Year: 2009
Type: article
Abstract: In this paper, 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 linear programming (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 for max-product optimality, but it is not sufficient. Motivated by this observation, we show that a simple modification of max product becomes gradient descent on (a smoothed version of) the dual of the LP, and converges to the dual optimum. We also develop a message-passing algorithm that recovers the primal MWIS solution from the output of the descent algorithm. We show that the MWIS estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the MWIS is unique. Finally, we show that any problem of maximuma posteriori(MAP) estimation for probability distributions over finite domains can be reduced to an MWIS problem. We believe this reduction will yield new insights and algorithms for MAP estimation.
Authors: Sujay Sanghavi, Devavrat Shah, Alan S. Willsky
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: Collaborative Research: Flow Level Models and the Design of Flow-aware Networks, G3671960635, G414638342, HSD: An Analytic Framework for Political and Social Change: Conflict, Beliefs, and Dynamics, CAREER: Implementable Network Algorithms via Randomization, Belief Propagation and Heavy Traffic, G7090116110, G7452299184
OpenAlex ID: W2117038364