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Tightness of LP via Max-product Belief Propagation

Sujay Sanghavi·Devavrat Shah
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Abstract

We investigate the question of tightness of linear programming (LP) relaxation for finding a maximum weight independent set (MWIS) in sparse random weighted graphs. We show that an edge-based LP relaxation is asymptotically tight for Erdos-Renyi graph $G(n,c/n)$ for $c \leq 2e$ and random regular graph $G(n,r)$ for $r\leq 4$ when node weights are i.i.d. with exponential distribution of mean 1. We establish these results, through a precise relation between the tightness of LP relaxation and convergence of the max-product belief propagation algorithm. We believe that this novel method of understanding structural properties of combinatorial problems through properties of iterative procedure such as the max-product should be of interest in its own right.

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Sujay Sanghavi, Devavrat Shah. 2008-04-12. Tightness of LP via Max-product Belief Propagation. https://arxiv.org/abs/cs/0508097

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