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2010 Edge cover and polymatroid flow problems
Martin Hessler, Johan Wästlund
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Electron. J. Probab. 15: 2200-2219 (2010). DOI: 10.1214/EJP.v15-846

Abstract

In an $n$ by $n$ complete bipartite graph with independent exponentially distributed edge costs, we ask for the minimum total cost of a set of edges of which each vertex is incident to at least one. This so-called minimum edge cover problem is a relaxation of perfect matching. We show that the large $n$ limit cost of the minimum edge cover is $W(1)^2+2W(1)\approx 1.456$, where $W$ is the Lambert $W$-function. In particular this means that the minimum edge cover is essentially cheaper than the minimum perfect matching, whose limit cost is $\pi^2/6\approx 1.645$. We obtain this result through a generalization of the perfect matching problem to a setting where we impose a (poly-)matroid structure on the two vertex-sets of the graph, and ask for an edge set of prescribed size connecting independent sets.

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Martin Hessler. Johan Wästlund. "Edge cover and polymatroid flow problems." Electron. J. Probab. 15 2200 - 2219, 2010. https://doi.org/10.1214/EJP.v15-846

Information

Accepted: 18 December 2010; Published: 2010
First available in Project Euclid: 1 June 2016

zbMATH: 1226.90124
MathSciNet: MR2748403
Digital Object Identifier: 10.1214/EJP.v15-846

Subjects:
Primary: 60C05
Secondary: 90C27, 90C35

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Vol.15 • 2010
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