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On the stability of solutions to random optimization problems under small perturbations

Sourav Chatterjee, Souvik Ray

Abstract

Consider the Euclidean traveling salesman problem with $n$ random points on the plane. Suppose that one of the points is shifted to a new random location. This gives us a new optimal path. Consider such shifts for each of the $n$ points. Do we get $n$ very different optimal paths? In this article, we show that this is not the case - in fact, the number of truly different paths can be at most $\mathcal{O}(1)$ as $n\to \infty$. The proof is based on a general argument which allows us to prove similar stability results in a number of other settings, such as branching random walk, the Sherrington-Kirkpatrick model of mean-field spin glasses, the Edwards-Anderson model of short-range spin glasses, and the Wigner ensemble of random matrices.

On the stability of solutions to random optimization problems under small perturbations

Abstract

Consider the Euclidean traveling salesman problem with random points on the plane. Suppose that one of the points is shifted to a new random location. This gives us a new optimal path. Consider such shifts for each of the points. Do we get very different optimal paths? In this article, we show that this is not the case - in fact, the number of truly different paths can be at most as . The proof is based on a general argument which allows us to prove similar stability results in a number of other settings, such as branching random walk, the Sherrington-Kirkpatrick model of mean-field spin glasses, the Edwards-Anderson model of short-range spin glasses, and the Wigner ensemble of random matrices.

Paper Structure

This paper contains 23 sections, 62 theorems, 361 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature review
  4. Notation
  5. Theoretical framework
  6. Main abstract results
  7. General results to prove condition \ref{['item:A']} of Theorem \ref{['strat']}
  8. General results to prove condition \ref{['item:B']} of Theorem \ref{['strat']}
  9. Examples
  10. Branching random walk
  11. The Sherrington--Kirkpatrick model
  12. The Edwards--Anderson model
  13. Graph optimization problems on weighted complete graphs
  14. The random assignment problem
  15. Wigner matrices
  16. ...and 8 more sections

Key Result

Theorem 1.1.1

Fix $d \geq 2$ and take $X_1,\ldots, X_n \stackrel{i.i.d.}{\sim} \mathrm{Uniform}([0,1]^d)$. For each $i \in \{1, \ldots,n\}$, take a copy $X_i^{\prime}$ of $X_i$, independent of $(X_1, \ldots,X_n)$. Define $X_j^{(i)}$ to be equal to $X_i^{\prime}$ if $j=i$ and to be equal to $X_j$ if $j \neq i$. Le where $\|\cdot \|_2$ denotes the $\ell_2$-norm, $\mathcal{G}_n$ is the collection of all Hamiltonia

Theorems & Definitions (152)

  • Theorem 1.1.1
  • Definition 1.4.1
  • Example 1.4.4
  • Remark 1.4.5
  • Definition 1.4.6
  • Definition 1.4.7
  • Theorem 1.4.8
  • proof : Proof of Theorem \ref{['strat']}
  • Remark 1.4.9
  • Remark 1.4.10
  • ...and 142 more