:31.320 --> 01:22:32.870 the appropriate planar version. 01:22:32.870 --> 01:22:36.290 You just have to be careful of ones that are not hard. 01:22:36.290 --> 01:22:38.130 As long as you avoid that and you 01:22:38.130 --> 01:22:40.210 have the right sort of graph setup, 01:22:40.210 --> 01:22:42.350 you don't have to add too many extra edges. 01:22:42.350 --> 01:22:44.110 Then you'll have planarity and not 01:22:44.110 --> 01:22:45.401 have to worry about crossovers. 01:22:45.401 --> 01:22:47.150 And crossover, it wouldn't have a clue 01:22:47.150 --> 01:22:50.839 how to do a crossover in a problem like this. 01:22:50.839 --> 01:22:53.380 And other times, it just makes your life a little bit easier, 01:22:53.380 --> 01:22:55.550 because you skip one gadget. 01:22:55.550 --> 01:22:58.130 But that is planar 3SAT.