### Improving Christofides' Algorithm for the s-t Path TSPImproving Christofides' Algorithm for the s-t Path TSP

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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