随机组合优化问题:平均场和有限维结果

E. Malatesta
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引用次数: 3

摘要

本博士论文组织如下。在前两章中,我将回顾一些无序系统统计物理的基本概念,如随机图论、平均场近似、自旋玻璃和组合优化。本文还介绍了复制方法,并将其应用于自旋玻璃最简单的平均场模型之一Sherrington-Kirkpatrick模型。论文的第二部分讨论了平均场组合优化问题。注意力将集中在随机整数匹配问题(第3章)和分数匹配问题(第4章)的有限大小修正的研究上。在第5章中,我将讨论连接多重重叠和腔磁化分布矩的非常一般的关系。在第三部分,我们考虑随机欧几里得优化问题。我将开始在一维中解决旅行推销员问题(TSP)的二部和单部版本(第6章)。在第7章中,我将讨论两因素问题的可能的最优解。在第8章中,我将在大量点的限制下求解二维的二部TSP问题。第九章是一些结论。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
RANDOM COMBINATORIAL OPTIMIZATION PROBLEMS: MEAN FIELD AND FINITE-DIMENSIONAL RESULTS
This PhD thesis is organized as follows. In the first two chapters I will review some basic notions of statistical physics of disordered systems, such as random graph theory, the mean-field approximation, spin glasses and combinatorial optimization. The replica method will also be introduced and applied to the Sherrington-Kirkpatrick model, one of the simplest mean-field models of spin-glasses. The second part of the thesis deals with mean-field combinatorial optimization problems. The attention will be focused on the study of finite-size corrections of random integer matching problems (chapter 3) and fractional ones (chapter 4). In chapter 5 I will discuss a very general relation connecting multi-overlaps and the moments of the cavity magnetization distribution. In the third part we consider random Euclidean optimization problems. I will start solving the traveling-salesman-problem (TSP) in one dimension both in its bipartite and monopartite version (chapter 6). In chapter 7 I will discuss the possible optimal solutions of the 2-factor problem. In chapter 8 I will solve the bipartite TSP in two dimensions, in the limit of large number of points. Chapter 9 contains some conclusions.
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