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An Experimental Evaluation of the Best-of-Many Christofides’ Algorithm for the Traveling Salesman Problem

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Abstract

Recent papers on approximation algorithms for the traveling salesman problem (TSP) have given a new variant of the well-known Christofides’ algorithm for the TSP, called the Best-of-Many Christofides’ algorithm. The algorithm involves sampling a spanning tree from the solution to the standard LP relaxation of the TSP, subject to the condition that each edge is sampled with probability at most its value in the LP relaxation. One then runs Christofides’ algorithm on the tree by computing a minimum-cost matching on the odd-degree vertices in the tree, and shortcutting a traversal of the resulting Eulerian graph to a tour. In this paper we perform an experimental evaluation of the Best-of-Many Christofides’ algorithm to see if there are empirical reasons to believe its performance is better than that of Christofides’ algorithm. Furthermore, several different sampling schemes have been proposed; we implement several different schemes to determine which ones might be the most promising for obtaining improved performance guarantees over that of Christofides’ algorithm. In our experiments, all of the implemented methods perform significantly better than Christofides’ algorithm; an algorithm that samples from a maximum entropy distribution over spanning trees seems to be particularly good, though there are others that perform almost as well.

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Acknowledgements

We thank Shayan Oveis Gharan for generously sharing his maximum entropy code with us; we used many of his implementation ideas in coding our own algorithm. We thank Frans Schalekamp and Anke van Zuylen for allowing us to include their example of a bad case for Best-of-Many Christofides. We thank Frans Schalekamp for several observations which he allowed us to use and include.

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Authors and Affiliations

  1. Department of Computer Science, Princeton University, Princeton, NJ, 08540, USA

    Kyle Genova

  2. School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, 14853, USA

    David P. Williamson

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  1. Kyle Genova
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  2. David P. Williamson
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Correspondence to David P. Williamson.

Additional information

This work was supported in part by the NSF Grant CCF-1115256. An extended abstract of this paper appeared at the 2015 European Symposium on Algorithms; see [10].

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Genova, K., Williamson, D.P. An Experimental Evaluation of the Best-of-Many Christofides’ Algorithm for the Traveling Salesman Problem. Algorithmica 78, 1109–1130 (2017). https://doi.org/10.1007/s00453-017-0293-5

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