Edge cover and polymatroid flow problems
Journal article, 2010

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) approximate to 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 approximate to 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.

Random graphs

Combinatorial optimization

Author

Martin Hessler

Johan Wästlund

University of Gothenburg

Chalmers, Mathematical Sciences, Mathematics

Electronic Journal of Probability

1083-6489 (ISSN)

Vol. 15 2200-2219

Subject Categories

Probability Theory and Statistics

More information

Created

10/6/2017