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.
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Acknowledgements
We are indebted to Martin Huesmann for pointing an error in the proof of a previous version of Theorem 2. This is also a pleasure to thank for its hospitality the Newton Institute where part of this work has been done (2011 Discrete Analysis program).
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Barthe, F., Bordenave, C. (2013). Combinatorial Optimization Over Two Random Point Sets. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_19
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