Electronic Journal of Probability

Belief propagation for minimum weight many-to-one matchings in the random complete graph

Mustafa Khandwawala

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In a complete bipartite graph with vertex sets of cardinalities $n$ and $n^\prime$, assign random weights from exponential distribution with mean 1, independently to each edge. We show that, as $n\rightarrow\infty$, with $n^\prime=\lceil n/\alpha\rceil$ for any fixed $\alpha>1$, the minimum weight of many-to-one matchings converges to a constant (depending on $\alpha$). Many-to-one matching arises as an optimization step in an algorithm for genome sequencing and as a measure of distance between finite sets. We prove that a belief propagation (BP) algorithm converges asymptotically to the optimal solution. We use the objective method of Aldous to prove our results. We build on previous works on minimum weight matching and minimum weight edge cover problems to extend the objective method and to further the applicability of belief propagation to random combinatorial optimization problems.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 112, 40 pp.

Accepted: 11 December 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) [See also 68W20, 68W40]

belief propagation local weak convergence many-to-one matching objective method random graph

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Khandwawala, Mustafa. Belief propagation for minimum weight many-to-one matchings in the random complete graph. Electron. J. Probab. 19 (2014), paper no. 112, 40 pp. doi:10.1214/EJP.v19-3491. https://projecteuclid.org/euclid.ejp/1465065754

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