Open Access
2014 Belief propagation for minimum weight many-to-one matchings in the random complete graph
Mustafa Khandwawala
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Electron. J. Probab. 19: 1-40 (2014). DOI: 10.1214/EJP.v19-3491


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.


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Mustafa Khandwawala. "Belief propagation for minimum weight many-to-one matchings in the random complete graph." Electron. J. Probab. 19 1 - 40, 2014.


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

zbMATH: 1321.60013
MathSciNet: MR3296528
Digital Object Identifier: 10.1214/EJP.v19-3491

Primary: 60C05
Secondary: 68Q87

Keywords: belief propagation , Local weak convergence , many-to-one matching , objective method , random graph

Vol.19 • 2014
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