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Author An, Hyung-Chan ♦ Kleinberg, Robert ♦ Shmoys, David B.
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2015
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Approximation algorithms ♦ TSP (traveling salesman problem) ♦ Linear programming relaxations and rounding algorithms
Abstract We present a deterministic (1+√5/2)-approximation algorithm for the $\textit{s}-\textit{t}$ path TSP for an arbitrary metric. Given a symmetric metric cost on $\textit{n}$ 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 has 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 ratio 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 article 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 $\textit{s}-\textit{t}$ path problem and the unit-weight graphical metric $\textit{s}-\textit{t}$ path TSP.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2015-11-02
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 62
Issue Number 5
Page Count 28
Starting Page 1
Ending Page 28


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Source: ACM Digital Library