research-article

2-Matchings, the Traveling Salesman Problem, and the Subtour LP: : A Proof of the Boyd-Carr Conjecture

Authors: Frans Schalekamp, David P. Williamson, Anke van ZuylenAuthors Info & Claims

Mathematics of Operations Research, Volume 39, Issue 2

Pages 403 - 417

https://doi.org/10.1287/moor.2013.0608

Published: 01 May 2014 Publication History

Abstract

Determining the precise integrality gap for the subtour linear programming (LP) relaxation of the traveling salesman problem is a significant open question, with little progress made in thirty years in the general case of symmetric costs that obey triangle inequality. Boyd and Carr [Boyd S, Carr R (2011) Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices. Discrete Optim. 8:525–539. Prior version accessed June 27, 2011, http://www.site.uottawa.ca/~sylvia/recentpapers/halftri.pdf.] observe that we do not even know the worst-case upper bound on the ratio of the optimal 2-matching to the subtour LP; they conjecture the ratio is at most 10/9.

In this paper, we prove the Boyd-Carr conjecture. In the case that the support of a fractional 2-matching has no cut edge, we can further prove that an optimal 2-matching has cost at most 10/9 times the cost of the fractional 2-matching.

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Cited By

Gupta ALee ELi JMucha MNewman HSarkar S(2024)Matroid-based TSP rounding for half-integral solutionsMathematical Programming: Series A and B10.1007/s10107-024-02065-4206:1-2(541-576)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s10107-024-02065-4
Jin BKlein NWilliamson D(2023)A 4/3-Approximation Algorithm for Half-Integral Cycle Cut Instances of the TSPInteger Programming and Combinatorial Optimization10.1007/978-3-031-32726-1_16(217-230)Online publication date: 21-Jun-2023
https://dl.acm.org/doi/10.1007/978-3-031-32726-1_16
Haddadan ANewman A(2022)Towards improving Christofides algorithm on fundamental classes by gluing convex combinations of toursMathematical Programming: Series A and B10.1007/s10107-022-01784-w198:1(595-620)Online publication date: 21-Mar-2022
https://dl.acm.org/doi/10.1007/s10107-022-01784-w
Show More Cited By

Index Terms

2-Matchings, the Traveling Salesman Problem, and the Subtour LP: A Proof of the Boyd-Carr Conjecture
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
  2. Mathematical analysis
    1. Mathematical optimization
      1. Continuous optimization
        Linear programming
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Continuous optimization
        Linear programming

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

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cover image Mathematics of Operations Research

Mathematics of Operations Research Volume 39, Issue 2

May 2014

368 pages

ISSN:0364-765X

DOI:10.1287/moor.2014.39.issue-2

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Copyright © 2014, INFORMS.

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INFORMS

Linthicum, MD, United States

Publication History

Published: 01 May 2014

Received: 07 June 2012

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Cited By

Gupta ALee ELi JMucha MNewman HSarkar S(2024)Matroid-based TSP rounding for half-integral solutionsMathematical Programming: Series A and B10.1007/s10107-024-02065-4206:1-2(541-576)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s10107-024-02065-4
Jin BKlein NWilliamson D(2023)A 4/3-Approximation Algorithm for Half-Integral Cycle Cut Instances of the TSPInteger Programming and Combinatorial Optimization10.1007/978-3-031-32726-1_16(217-230)Online publication date: 21-Jun-2023
https://dl.acm.org/doi/10.1007/978-3-031-32726-1_16
Haddadan ANewman A(2022)Towards improving Christofides algorithm on fundamental classes by gluing convex combinations of toursMathematical Programming: Series A and B10.1007/s10107-022-01784-w198:1(595-620)Online publication date: 21-Mar-2022
https://dl.acm.org/doi/10.1007/s10107-022-01784-w
Karlin AKlein NGharan SKhuller SVassilevska Williams V(2021)A (slightly) improved approximation algorithm for metric TSPProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451009(32-45)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451009
Qian JSchalekamp FWilliamson DZuylen A(2015)On the integrality gap of the subtour LP for the 1,2-TSPMathematical Programming: Series A and B10.1007/s10107-014-0835-4150:1(131-151)Online publication date: 1-Apr-2015
https://dl.acm.org/doi/10.1007/s10107-014-0835-4

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