Electronic Journal of Probability

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

Mustafa Khandwawala

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065754

Digital Object Identifier
doi:10.1214/EJP.v19-3491

Mathematical Reviews number (MathSciNet)
MR3296528

Zentralblatt MATH identifier
1321.60013

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

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

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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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