[Paper Review] Minimum Weight Perfect Matching via Blossom Belief Propagation

Sungsoo Ahn, Sejun Park+2 more|arXiv (Cornell University)|Sep 23, 2015

Error Correcting Code Techniques26 references3 citations

TL;DR

This paper introduces Blossom-BP, a novel distributed algorithm that solves the minimum weight perfect matching problem on arbitrary graphs by sequentially applying belief propagation (BP) over modified graphs through blossom contractions and expansions. It guarantees convergence in O(n²) BP runs and provides the first systematic BP-based solution for problems with integrality gaps, offering a distributed interpretation of Edmonds' blossom algorithm.

ABSTRACT

Max-product Belief Propagation (BP) is a popular message-passing algorithm for computing a Maximum-A-Posteriori (MAP) assignment over a distribution represented by a Graphical Model (GM). It has been shown that BP can solve a number of combinatorial optimization problems including minimum weight matching, shortest path, network flow and vertex cover under the following common assumption: the respective Linear Programming (LP) relaxation is tight, i.e., no integrality gap is present. However, when LP shows an integrality gap, no model has been known which can be solved systematically via sequential applications of BP. In this paper, we develop the first such algorithm, coined Blossom-BP, for solving the minimum weight matching problem over arbitrary graphs. Each step of the sequential algorithm requires applying BP over a modified graph constructed by contractions and expansions of blossoms, i.e., odd sets of vertices. Our scheme guarantees termination in O(n^2) of BP runs, where n is the number of vertices in the original graph. In essence, the Blossom-BP offers a distributed version of the celebrated Edmonds' Blossom algorithm by jumping at once over many sub-steps with a single BP. Moreover, our result provides an interpretation of the Edmonds' algorithm as a sequence of LPs.

Motivation & Objective

• To develop a systematic, distributed message-passing algorithm for solving minimum weight perfect matching in graphs where LP relaxations exhibit integrality gaps.
• To extend belief propagation (BP) beyond tree-structured models to combinatorial optimization problems with non-tight LP relaxations.
• To provide a BP-based counterpart to Edmonds' classic blossom algorithm, enabling parallel and distributed computation.
• To establish a polynomial-time guarantee for BP-based solvers in integer programming by introducing intermediate LPs and BP applications.
• To offer a new interpretation of Edmonds' algorithm as a sequence of tractable linear programs, enhancing theoretical and practical understanding.

Proposed method

• The algorithm sequentially applies BP over a series of modified graphs formed by contracting and occasionally expanding blossoms—odd sets of vertices that violate LP tightness.
• It introduces Blossom-LP, a sequence of intermediate LPs that maintain equivalence to the Blossom-V algorithm but are designed to be solvable via BP using a recent LP-BP correspondence framework.
• Each BP run operates on a contracted graph where vertices represent blossoms, and edge weights are adjusted using dual variables to reflect reduced costs.
• The method uses cycle-claw-matching decompositions to track structural components of the graph and ensure correctness at each iteration.
• Complementary slackness conditions are enforced between primal (matching) and dual (variables) solutions to verify optimality at each step.
• The algorithm terminates in O(n²) BP iterations by mimicking the step-by-step structure of the Blossom-V algorithm, but with BP replacing sequential tree updates.

Experimental results

Research questions

• RQ1 Can belief propagation be systematically applied to solve integer programming problems when the LP relaxation has an integrality gap?
• RQ2 Is there a way to simulate the classical Edmonds' blossom algorithm using a sequence of BP runs on modified graphs?
• RQ3 Can intermediate LPs be designed such that they are amenable to BP solution while preserving equivalence to the original optimization problem?
• RQ4 Does a distributed, message-passing algorithm exist that replicates the behavior of the deterministic Blossom-V algorithm with polynomial complexity?
• RQ5 Can the Edmonds' algorithm be interpreted as a sequence of tractable linear programs, each solvable via BP?

Key findings

• Blossom-BP guarantees termination in O(n²) belief propagation runs, where n is the number of vertices in the original graph.
• The algorithm achieves correctness by maintaining equivalence to a variant of the Blossom-V algorithm, ensuring optimal solutions are found.
• The method provides a distributed implementation of Edmonds' blossom algorithm by replacing sequential tree updates with BP over contracted graphs.
• Blossom-LP is proven to be equivalent to the intermediate steps of Blossom-V, with each LP solvable via BP using the framework of [16].
• The cycle-claw-matching decomposition in the algorithm corresponds exactly to half-integral solutions of the LP relaxation, ensuring complementary slackness and optimality.
• The approach establishes the first known polynomial-time BP-based solver for integer programs with integrality gaps, extending BP’s applicability beyond tight LP relaxations.