Analysis of the Min-Sum Algorithm for Packing and Covering Problems via Linear Programming

TL;DR

This paper analyzes the min-sum message-passing algorithm's effectiveness for packing and covering problems modeled as linear programs, revealing conditions under which it computes optimal solutions and extending previous results.

Contribution

It provides theoretical insights into when the min-sum algorithm finds optimal solutions for packing and covering LPs, especially regarding fractional solutions and boundary conditions.

Findings

Min-sum algorithm matches LP solutions only with unique integral optima.

If LP has fractional solutions, min-sum oscillates or finds multiple solutions.

For certain boundary cases, min-sum computes solutions in pseudo-polynomial time.

Abstract

Message-passing algorithms based on belief-propagation (BP) are successfully used in many applications including decoding error correcting codes and solving constraint satisfaction and inference problems. BP-based algorithms operate over graph representations, called factor graphs, that are used to model the input. Although in many cases BP-based algorithms exhibit impressive empirical results, not much has been proved when the factor graphs have cycles. This work deals with packing and covering integer programs in which the constraint matrix is zero-one, the constraint vector is integral, and the variables are subject to box constraints. We study the performance of the min-sum algorithm when applied to the corresponding factor graph models of packing and covering LPs. We compare the solutions computed by the min-sum algorithm for packing and covering problems to the optimal solutions of the corresponding linear programming (LP) relaxations. In particular, we prove that if the LP has an optimal fractional solution, then for each fractional component, the min-sum algorithm either computes multiple solutions or the solution oscillates below and above the fraction. This implies that the min-sum algorithm computes the optimal integral solution only if the LP has a unique optimal solution that is integral. The converse is not true in general. For a special case of packing and covering problems, we prove that if the LP has a unique optimal solution that is integral and on the boundary of the box constraints, then the min-sum algorithm computes the optimal solution in pseudo-polynomial time. Our results unify and extend recent results for the maximum weight matching problem by [Sanghavi et al.,'2011] and [Bayati et al., 2011] and for the maximum weight independent set problem [Sanghavi et al.'2009].