Abstract
Recent papers on approximation algorithms for the traveling salesman problem (TSP) have given a new variant on 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, and running Christofides’ algorithm on the sampled tree. 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. In our experiments, all of the implemented variants of the Best-of-Many Christofides’ algorithm perform significantly better than Christofides’ algorithm; an algorithm that samples from a maximum entropy distribution over spanning trees seems to be particularly good.
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References
An, H.C.: Approximation Algorithms for Traveling Salesman Problems Based on Linear Programming Relaxations. Ph.D. thesis, Department of Computer Science, Cornell University (August 2012)
An, H.C., Kleinberg, R., Shmoys, D.B.: Improving Christofides’ algorithm for the s-t path TSP. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pp. 875–886 (2012)
Applegate, D., Bixby, R., Chvátal, V., Cook, W.: Concorde 03.12.19. http://www.math.uwaterloo.ca/tsp/concorde/index.html
Asadpour, A., Goemans, M.X., Madry, A., Oveis Gharan, S., Saberi, A.: An O(logn/loglogn)-approximation algorithm for the asymmetric traveling salesman problem. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 379–389 (2010)
Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 575–584 (2010), see full version at arxiv.0909:4348
Christofides, N.: Worst case analysis of a new heuristic for the traveling salesman problem. Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA (1976)
Frank, A.: Connections in Combinatorial Optimization. Oxford University Press, Oxford (2011)
Frieze, A., Galbiati, G., Maffioli, F.: On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 12, 23–39 (1982)
Genova, K., Williamson, D.P.: An experimental evaluation of the Best-of-Many Christofides’ algorithm for the traveling salesman problem, CORR abs/1506.07776 (2015)
Gurobi Optimization: Gurobi 5.6.3 (2014). http://www.gurobi.com
Johnson, D.S., McGeoch, L.A.: Experimental analysis of heuristics for the STSP. In: Gutin, G., Punnen, A. (eds.) The Traveling Salesman Problem and its Variants, pp. 369–443. Kluwer Academic Publishers (2002)
Kolmogorov, V.: Blossom V: a new implementation of a minimum cost perfect matching algorithm. Mathematical Programming Computation 1, 43–67 (2009). http://pub.ist.ac.at/~vnk/software.html
Kunegis, J.: KONECT – the Koblenz network collection. In: Proceedings of the International Web Observatory Workshop, pp. 1343–1350 (2013)
Lin, S., Kernighan, B.W.: An effective heuristic algorithm for the traveling-salesman problem. Operations Research 21, 498–516 (1973)
Lovász, L.: On some connectivity properties of Eulerian graphs. Acta Math. Acad. Sci. Hungar. 28, 129–138 (1976)
Mömke, T., Svensson, O.: Approximating graphic TSP by matchings. In: Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, pp. 560–569 (2011)
Mucha, M.: 13/9-approximation for graphic TSP. Theory of Computing Systems 55, 640–657 (2014)
Nagamochi, H., Ibaraki, T.: Deterministic Õ(mn) time edge-splitting in undirected graphs. Journal of Combinatorial Optimization 1, 5–46 (1997)
Oveis Gharan, S.: New Rounding Techniques for the Design and Analysis of Approximation Algorithms. Ph.D. thesis, Department of Management Science and Engineering, Stanford University (May 2013)
Oveis Gharan, S.: Personal communication (2014)
Oveis Gharan, S., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, pp. 550–559 (2011)
Reinelt, G.: TSPLIB – a traveling salesman problem library. ORSA Journal on Computing, 376–384 (1991)
Rohe, A.: Instances found at http://www.math.uwaterloo.ca/tsp/vlsi/index.html (Accessed December 16, 2014)
Sebő, A., Vygen, J.: Shorter tours by nicer ears: 7/5-approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. Combinatorica 34, 597–629 (2014)
Shewchuk, J.R.: Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. In: Lin, M.C., Manocha, D. (eds.) FCRC-WS 1996 and WACG 1996. LNCS, vol. 1148, pp. 203–222. Springer, Heidelberg (1996)
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Genova, K., Williamson, D.P. (2015). An Experimental Evaluation of the Best-of-Many Christofides’ Algorithm for the Traveling Salesman Problem. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_48
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DOI: https://doi.org/10.1007/978-3-662-48350-3_48
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