Minimum Weight Perfect Matching via Blossom Belief

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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

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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: 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.<br>
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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. The LP relaxation of the MWPM problem is not tight! Dual<br>