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Solving a Random Asymmetric TSP Exactly in Quasi-Polynomial Time w.h.p

Tolson Bell, Alan Frieze

TL;DR

The paper addresses exact solvability of the asymmetric TSP when edge costs are i.i.d. from a broad class of continuous distributions, establishing an algorithm that runs in $e^{\log^{2+o(1)}n}$ time with high probability. It connects the Assignment Problem to ATSP via LP duality and alternating cycles, and leverages concentration inequalities to bound reduced costs and dual variables, thereby restricting the ATSP solution to a small set of non-basic edges. The main contribution is a quasi-polynomial-time algorithm that enumerates all plausible ATSP augmentations of the AP solution, using structural results on alternating paths and spanning-tree bases, plus reductions to general cost distributions through coupling. The work also outlines implications for branch-and-bound strategies and poses open questions regarding polynomial-time w.h.p. solvability and extensions to the symmetric TSP.

Abstract

Let the costs $C(i,j)$ for an instance of the Asymmetric Traveling Salesperson Problem (ATSP) be independent copies of a non-negative random variable $C$ from a class of distributions that include the uniform $[0,1]$ distribution and the exponential mean 1 distribution with mean 1. We describe an algorithm that solves ATSP exactly in time $e^{\log^{2+o(1)}n}$, w.h.p.

Solving a Random Asymmetric TSP Exactly in Quasi-Polynomial Time w.h.p

TL;DR

The paper addresses exact solvability of the asymmetric TSP when edge costs are i.i.d. from a broad class of continuous distributions, establishing an algorithm that runs in time with high probability. It connects the Assignment Problem to ATSP via LP duality and alternating cycles, and leverages concentration inequalities to bound reduced costs and dual variables, thereby restricting the ATSP solution to a small set of non-basic edges. The main contribution is a quasi-polynomial-time algorithm that enumerates all plausible ATSP augmentations of the AP solution, using structural results on alternating paths and spanning-tree bases, plus reductions to general cost distributions through coupling. The work also outlines implications for branch-and-bound strategies and poses open questions regarding polynomial-time w.h.p. solvability and extensions to the symmetric TSP.

Abstract

Let the costs for an instance of the Asymmetric Traveling Salesperson Problem (ATSP) be independent copies of a non-negative random variable from a class of distributions that include the uniform distribution and the exponential mean 1 distribution with mean 1. We describe an algorithm that solves ATSP exactly in time , w.h.p.
Paper Structure (18 sections, 22 theorems, 56 equations)

This paper contains 18 sections, 22 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Chernoff bounds
  4. The Assignment Problem and Nearby Permutations
  5. AP as a linear program
  6. Outline Proof of Theorem \ref{['th1']}
  7. Trees and bases
  8. Alternating paths
  9. Finishing the proof of Theorem \ref{['th1']}
  10. Specifying a possible ATSP
  11. Iterating through Possible ATSP Solutions
  12. Reducing the Space Complexity
  13. Properties of the assignment problem
  14. $M^*$ only has low cost edges
  15. From exponential mean one to more general distributions
  16. ...and 3 more sections

Key Result

Theorem 1

Let the costs for ATSP, $(C(i,j))$, be independent copies of $C$, where $C$ has a distribution satisfying (i),(ii),(iii) above. There is an algorithm that solves ATSP exactly in $e^{\log^{2+o(1)}n}$ time, with high probability over the choices of random costs.

Theorems & Definitions (23)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Remark 4
  • Proposition 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Corollary 10
  • ...and 13 more