Message-passing for Maximum Weight Independent Set
Sujay Sanghavi, Devavrat Shah, Alan Willsky

TL;DR
This paper explores message-passing algorithms for solving the maximum weight independent set problem, establishing conditions for their correctness, and connecting them to LP relaxations and dual optimization, with implications for MAP estimation.
Contribution
It demonstrates that max-product message passing can solve LP relaxations of MWIS, introduces a gradient descent approach on the dual, and reduces MAP estimation to MWIS.
Findings
Max-product fixed points relate to LP extreme points.
Starting max-product from uninformative messages solves the LP if it converges.
The combined algorithms correctly solve MWIS on bipartite graphs with unique solutions.
Abstract
We investigate the use of message-passing algorithms for the problem of finding the max-weight independent set (MWIS) in a graph. First, we study the performance of the classical loopy max-product belief propagation. We show that each fixed point estimate of max-product can be mapped in a natural way to an extreme point of the LP polytope associated with the MWIS problem. However, this extreme point may not be the one that maximizes the value of node weights; the particular extreme point at final convergence depends on the initialization of max-product. We then show that if max-product is started from the natural initialization of uninformative messages, it always solves the correct LP -- if it converges. This result is obtained via a direct analysis of the iterative algorithm, and cannot be obtained by looking only at fixed points. The tightness of the LP relaxation is thus necessary…
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