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In a complete bipartite graph on $m$ by $n$ vertices, the $m\cdot n$ edges are assigned random costs, and we ask for the minimum total cost of a set of $k$ edges where no two share a vertex. This random model of combinatorial optimisation has been studied extensively as a test case for algorithms and as a toy model of a physical system.
It has been known since the 1990’s that taking the edge-costs to be independent mean 1 exponential permits explicit formulas for things like the expected minimum cost, the most famous case being the $\pi^2/6$ limit for the case $k=m=n$ (perfect matching).
In this talk I will show how the distribution, not just its expectation, can be computed explicitly and efficiently from a certain recursion. The recursion implies a Gaussian limit (after proper rescaling) of the cost of matching when a positive proportion of the vertices remain unmatched, although such a “central limit theorem” for perfect matching remains elusive.
As time permits, I will describe how the results can be generalised to other optimisation problems like minimum 2-factor and minimum cover.
