18.997 | Spring 2004 | Graduate
Topics in Combinatorial Optimization


1 Non-Bipartite Matching: Tutte-Berge Formula, Gallai-Edmonds Decomposition, Blossoms
2 Non-Bipartite Matching: Edmonds’ Cardinality Algorithm and Proofs of Tutte-Berge Formulas and Gallai-Edmonds Decomposition
3 Cubic Graphs and Matchings, Factor-Critical Graphs, Ear Decompositions
4 The Matching Polytope, Total Dual Integrality, and Hilbert Bases
5 Total Dual Integrality, Totally Unimodularity

Matching Polytope and the Cunningham-Marsh Formula Showing TDI

6 Posets and Dilworth Theorem

Deduce Konig’s Theorem for Bipartite Matchings

Weighted Posets and the Chain and Antichain Polytopes

7 Partitioning Digraphs by Paths and Covering them by Cycles

Gallai-Milgram and Bessy-Thomasse Theorems

Cyclic Orderings

8 Proof of the Bessy-Thomasse Result

The Cyclic Stable Set Polytope

9 Matroids: Defs, Dual, Minor, Representability
10 Matroids: Representability, Greedy Algorithm, Matroid Polytope
11 Matroid Intersection
12 Matroid Intersection, Matroid Union, Shannon Switching Game
13 Matroid Intersection Polytope, Matroid Union
14 Matroid Union, Packing and Covering with Spanning Trees, Strong Basis Exchange Properties
15 Matroid Matching: Examples, Complexity, Lovasz’s Minmax Relation for Linear Matroids
16 Jump Systems: Definitions, Examples, Operations, Optimization, and Membership
17 Jump Systems: Membership (cont.)
18 Graph Orientations, Directed Cuts (Lucchesi-Younger Theorem), Submodular Flows
19 Submodular Flows: Examples, Edmonds-Giles Theorem, Reduction to Matroid Intersection in Special Cases
20 Splitting Off

$k$-Connectivity Orientations and Augmentations

21 Proof of Splitting-Off

Submodular Function Minimization

22 Multiflow and Disjoint Path Problems

Two-Commodity Flows

23 The Okamura-Seymour Theorem and the Wagner-Weihe Linear-Time Algorithm
Course Info
As Taught In
Spring 2004