Message passing for the coloring problem: Gallager meets Alon and Kahale

TL;DR

This paper connects message passing algorithms used in graph coloring and decoding, providing theoretical insights and demonstrating the effectiveness of message passing in solving the coloring problem.

Contribution

It interprets Alon and Kahale's coloring algorithm through Gallager's decoding framework, establishing a link between coloring and decoding problems.

Findings

Connection established between coloring and decoding algorithms

Rigorous evidence supporting message passing in graph coloring

Techniques applicable to other combinatorial optimization problems

Abstract

Message passing algorithms are popular in many combinatorial optimization problems. For example, experimental results show that {\em survey propagation} (a certain message passing algorithm) is effective in finding proper $k$-colorings of random graphs in the near-threshold regime. In 1962 Gallager introduced the concept of Low Density Parity Check (LDPC) codes, and suggested a simple decoding algorithm based on message passing. In 1994 Alon and Kahale exhibited a coloring algorithm and proved its usefulness for finding a $k$-coloring of graphs drawn from a certain planted-solution distribution over $k$-colorable graphs. In this work we show an interpretation of Alon and Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus showing a connection between the two problems - coloring and decoding. This also provides a rigorous evidence for the usefulness of the message passing paradigm for the graph coloring problem. Our techniques can be applied to several other combinatorial optimization problems and networking-related issues.