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Edge cover and polymatroid flow problems. (English) Zbl 1226.90124
Summary: 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.

90C35 Programming involving graphs or networks
90C27 Combinatorial optimization
60C05 Combinatorial probability
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