Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem

Hyung Chan An; Robert Kleinberg; David B. Shmoys

doi:10.1145/3478537

Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem

Hyung Chan An, Robert Kleinberg, David B. Shmoys

College of Computing

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O(log n/ log log n) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, M ...dry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem.

Original language	English
Article number	35
Journal	ACM Transactions on Algorithms
Volume	17
Issue number	4
DOIs	https://doi.org/10.1145/3478537
Publication status	Published - 2021 Oct

Bibliographical note

Funding Information:
Hyung-Chan An research supported in part by NSF under grants no. CCR-0635121,DMS-0732196, CCF-0832782, CCF-0729102 and the Korea Foundation for Advanced Studies. Part of this research was conducted while the author was a PhD student at Cornell University. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1C1C1008934). Robert Kleinberg supported by NSF grants no. CCF-0643934 and CCF-0729102, a grant from the Air Force Office of Scientific Research, a Microsoft Research New Faculty Fellowship, and an Alfred P. Sloan Foundation Fellowship. David B. Shmoys research supported in part by NSF under grants no. CCR-0635121, DMS-0732196, CCF-0832782.

Funding Information:
A preliminary version of this work was presented in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and part of the thesis of the first author. Hyung-Chan An research supported in part by NSF under grants no. CCR-0635121,DMS-0732196, CCF-0832782, CCF-0729102 and the Korea Foundation for Advanced Studies. Part of this research was conducted while the author was a PhD student at Cornell University. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1C1C1008934). Robert Kleinberg supported by NSF grants no. CCF-0643934 and CCF-0729102, a grant from the Air Force Office of Scientific Research, a Microsoft Research New Faculty Fellowship, and an Alfred P. Sloan Foundation Fellowship. David B. Shmoys research supported in part by NSF under grants no. CCR-0635121, DMS-0732196, CCF-0832782. Authors’ addresses: H.-C. An, Department of Computer Science, Yonsei University, 50 Seodaemun-gu, Seoul, 03722, South Korea; email: hyung-chan.an@yonsei.ac.kr; R. Kleinberg, Department of Computer Science, Cornell University, Ithaca,NY, 14853, United States; email: rdk@cs.cornell.edu; D. B. Shmoys, School of ORIE and Department of Computer Science, Cornell University, Ithaca, NY, 14853, United States; emails: rdk@cs.cornell.edu, dbs10@cornell.edu. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2021 Association for Computing Machinery. 1549-6325/2021/10-ART35 $15.00 https://doi.org/10.1145/3478537

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© 2021 Association for Computing Machinery.

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Mathematics (miscellaneous)

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10.1145/3478537

Cite this

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title = "Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem",

abstract = "We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O(log n/ log log n) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, M ...dry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem.",

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note = "Funding Information: Hyung-Chan An research supported in part by NSF under grants no. CCR-0635121,DMS-0732196, CCF-0832782, CCF-0729102 and the Korea Foundation for Advanced Studies. Part of this research was conducted while the author was a PhD student at Cornell University. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1C1C1008934). Robert Kleinberg supported by NSF grants no. CCF-0643934 and CCF-0729102, a grant from the Air Force Office of Scientific Research, a Microsoft Research New Faculty Fellowship, and an Alfred P. Sloan Foundation Fellowship. David B. Shmoys research supported in part by NSF under grants no. CCR-0635121, DMS-0732196, CCF-0832782. Funding Information: A preliminary version of this work was presented in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and part of the thesis of the first author. Hyung-Chan An research supported in part by NSF under grants no. CCR-0635121,DMS-0732196, CCF-0832782, CCF-0729102 and the Korea Foundation for Advanced Studies. Part of this research was conducted while the author was a PhD student at Cornell University. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2019R1C1C1008934). Robert Kleinberg supported by NSF grants no. CCF-0643934 and CCF-0729102, a grant from the Air Force Office of Scientific Research, a Microsoft Research New Faculty Fellowship, and an Alfred P. Sloan Foundation Fellowship. David B. Shmoys research supported in part by NSF under grants no. CCR-0635121, DMS-0732196, CCF-0832782. Authors{\textquoteright} addresses: H.-C. An, Department of Computer Science, Yonsei University, 50 Seodaemun-gu, Seoul, 03722, South Korea; email: hyung-chan.an@yonsei.ac.kr; R. Kleinberg, Department of Computer Science, Cornell University, Ithaca,NY, 14853, United States; email: rdk@cs.cornell.edu; D. B. Shmoys, School of ORIE and Department of Computer Science, Cornell University, Ithaca, NY, 14853, United States; emails: rdk@cs.cornell.edu, dbs10@cornell.edu. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. {\textcopyright} 2021 Association for Computing Machinery. 1549-6325/2021/10-ART35 $15.00 https://doi.org/10.1145/3478537 Publisher Copyright: {\textcopyright} 2021 Association for Computing Machinery.",

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Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem

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