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

Message passing algorithms are popular in many combinatorial optimization problems. Forexample, experimental results show that survey propagation (a certain message passing algorithm)is effective in finding proper k-colorings of random graphs in the near-threshold regime. Gallagerin 1962 introduced the concept of Low Density Parity Check (LDPC) codes, and suggested asimple decoding algorithm based on message passing. In 1994 Alon and Kahale exhibited acoloring algorithm and proved its usefulness for finding a k-coloring of graphs drawn from a certainplanted-solution distribution over k-colorable graphs. In this work we show an interpretation ofAlon and Kahale’s coloring algorithm in light of Gallager’s decoding algorithm, thus showinga connection between the two problems coloring and decoding. This also provides a rigorousevidence for the usefulness of the message passing paradigm for the graph coloring problem.