Abstract
We study the asymptotic behaviour of the expected cost of the random matching problem on a $2$-dimensional compact manifold, improving in several aspects the results of L. Ambrosio, F. Stra and D. Trevisan (A PDE approach to a 2-dimensional matching problem). In particular, we simplify the original proof (by treating at the same time upper and lower bounds) and we obtain the coefficient of the leading term of the asymptotic expansion of the expected cost for the random bipartite matching on a general 2-dimensional closed manifold. We also sharpen the estimate of the error term given by M. Ledoux (On optimal matching of Gaussian samples II) for the semi-discrete matching. As a technical tool, we develop a refined contractivity estimate for the heat flow on random data that might be of independent interest.
Highlights
The bipartite matching problem is a very classical problem in computer science
Given two families (Xi)[1] i n and (Yi)[1] i n of independent random points m-uniformly distributed on M, study the value of the expected matching cost min σ∈Sn n n dp
Possibly replacing the empirical measures with a Poisson point process, the semi-discrete matching problem can be connected to the Lebesgue-to-Poisson transport problem of [HS13], see [GHO18] where the large scale behaviour of the optimal maps is deeply analyzed
Summary
The bipartite matching problem is a very classical problem in computer science. It asks to find, among all possible matching in a bipartite weighted graph, the one that minimizes the sum of the costs of the chosen edges. Given two families (Xi)[1] i n and (Yi)[1] i n of independent random points m-uniformly distributed on M , study the value of the expected matching cost. Possibly replacing the empirical measures with a Poisson point process, the semi-discrete matching problem can be connected to the Lebesgue-to-Poisson transport problem of [HS13], see [GHO18] where the large scale behaviour of the optimal maps is deeply analyzed. From the proof contained in [AST18], we do not need the duality theory of optimal transport as it is completely encoded in the mentioned theorem stating that a small map is optimal In this way, we do not need to manage the upper-bound and the lower-bound of the expected cost separately. Trevisan for useful comments, during the development of the paper, and the paper’s reviewers for their constructive and detailed observations
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