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.
Highlights
In the current paper we investigate the minimum edge cover problem
The minimum edge cover problem asks for the edge cover of minimum total cost
For n ≥ 3, it may happen that an edge cover with more than n edges is cheaper than the minimum perfect matching
Summary
For n ≥ 3, it may happen that an edge cover with more than n edges is cheaper than the minimum perfect matching Such situations clearly have positive probability, and the expected cost of the minimum edge cover is strictly smaller than (1) as soon as n ≥ 3. C would be the average vertex degree in the minimum edge cover, in other words the average number of edges incident to a particular vertex We will answer these questions by establishing the following two theorems. In particular it turns out that the edge cover problem with specified number of edges belongs to this class of problems This allows us in Section to prove Theorems 1.1 and 1.2, roughly speaking by optimizing the number of edges to minimize the limit cost.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
