Abstract
Recently, Mömke and Svensson presented a beautiful new approach for the traveling salesman problem on a graph metric (graph-TSP), which yielded a \(\frac{4}{3}\)-approximation guarantee on subcubic graphs as well as a substantial improvement over the \(\frac{3}{2}\)-approximation guarantee of Christofides’ algorithm on general graphs. The crux of their approach is to compute an upper bound on the minimum cost of a circulation in a particular network, C(G,T), where G is the input graph and T is a carefully chosen spanning tree. The cost of this circulation is directly related to the number of edges in a tour output by their algorithm. Mucha subsequently improved the analysis of the circulation cost, proving that Mömke and Svensson’s algorithm for graph-TSP has an approximation ratio of at most \(\frac{13}{9}\) on general graphs.
This analysis of the circulation is local, and vertices with degree four and five can contribute the most to its cost. Thus, hypothetically, there could exist a subquartic graph (a graph with degree at most four at each vertex) for which Mucha’s analysis of the Mömke-Svensson algorithm is tight. In this paper, we show that this is not the case and that Mömke and Svensson’s algorithm for graph-TSP has an approximation guarantee of at most \(\frac{46}{33}\) on subquartic graphs. To prove this, we present a different method to upper bound the minimum cost of a circulation on the network C(G,T). Our approximation guarantee actually holds for all graphs that have an optimal solution to a standard linear programming relaxation of graph-TSP with subquartic support.
Supported in part by LabEx PERSYVAL-Lab (ANR–11-LABX-0025).
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References
An, H.-C., Kleinberg, R., Shmoys, D.B.: Improving Christofides’ algorithm for the s-t-path TSP. In: Proceedings of the 44th Symposium on Theory of Computing, pp. 875–886. ACM (2012)
Boyd, S., Sitters, R., van der Ster, S., Stougie, L.: TSP on cubic and subcubic graphs. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 65–77. Springer, Heidelberg (2011)
Cornuéjols, G., Fonlupt, J., Naddef, D.: The traveling salesman problem on a graph and some related integer polyhedra. Mathematical Programming 33(1), 1–27 (1985)
Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report, DTIC Document (1976)
Gao, Z.: An LP-based-approximation algorithm for the s-t path graph traveling salesman problem. Operations Research Letters 41(6), 615–617 (2013)
Gamarnik, D., Lewenstein, M., Sviridenko, M.: An improved upper bound for the TSP in cubic 3-edge-connected graphs. Operations Research Letters 33(5), 467–474 (2005)
Gharan, S.O., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 550–559. IEEE (2011)
Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Operations Research 18(6), 1138–1162 (1970)
Hoogeveen, J.A.: Analysis of Christofides’ heuristic: some paths are more difficult than cycles. Operations Research Letters 10(5), 291–295 (1991)
Mömke, T., Svensson, O.: Approximating graphic TSP by matchings. In: IEEE 52nd Annual Symposium on Foundations of Computer Science, pp. 560–569 (2011)
Mucha, M.: 13/9-approximation for graphic TSP. In: Dürr, C., Wilke, T. (eds.) STACS. LIPIcs, vol. 14, pp. 30–41. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)
Naddef, D., Pulleyblank, W.R.: Matchings in regular graphs. Discrete Mathematics 34(3), 283–291 (1981)
Sebő, A.: Eight-fifth approximation for the path TSP. In: Goemans, M., Correa, J. (eds.) IPCO 2013. LNCS, vol. 7801, pp. 362–374. Springer, Heidelberg (2013)
Sebő, A., Vygen, J.: Shorter tours by nicer ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. arXiv:1201.1870 (2012)
Vishnoi, N.K.: A permanent approach to the traveling salesman problem. In: 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pp. 76–80. IEEE (2012)
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Newman, A. (2014). An Improved Analysis of the Mömke-Svensson Algorithm for Graph-TSP on Subquartic Graphs. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_61
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