  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.
