research-article

The Salesman’s Improved Paths through Forests

Authors:

Anke Van ZuylenAuthors Info & Claims

Journal of the ACM (JACM), Volume 66, Issue 4

Article No.: 28, Pages 1 - 16

https://doi.org/10.1145/3326123

Published: 05 June 2019 Publication History

Abstract

We give a new, strongly polynomial-time algorithm and improved analysis for the metric s-t path Traveling Salesman Problem (TSP). It finds a tour of cost less than 1.53 times the optimum of the subtour elimination linear program (LP), while known examples show that 1.5 is a lower bound for the integrality gap.

A key new idea is the deletion of some edges of the spanning trees used in the best-of-many Christofides-Serdyukov-algorithm, which is then accompanied by novel arguments of the analysis: edge-deletion disconnects the trees, and the arising forests are then partly reconnected by “parity correction.” We show that the arising “connectivity correction” can be achieved for a minor extra cost.

On the one hand, this algorithm and analysis extend previous tools such as the best-of-many Christofides-Serdyukov-algorithm. On the other hand, powerful new tools are solicited, such as a flow problem for analyzing the reconnection cost, and the construction of a set of more and more restrictive spanning trees, each of which can still be found by the greedy algorithm. We show that these trees, which are easy to compute, can replace the spanning trees of the best-of-many Christofides-Serdyukov-algorithm.

These new methods lead to improving the integrality ratio and approximation guarantee below 1.53, as was shown in the preliminary, shortened version of this article that appeared in FOCS 2016. The algorithm and analysis have been significantly simplified in the current article, while details and explanations have been added.

References

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Cited By

Nägele MZenklusen R(2024)A New Dynamic Programming Approach for Spanning Trees with Chain Constraints and BeyondMathematics of Operations Research10.1287/moor.2023.001249:4(2078-2108)Online publication date: Nov-2024
https://doi.org/10.1287/moor.2023.0012
Sun JGutin GLi PShi PZhang X(2023)An LP-based approximation algorithm for the generalized traveling salesman path problemTheoretical Computer Science10.1016/j.tcs.2022.11.013941(180-190)Online publication date: Jan-2023
https://doi.org/10.1016/j.tcs.2022.11.013
Bérczi KMnich MVincze R(2022)A 3/2-Approximation for the Metric Many-Visits Path TSPSIAM Journal on Discrete Mathematics10.1137/22M148341436:4(2995-3030)Online publication date: 8-Dec-2022
https://doi.org/10.1137/22M1483414
Show More Cited By

Index Terms

The Salesman’s Improved Paths through Forests
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
    2. Graph theory

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 66, Issue 4

Networking, Computational Complexity, Design and Analysis of Algorithms, Real Computation, Algorithms, Online Algorithms and Computer-aided Verification

August 2019

299 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3338848

Editor:
Éva Tardos
Cornell University

Issue’s Table of Contents

Copyright © 2019 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 05 June 2019

Accepted: 01 April 2019

Revised: 01 April 2019

Received: 01 August 2018

Published in JACM Volume 66, Issue 4

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Cited By

Nägele MZenklusen R(2024)A New Dynamic Programming Approach for Spanning Trees with Chain Constraints and BeyondMathematics of Operations Research10.1287/moor.2023.001249:4(2078-2108)Online publication date: Nov-2024
https://doi.org/10.1287/moor.2023.0012
Sun JGutin GLi PShi PZhang X(2023)An LP-based approximation algorithm for the generalized traveling salesman path problemTheoretical Computer Science10.1016/j.tcs.2022.11.013941(180-190)Online publication date: Jan-2023
https://doi.org/10.1016/j.tcs.2022.11.013
Bérczi KMnich MVincze R(2022)A 3/2-Approximation for the Metric Many-Visits Path TSPSIAM Journal on Discrete Mathematics10.1137/22M148341436:4(2995-3030)Online publication date: 8-Dec-2022
https://doi.org/10.1137/22M1483414
Traub VVygen J(2022)An Improved Approximation Algorithm for The Asymmetric Traveling Salesman ProblemSIAM Journal on Computing10.1137/20M133931351:1(139-173)Online publication date: 22-Feb-2022
https://doi.org/10.1137/20M1339313
Zhang XDu DGutin GMing QSun J(2022)Approximation algorithms with constant ratio for general cluster routing problemsJournal of Combinatorial Optimization10.1007/s10878-021-00772-844:4(2499-2514)Online publication date: 1-Nov-2022
https://dl.acm.org/doi/10.1007/s10878-021-00772-8
Traub VVygen JZenklusen R(2021)Reducing Path TSP to TSPSIAM Journal on Computing10.1137/20M135594X51:3(STOC20-24-STOC20-53)Online publication date: 18-Oct-2021
https://doi.org/10.1137/20M135594X
Sun JGutin GZhang X(2021)A LP-based Approximation Algorithm for generalized Traveling Salesperson Path ProblemCombinatorial Optimization and Applications10.1007/978-3-030-92681-6_50(641-652)Online publication date: 17-Dec-2021
https://dl.acm.org/doi/10.1007/978-3-030-92681-6_50
Traub V(2020) Improving on Best-of-Many-Christofides for -tours Operations Research Letters10.1016/j.orl.2020.09.009Online publication date: Oct-2020
https://doi.org/10.1016/j.orl.2020.09.009
Zhong X(2020) Slightly improved upper bound on the integrality ratio for the Path TSP Operations Research Letters10.1016/j.orl.2020.07.015Online publication date: Aug-2020
https://doi.org/10.1016/j.orl.2020.07.015
van Bevern RSlugina V(2020)A historical note on the 3/2-approximation algorithm for the metric traveling salesman problemHistoria Mathematica10.1016/j.hm.2020.04.00353(118-127)Online publication date: Nov-2020
https://doi.org/10.1016/j.hm.2020.04.003
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