research-article

Characterizing the Integrality Gap of the Subtour LP for the Circulant Traveling Salesman Problem

Authors:

Samuel C. Gutekunst,

David P. WilliamsonAuthors Info & Claims

SIAM Journal on Discrete Mathematics, Volume 33, Issue 4

Pages 2452 - 2478

https://doi.org/10.1137/19M1245840

Published: 01 January 2019 Publication History

Abstract

We consider the integrality gap of the subtour linear program (LP) relaxation of the traveling salesman problem (TSP) restricted to circulant instances. De Klerk and Dobre [Discrete Appl. Math., 159 (2011), pp. 1815--1826] conjectured that the value of the optimal solution to the subtour LP in these instances is equal to an entirely combinatorial lower bound from Van der Veen, Van Dal, and Sierksma [The Symmetric Circulant Traveling Salesman Problem, Research Memorandum 429, Institute of Economic Research, University of Groningen, 1991]. We prove this conjecture by giving an explicit optimal solution to the subtour LP. We then show that the integrality gap of the subtour LP is 2 on circulant instances, making such instances one of the few nontrivial classes of TSP instances for which the integrality gap of the subtour LP is exactly known. We also show that the degree constraints do not strengthen the subtour LP on circulant instances, mimicking the parsimonious property of metric, symmetric TSP instances shown in Goemans and Bertsimas [Math. Programming, 60 (1993), pp. 145--166] in a distinctly nonmetric set of instances.

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Cited By

Gutekunst SWilliamson D(2023)The Circlet InequalitiesMathematics of Operations Research10.1287/moor.2022.126548:1(393-418)Online publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1287/moor.2022.1265
Gutekunst SJin BWilliamson D(2022)The Two-Stripe Symmetric Circulant TSP is in PInteger Programming and Combinatorial Optimization10.1007/978-3-031-06901-7_24(319-332)Online publication date: 27-Jun-2022
https://dl.acm.org/doi/10.1007/978-3-031-06901-7_24

Index Terms

Characterizing the Integrality Gap of the Subtour LP for the Circulant Traveling Salesman Problem
1. Theory of computation
  1. Design and analysis of algorithms

Index terms have been assigned to the content through auto-classification.

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cover image SIAM Journal on Discrete Mathematics

SIAM Journal on Discrete Mathematics Volume 33, Issue 4

2019

710 pages

ISSN:0895-4801

DOI:10.1137/sjdmec.33.4

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© 2019, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

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Published: 01 January 2019

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Cited By

Gutekunst SWilliamson D(2023)The Circlet InequalitiesMathematics of Operations Research10.1287/moor.2022.126548:1(393-418)Online publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1287/moor.2022.1265
Gutekunst SJin BWilliamson D(2022)The Two-Stripe Symmetric Circulant TSP is in PInteger Programming and Combinatorial Optimization10.1007/978-3-031-06901-7_24(319-332)Online publication date: 27-Jun-2022
https://dl.acm.org/doi/10.1007/978-3-031-06901-7_24

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