Combinatorial Optimization Over Two Random Point Sets

Abstract

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 Dobrić and Yukich, and Boutet de Monvel and Martin.