research-article

Approaching 3/2 for the s-t-path TSP

Authors:

Jens VygenAuthors Info & Claims

Journal of the ACM (JACM), Volume 66, Issue 2

Article No.: 14, Pages 1 - 17

https://doi.org/10.1145/3309715

Published: 07 March 2019 Publication History

Abstract

We show that there is a polynomial-time algorithm with approximation guarantee 3/2+ε for the s-t-path TSP, for any fixed ε > 0.

It is well-known that Wolsey’s analysis of Christofide algorithm also works for the s-t-path TSP with its natural LP relaxation, except for the narrow cuts (in which the LP solution has a value less than two). A fixed optimum tour has either a single edge in a narrow cut (then call the edge and the cut lonely) or at least three (then call the cut busy). Our algorithm “guesses” (by dynamic programming) lonely cuts and edges. Then, we partition the instance into smaller instances and strengthen the LP, requiring a value of at least three for busy cuts. By setting up a k-stage recursive dynamic program, we can compute a spanning tree (V,S) and an LP solution y such that (½+O(2^−k))y is in the T-join polyhedron, where T is the set of vertices whose degree in S has the wrong parity.

References

[1]

H.-C. An, R. Kleinberg, and D. B. Shmoys. 2015. Improving Christofides’ algorithm for the s-t path TSP. J. ACM 62, Article 34 (2015).

Digital Library

[2]

N. Christofides. 1976. Worst-case Analysis of a New Heuristic for the Traveling Salesman Problem. Technical Report 388. Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh.

[3]

J. Edmonds and E. L. Johnson. 1973. Matching, Euler tours and the Chinese postman. Math. Program. 5 (1973), 88--124.

Digital Library

[4]

M. X. Goemans and D. J. Bertsimas. 1993. Survivable networks, linear programming relaxations and the parsimonious property. Math. Program. 60 (1993), 145--166.

Digital Library

[5]

C. Gottschalk and J. Vygen. 2018. Better s-t-tours by Gao trees. Math. Program. B 172 (2018), 191--207.

Digital Library

[6]

M. Held and R. M. Karp. 1970. The traveling-salesman problem and minimum spanning trees. Oper. Res. 18 (1970), 1138--1162.

Digital Library

[7]

J. A. Hoogeveen. 1991. Analysis of Christofides’ heuristic: Some paths are more difficult than cycles. Oper. Res. Lett. 10 (1991), 291--295.

Digital Library

[8]

L. Lovász. 1976. On some connectivity properties of Eulerian graphs. Acta Mathematica Academiae Scientiarum Hungaricae 28 (1976), 129--138.

[9]

C. L. Monma, B. S. Munson, and W. R. Pulleyblank. 1990. Minimum-weight two-connected spanning networks. Math. Program. 46 (1990), 153--171.

Digital Library

[10]

A. Sebő. 2013. Eight fifth approximation for TSP paths. In Proceedings of the 16th IPCO Conference on Integer Programming and Combinatorial Optimization, M. X. Goemans J. Correa (Ed.). Springer, 362--374.

Digital Library

[11]

A. Sebő and A. van Zuylen. 2016. The salesman’s improved paths: 3/2+1/34 integrality gap and approximation ratio. In Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS’16). 118--127.

[12]

A. I. Serdjukov. 1978. Some extremal bypasses in graphs {in Russian}. Upravlyaemye Sistemy 17 (1978), 76--79.

[13]

V. Traub and J. Vygen. 2018. An improved upper bound on the integrality ratio for the s-t-path TSP. Operations Research Letters, to appear.

[14]

J. Vygen. 2016. Reassembling trees for the traveling salesman. SIAM J. Discrete Math. 30 (2016), 875--894.

[15]

L. A. Wolsey. 1980. Heuristic analysis, linear programming and branch and bound. Math. Program. Study 13 (1980), 121--134.

[16]

R. Zenklusen. 2019. A 1.5-approximation for path TSP. In Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA’19). 1539--1549.

Digital Library

Cited By

Balkanski EFaenza YKubik M(2024)The simultaneous semi-random model for TSPMathematical Programming: Series A and B10.1007/s10107-023-02011-w206:1-2(305-332)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s10107-023-02011-w
Nägele MZenklusen R(2023)A New Dynamic Programming Approach for Spanning Trees with Chain Constraints and BeyondMathematics of Operations Research10.1287/moor.2023.0012Online publication date: 13-Oct-2023
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
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Index Terms

Approaching 3/2 for the s-t-path TSP
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial optimization

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 66, Issue 2

April 2019

260 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3318168

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: 07 March 2019

Accepted: 01 January 2019

Revised: 01 December 2018

Received: 01 January 2018

Published in JACM Volume 66, Issue 2

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Cited By

Balkanski EFaenza YKubik M(2024)The simultaneous semi-random model for TSPMathematical Programming: Series A and B10.1007/s10107-023-02011-w206:1-2(305-332)Online publication date: 1-Jul-2024
https://dl.acm.org/doi/10.1007/s10107-023-02011-w
Nägele MZenklusen R(2023)A New Dynamic Programming Approach for Spanning Trees with Chain Constraints and BeyondMathematics of Operations Research10.1287/moor.2023.0012Online publication date: 13-Oct-2023
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(2023)Approximations for many-visits multiple traveling salesman problemsOmega10.1016/j.omega.2022.102816116(102816)Online publication date: Apr-2023
https://doi.org/10.1016/j.omega.2022.102816
Zhang JGao SHou BLiu W(2023)An approximation algorithm for the clustered path travelling salesman problemJournal of Combinatorial Optimization10.1007/s10878-023-01029-245:4Online publication date: 30-Apr-2023
https://dl.acm.org/doi/10.1007/s10878-023-01029-2
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
Hsu YGau R(2022)Reinforcement Learning-Based Collision Avoidance and Optimal Trajectory Planning in UAV Communication NetworksIEEE Transactions on Mobile Computing10.1109/TMC.2020.300363921:1(306-320)Online publication date: 1-Jan-2022
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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
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Zhang JGao SHou BLiu W(2022)An Approximation Algorithm for the Clustered Path Travelling Salesman ProblemAlgorithmic Aspects in Information and Management10.1007/978-3-031-16081-3_2(15-27)Online publication date: 13-Aug-2022
https://dl.acm.org/doi/10.1007/978-3-031-16081-3_2
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