Belief Propagation Min-Sum Algorithm for Generalized Min-Cost Network Flow

TL;DR

This paper proves that the Belief Propagation Min-Sum algorithm converges to the exact solution for a generalized min-cost network flow problem with relaxed constraints, extending theoretical guarantees beyond traditional cases.

Contribution

It generalizes the Min-Sum Network Flow formulation by relaxing flow conservation constraints and proves convergence of Belief Propagation in this broader setting.

Findings

Belief Propagation converges to the exact solution in the generalized setting.

The paper extends theoretical guarantees for Belief Propagation beyond traditional cases.

It relaxes flow conservation constraints in the network flow problem.

Abstract

Belief Propagation algorithms are instruments used broadly to solve graphical model optimization and statistical inference problems. In the general case of a loopy Graphical Model, Belief Propagation is a heuristic which is quite successful in practice, even though its empirical success, typically, lacks theoretical guarantees. This paper extends the short list of special cases where correctness and/or convergence of a Belief Propagation algorithm is proven. We generalize formulation of Min-Sum Network Flow problem by relaxing the flow conservation (balance) constraints and then proving that the Belief Propagation algorithm converges to the exact result.