Combinatorial Optimization Over Two Random Point Sets

Let \((\mathcal{X},\mathcal{Y})\) be a pair of random point sets in \({\mathbb{R}}^{d}\) of equal cardinal obtained by sampling independently 2n points from a common probability distribution μ. In this paper, we are interested by functions L of \((\mathcal{X},\mathcal{Y})\) which appear in combinatorial optimization. Typical examples include the minimal length of a matching of \(\mathcal{X}\) and \(\mathcal{Y}\), the length of a traveling salesperson tour constrained to alternate between points of each set, or the minimal length of a connected bipartite r-regular graph with vertex set \((\mathcal{X},\mathcal{Y})\). As the size n of the point sets goes to infinity, we give sufficient conditions on the function L and the probability measure μ which guarantee the convergence of \(L(\mathcal{X},\mathcal{Y})\) under a suitable scaling. In the case of the minimal length matching, we extend results of Dobric and Yukich, and Boutet de Monvel and Martin.