Lecture Notes

A. Second Moment Method: The second moment method is a simple probabilistic tool to establish existence of non-rare events in a complex setup.
1 A model of random k-SAT (random instance of a boolean constraint satisfaction problem) is considered where the size k of each clause is growing as function of the number of variables n. A threshold for satisfiability is obtained using the second moment method.
2 The authors consider the random sparse regular graph model and the problem of coloring such graphs. For each graph q they obtain the largest connectivity parameter r = r(q) such that the graph is q colorable. They identify r up to two possible values.
B. Local Weak Convergence and Correlation Decay: The notion of local weak convergence allows one to study asymptotic spatial structural properties of probabilistic combinatorial optimization problems. The property of correlation decay facilitates one to establish these properties.
3 A queuing model is considered where the customers access database over time. A certain phase transition result is established depending on the arrival rate of the customer requests. This result is related to certain statistical physics models, specifically in the context of hard-core model on Bethe lattices.
4 A new method for solving approximately a class of counting problems is introduced. The method is based on correlation decay property, originating from statistical physics. The method is applied to approximately count the number of independent sets in arbitrary graphs with degree at most five.
5 A survey of local weak convergence method is discussed with applications to random matching, spanning trees and various other related models.
6 A local weak convergence method is used to compute the expected size of the largest weighted independent set and matching on regular graphs with large girth.
C. Belief Propagation and LDPC Codes: Belief propagation is an iterative distributed heuristic algorithm for combinatorial optimization problem. It is expected to perform very well in the context of probabilistic setup.
7-8 A framework for using the belief propagation algorithm is discussed for several applications including Markov random fields and Bayesian (belief) networks.
9 Belief propagation algorithm is proven to solve the problem of finding the largest weighted matching of a bi-partite graph.
10 First two chapters of an important textbook on a coding theory are discussed. Topics include linear codes, ML decoding, Channel coding theory, factor graph representation, decoding via message passing.