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Random optimization problems at fixed temperatures

Partha S. Dey, Grigory Terlov

Abstract

This article considers a class of disordered mean-field combinatorial optimization problems. We focus on the Gibbs measure, where the inverse temperature does not vary with the size of the graph and the edge weights are sampled from a general distribution under mild assumptions. Our results consist of the Law of Large Numbers and Central Limit Theorems for the log-partition function, the weight of a typical configuration, and the Gibbs average in both quenched and annealed forms. We also derive quenched Poisson convergence for the size of the intersection of two independent samples, yielding replica symmetry of the model. Applications cover popular models from the literature, such as the Minimal Matching Problem, Traveling Salesman Problem, and Minimal Spanning Tree Problem, on a sequence of deterministic and random dense block graphs of increasing size.

Random optimization problems at fixed temperatures

Abstract

This article considers a class of disordered mean-field combinatorial optimization problems. We focus on the Gibbs measure, where the inverse temperature does not vary with the size of the graph and the edge weights are sampled from a general distribution under mild assumptions. Our results consist of the Law of Large Numbers and Central Limit Theorems for the log-partition function, the weight of a typical configuration, and the Gibbs average in both quenched and annealed forms. We also derive quenched Poisson convergence for the size of the intersection of two independent samples, yielding replica symmetry of the model. Applications cover popular models from the literature, such as the Minimal Matching Problem, Traveling Salesman Problem, and Minimal Spanning Tree Problem, on a sequence of deterministic and random dense block graphs of increasing size.
Paper Structure (32 sections, 16 theorems, 132 equations)

This paper contains 32 sections, 16 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Road map
  3. Review of random optimization problems
  4. Minimal Matching Problem (MMP)
  5. Traveling Salesman Problem (TSP)
  6. Minimal Spanning Tree Problem (MSTP)
  7. Minimal $k$-factor problem
  8. Highlight of the main results in application to MMP on $K_{n,n}$
  9. Cluster expansion
  10. Setup and Assumptions
  11. Heuristic
  12. Notations
  13. Main results
  14. Application to multipartite graphs
  15. Application to Random graphs
  16. ...and 17 more sections

Key Result

Theorem 1.1

In the set up as above for any fixed $\beta\in (0,\infty)$ the following limits hold as $n\to \infty$,

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3: $\left\vert\pi_1\cap\pi_2\right\vert$ at $\beta=0$
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Theorem 2.2: CLT for the log-partition function
  • Corollary 2.3
  • proof
  • Theorem 2.4: Size of the intersection of two Gibbs samples
  • ...and 21 more