Random Combinatorial Optimization Problems: Mean Field and Finite-Dimensional Results

TL;DR

This thesis explores statistical physics methods applied to various random combinatorial optimization problems, analyzing mean-field and finite-dimensional cases, with results on finite-size corrections and solutions to TSP variants.

Contribution

It provides new insights into finite-size effects in random matching problems and solutions to Euclidean TSP in one and two dimensions, connecting physics concepts with optimization.

Findings

Finite-size corrections characterized for random integer and fractional matching problems.

Solutions and analysis of the one-dimensional bipartite and monopartite TSP.

Large-scale bipartite TSP solutions in two dimensions.

Abstract

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.