Optimal transport methods for combinatorial optimization over two random point sets

IF 1.5 1区数学 Q2 STATISTICS & PROBABILITY

Probability Theory and Related Fields Pub Date : 2023-11-07 DOI:10.1007/s00440-023-01245-1

Michael Goldman, Dario Trevisan

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引用次数: 2

Abstract

Abstract We investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in $$\mathbb {R}^d$$ R d where the edge cost between two points is given by a p th power of their Euclidean distance. This includes e.g. the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as n grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and $$d \ge 3, 1\le p d ≥ 3 , 1 ≤ p < d . Previous results were limited to the range $$p p < d / 2 . Our proofs are based on subadditivity methods and build upon new bounds for random instances of the Euclidean bipartite matching problem, obtained through its optimal transport relaxation and functional analytic techniques.

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两个随机点集组合优化的最优传输方法

研究了在$$\mathbb {R}^d$$ R d中随机二部几何图上的一类组合优化问题的最小代价，其中两点之间的边代价由它们的欧几里得距离的p次幂给出。这包括旅行销售问题和有界度最小生成树。当点均匀分布且$$d \ge 3, 1\le p<d$$ d≥3,1≤p ＆lt;D。先前的结果仅限于$$p<d/2$$ p ＆lt;D / 2。我们的证明基于子可加性方法，并建立在欧几里得二部匹配问题随机实例的新边界上，通过其最优传输松弛和泛函分析技术获得。

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来源期刊

Probability Theory and Related Fields 数学-统计学与概率论

CiteScore

3.70

自引率

5.00%

发文量

审稿时长

6-12 weeks

期刊介绍： Probability Theory and Related Fields publishes research papers in modern probability theory and its various fields of application. Thus, subjects of interest include: mathematical statistical physics, mathematical statistics, mathematical biology, theoretical computer science, and applications of probability theory to other areas of mathematics such as combinatorics, analysis, ergodic theory and geometry. Survey papers on emerging areas of importance may be considered for publication. The main languages of publication are English, French and German.