1. Minimum Weight Perfect Matching via Blossom Belief Propagation
Sungsoo Ahn (speaker)1, Sejun Park1, Michael Chertkov2, Jinwoo Shin1
1Korea Advanced Institute of Science and Technology, 2Los Alamos National Laboratory
Minimum Weight Perfect Matching (MWPM) Problem:
Given a graph 𝐺= 𝑉,𝐸 with weight function, find the MWPM.
Our Goal
Use Belief Propagation (BP) in order to develop a simple, fast and distributable algorithm for the MWPM problem.
Past works show that BP converge to the solution of the Linear Programming (LP) relaxations of several combinatorial optimizations.
e.g., maximum weight matching [Sharma et al. 2005], [Willsky et al. 2007], [Jebra et al 2007], perfect matching [Zecchina et al. 2011], shortest path [Tatikonda et al. 2008], independent set [Willsky et al. 2007], network flow [Wei et al. 2010] and vertex cover [Shin et al. 2015]
However, past works are not applicable to the case when LPs are not tight.

2. The LP relaxation of the MWPM problem is not tight!
Dual
The famous Edmond’s Blossom algorithm uses the primal-dual method to solve the MWPM problem in 𝑂( 𝑉 2 𝐸 ) running time.
Our Approach
Design a sequence of LPs where each intermediate LP is solvable via BP.
There exist past attempts to solve the MWPM problem via LP.
[Trick. 1978], [Padberg, Rao. 1982], [Grotschel, Holland. 1978] and [Fischetti, Lodi. 1978] attempts to solve MWPM via LP.
Recent work of [Vempala et al. 2012] provides a polynomial-time cutting-plane scheme with sequence of intermediate LPs. (However the LPs are quite complex to be solved via BP)
We develop such an algorithm, coined Blossom-LP.

3. Our Contribution Some notes on Blossom-LP:
Blossom-LP is an iteratively primal-dual algorithm iteratively solving LPs (cutting-plane-like method) in order to output the exact solution of the MWPM problem.
We prove that the intermediate LPs are solvable in 𝑂(|𝑉|) iterations by BP using the work of [Park, Shin. 2015].
We prove that it is implicitly equivalent to a variant of Edmond’s Blossom algorithm.
Our Contribution
The Blossom-BP solves the MWPM problem in 𝑂( 𝑉 2 ) BP runs.
In simulation, usually takes 2 ~ 3 BP runs.
Each BP take 𝑂(|𝑉|) iterations to converge.
We show that the Blossom-BP offers a simple, distributed version of Edmond’s Blossom algorithm.
The Blossom-BP jumps over many sub-steps of the Edmond’s algorithm with a single BP.
Blossom-BP is the first BP-based algorithm to solve a class of Integer Programming.
1 iteration of Blossom-BP + - 1
Edmond’s Blossom Algorithm: