Improving Christofides' algorithm for the s-t path TSP

Hyung Chan An, Robert Kleinberg, David B. Shmoys

Research output: Chapter in Book/Report/Conference proceedingConference contribution

27 Citations (Scopus)

Abstract

We present a deterministic (1+√5/2)-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact had been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of 1+√5/2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP.

Original languageEnglish
Title of host publicationSTOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing
Pages875-885
Number of pages11
DOIs
Publication statusPublished - 2012 Jun 26
Event44th Annual ACM Symposium on Theory of Computing, STOC '12 - New York, NY, United States
Duration: 2012 May 192012 May 22

Other

Other44th Annual ACM Symposium on Theory of Computing, STOC '12
CountryUnited States
CityNew York, NY
Period12/5/1912/5/22

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Approximation algorithms
Hamiltonians
Costs

All Science Journal Classification (ASJC) codes

  • Software

Cite this

An, H. C., Kleinberg, R., & Shmoys, D. B. (2012). Improving Christofides' algorithm for the s-t path TSP. In STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing (pp. 875-885) https://doi.org/10.1145/2213977.2214055
An, Hyung Chan ; Kleinberg, Robert ; Shmoys, David B. / Improving Christofides' algorithm for the s-t path TSP. STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing. 2012. pp. 875-885
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An, HC, Kleinberg, R & Shmoys, DB 2012, Improving Christofides' algorithm for the s-t path TSP. in STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing. pp. 875-885, 44th Annual ACM Symposium on Theory of Computing, STOC '12, New York, NY, United States, 12/5/19. https://doi.org/10.1145/2213977.2214055

Improving Christofides' algorithm for the s-t path TSP. / An, Hyung Chan; Kleinberg, Robert; Shmoys, David B.

STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing. 2012. p. 875-885.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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An HC, Kleinberg R, Shmoys DB. Improving Christofides' algorithm for the s-t path TSP. In STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing. 2012. p. 875-885 https://doi.org/10.1145/2213977.2214055