NDLI: Equivalence of LP Relaxation and Max-Product for Weighted Matching in General Graphs

Content Provider	IEEE Xplore Digital Library
Author	Sanghavi, S.
Copyright Year	2007
Description	Author affiliation: Massachusetts Inst. of Technol., Cambridge (Sanghavi, S.)
Abstract	Max-product belief propagation is a local, iterative algorithm find the mode/MAP estimate of a probability distribution. While it has been successfully employed in a wide variety of applications, there are relatively few theoretical guarantees of convergence and correctness for general loopy graphs that may have many short cycles. Of these, even fewer provide exact "necessary and sufficient" characterizations. In this paper we investigate the problem of using max-product to find the maximum weight matching in an arbitrary graph with edge weights. This is done by first constructing a probability distribution whose mode corresponds to the optimal matching, and then running max-product. Weighted matching can also be posed as an integer program, for which there is an LP relaxation. This relaxation is not always tight. In this paper we show that 1) If the LP relaxation is tight, then max-product always converges, and that too to the correct answer. 2) If the LP relaxation is loose, then max-product does not converge. This provides an exact, data-dependent characterization of max-product performance, and a precise connection to LP relaxation, which is a well-studied optimization technique. Also, since LP relaxation is known to be tight for bipartite graphs, our results generalize other recent results on using max-product to find weighted matching in bipartite graphs.
Starting Page	242
Ending Page	247
File Size	244803
Page Count	6
File Format	PDF
ISBN	1424415632
DOI	10.1109/ITW.2007.4313081
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2007-09-02
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Sufficient conditions Lakes Linear programming Iterative algorithms Probability distribution Inference algorithms Bipartite graph Iterative decoding Convergence Belief propagation
Content Type	Text
Resource Type	Article

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