Abstract
AbstractIt is known that for Kn,n equipped with i.i.d. exp (1) edge costs, the minimum total cost of a perfect matching converges to $\zeta(2)=\pi^2/6$ in probability. Similar convergence has been established for all edge cost distributions of pseudo-dimension $q \geq 1$ . In this paper we extend those results to all real positive q, confirming the Mézard–Parisi conjecture in the last remaining applicable case.
Highlights
There has been substantial interest over the past few decades in the minimum matching problem: given a graph G, and a positive cost associated to each edge of G, we want to find a perfect matching of minimal total cost M(G)
Of special interest is minimum matching on the complete graph Kn on n vertices or the complete bipartite graph Kn,n on n + n vertices, with random edge costs given by independent exp (1) variables
A natural question is whether these results extend to other edge cost distributions
Summary
There has been substantial interest over the past few decades in the minimum matching problem: given a graph G, and a positive cost (or weight) associated to each edge of G, we want to find a perfect matching of minimal total cost M(G). Of special interest is minimum matching on the complete graph Kn on n vertices or the complete bipartite graph Kn,n on n + n vertices, with random edge costs given by independent exp (1) variables. The latter is sometimes referred to as the random assignment problem. Aldous [1] proved that the limit exists, and later confirmed the conjecture [2] Both of these papers used what is sometimes called the ‘objective method’ [3], and worked with matchings on an infinite limit object.
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