An Experimental Evaluation of the Best-of-Many Christofides’ Algorithm for the Traveling Salesman Problem
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 re...
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| Published in | Algorithmica Vol. 78; no. 4; pp. 1109 - 1130 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.08.2017
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0178-4617 1432-0541 |
| DOI | 10.1007/s00453-017-0293-5 |
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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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| AbstractList | 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. 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. |
| Author | Genova, Kyle Williamson, David P. |
| Author_xml | – sequence: 1 givenname: Kyle surname: Genova fullname: Genova, Kyle organization: Department of Computer Science, Princeton University – sequence: 2 givenname: David P. orcidid: 0000-0002-2884-0058 surname: Williamson fullname: Williamson, David P. email: dpw@cs.cornell.edu organization: School of Operations Research and Information Engineering, Cornell University |
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| Cites_doi | 10.1007/s00224-012-9439-7 10.1145/2739008 10.1137/15M1010531 10.1007/978-3-662-48350-3_48 10.1007/978-3-642-36694-9_31 10.1007/s00493-014-2960-3 10.1145/2818310 10.1016/0167-6377(91)90016-I 10.1109/FOCS.2011.80 10.1145/2487788.2488173 10.1007/978-3-319-33461-5_11 10.1007/BFb0014497 10.1023/A:1009739202898 10.1287/opre.18.6.1138 10.1007/978-1-4612-1098-6 10.1007/0-306-48213-4_9 10.1137/1.9781611973075.32 10.1002/net.3230120103 10.1007/BF01902503 10.1109/FOCS.2016.21 10.1137/14096712X 10.1287/opre.21.2.498 |
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| Keywords | Christofides’ algorithm Traveling salesman problem Approximation algorithms Maximum entropy |
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| References_xml | – reference: LinSKernighanBWAn effective heuristic algorithm for the traveling-salesman problemOper. Res.19732149851635974210.1287/opre.21.2.4980256.90038 – reference: Rohe, A.: Instances found at http://www.math.uwaterloo.ca/tsp/vlsi/index.html (2002). Accessed 16 Dec 2014 – reference: Shewchuk, J.R.: Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. In: Lin, M.C., Manocha, D. (eds.) Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Computer Science, vol. 1148, pp. 203–222. Springer, Berlin (1996) – reference: MuchaM13/9-approximation for graphic TSPTheory Comput. Syst.201455640657325949610.1007/s00224-012-9439-71319.68255 – reference: An, H.C.: Approximation Algorithms for Traveling Salesman Problems Based on Linear Programming Relaxations. Ph.D. thesis, Department of Computer Science, Cornell University (2012) – reference: NagamochiHIbarakiTDeterministic O~(mn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{O}(mn)$$\end{document} time edge-splitting in undirected graphsJ. Comb. Optim.19971546160617310.1023/A:10097392028980895.90172 – reference: Reinelt, G.: TSPLIB—a traveling salesman problem library. ORSA J. Comput. 3, 376–384 (1991) – reference: Gottschalk, C., Vygen, J.: Better s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s$$\end{document}–t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document}-tours by Gao trees. In: Loveaux, Q., Skutella, M. (eds.) Integer Programming and Combinatorial Optimization, 18th International Conference, IPCO 2016, Lecture Notes in Computer Science, vol. 9682, pp. 126–137. Springer, Berlin (2016) – reference: 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 (2013) – reference: 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, Dordrecht (2002) – reference: Mömke, T., Svensson, O.: Removing and adding edges for the traveling salesman problem. J. ACM 63 (2016). Article 2 – reference: SchrijverACombinatorial Optimization: Polyhedra and Efficiency2003BerlinSpringer1041.90001 – reference: GaoZOn the metric s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s$$\end{document}–t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} path traveling salesman problemSIAM J. Discret. Math.20152911331149336998910.1137/14096712X1331.68293 – reference: Kolmogorov, V., Blossom, V.: A new implementation of a minimum cost perfect matching algorithm. Math. Program. Comput. 1, 43–67 (2009). http://pub.ist.ac.at/~vnk/software.html – reference: VygenJReassembling trees for the traveling salesmanSIAM J. Discret. Math.201630875894349395410.1137/15M10105311345.90079 – reference: FrankAConnections in Combinatorial Optimization2011OxfordOxford University Press1228.90001 – reference: Applegate, D., Bixby, R., Chvátal, V., Cook, W.: Concorde 03.12.19 (2003). http://www.math.uwaterloo.ca/tsp/concorde/index.html – reference: HoogeveenJAAnalysis of Christofides’ heuristic: some paths are more difficult than cyclesOper. Res. Lett.199110291295112233210.1016/0167-6377(91)90016-I0748.90071 – reference: HeldMKarpRMThe traveling-salesman problem and minimum spanning treesOper. Res.1971181138116227871010.1287/opre.18.6.11380226.90047 – reference: Walter, M., Wegmann, N.: Computational study of graphic TSP approximation algorithms (2014). Poster presented at IPCO 2014 – reference: FriezeAGalbiatiGMaffioliFOn the worst-case performance of some algorithms for the asymmetric traveling salesman problemNetworks198212233966726210.1002/net.32301201030478.90070 – reference: LovászLOn some connectivity properties of Eulerian graphsActa Math. Acad. Sci. Hung.19762812913843739110.1007/BF019025030337.05124 – reference: Sebő, A.: Eight-fifth approximation for the path TSP. In: Goemans, M.X., Correa, J.R. (eds.) Integer Programming and Combinatorial Optimization—16th International Conference (IPCO 2013), Lecture Notes in Computer Science, vol. 7801, pp. 362–374. Springer, Berlin (2013) – reference: Christofides, N.: Worst Case Analysis of a New Heuristic for the Traveling Salesman Problem. 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Accessed 28 Jan 2015 – reference: SebőAVygenJShorter tours by nicer ears: 7/5-approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphsCombinatorica201434597629328591010.1007/s00493-014-2960-31340.90201 – reference: Kunegis, J.: KONECT—the Koblenz network collection. 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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Approximation Computer Science Computer Systems Organization and Communication Networks Computing costs Cost engineering Data Structures and Information Theory Graph theory Mathematics of Computing Maximum entropy Sampling Theory of Computation Traveling salesman problem |
| Title | An Experimental Evaluation of the Best-of-Many Christofides’ Algorithm for the Traveling Salesman Problem |
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