Abstract
We generalize the tree doubling and Christofides algorithm to parameterized approximations for ATSP. The parameters we consider for the respective generalizations are upper bounded by the number of asymmetric distances, which yields algorithms to efficiently compute good approximations also for moderately asymmetric TSP instances. As generalization of the Christofides algorithm, we derive a parameterized 2.5-approximation, where the parameter is the size of a vertex cover for the subgraph induced by the asymmetric distances. Our generalization of the tree doubling algorithm gives a parameterized 3-approximation, where the parameter is the minimum number of asymmetric distances in a minimum spanning arborescence. Further, we combine these with a notion of symmetry relaxation which allows to trade approximation guarantee for runtime. Since the two parameters we consider are theoretically incomparable, we present experimental results which show that generalized tree doubling frequently outperforms generalized Christofides with respect to parameter size.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Applegate, D., Bixby, R., Cook, W., Chvátal, V.: On the solution of traveling salesman problems. Doc. Math. 111, 645–656 (1998)
Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An O(log n/log log n)-approximation algorithm for the asymmetric traveling salesman problem. Oper. Res. 65(4), 1043–1061 (2017). https://doi.org/10.1287/opre.2017.1603
Behrendt, L., Casel, K., Friedrich, T., Lagodzinski, J.A.G., Löser, A., Wilhelm, M.: From symmetry to asymmetry: generalizing TSP approximations by parametrization. CoRR abs/1911.02453 (2019). http://arxiv.org/abs/1911.02453
van Bevern, R., Slugina, V.A.: A historical note on the 3/2-approximation algorithm for the metric traveling salesman problem. Hist. Math. 53, 118–127 (2020). https://www.sciencedirect.com/science/article/pii/S0315086020300240
Bläser, M., Manthey, B., Sgall, J.: An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality. J. Discret. Algorithms 4(4), 623–632 (2006). https://doi.org/10.1016/j.jda.2005.07.004
Böckenhauer, H., Hromkovic, J., Klasing, R., Seibert, S., Unger, W.: Towards the notion of stability of approximation for hard optimization tasks and the traveling salesman problem. Theor. Comput. Sci. 285(1), 3–24 (2002). https://doi.org/10.1016/S0304-3975(01)00287-0
Böckenhauer, H., Hromkovic, J., Kneis, J., Kupke, J.: The parameterized approximability of TSP with deadlines. Theor. Comput. Sci. 41(3), 431–444 (2007). https://doi.org/10.1007/s00224-007-1347-x
Bonnet, É., Lampis, M., Paschos, V.T.: Time-approximation trade-offs for inapproximable problems. J. Comput. Syst. Sci. 92, 171–180 (2018). https://doi.org/10.1016/j.jcss.2017.09.009
Chandran, L.S., Ram, L.S.: Approximations for ATSP with parametrized triangle inequality. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 227–237. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45841-7_18
Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report 388, Graduate School of Industrial Administration, Carnegie Mellon University (1976)
Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3
Dezső, B., Jüttner, A., Kovács, P.: LEMON - an open source C++ graph template library. Electron. Notes Theor. Comput. Sci. 264(5), 23–45 (2011). https://doi.org/10.1016/j.entcs.2011.06.003
Edmonds, J.: Optimum branchings. J. Res. Natl. Bur. Stan. Sect. B Math. Math. Phys. 71B(4), 233 (1967)
Feige, U., Singh, M.: Improved approximation ratios for traveling salesperson tours and paths in directed graphs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) APPROX/RANDOM -2007. LNCS, vol. 4627, pp. 104–118. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74208-1_8
Gabow, H.N., Galil, Z., Spencer, T.H., Tarjan, R.E.: Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6(2), 109–122 (1986). https://doi.org/10.1007/BF02579168
Hagberg, A., Schult, D., Swart, P.: Exploring network structure, dynamics, and function using NetworkX. In: Proceedings of the 7th Python in Science Conference, pp. 11–15 (2008). http://conference.scipy.org/proceedings/SciPy2008/paper_2/
Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. Soc. Ind. Appl. Math. 10(1), 196–210 (1962)
Henry-Labordère, A.L.: The record balancing problem: a dynamic programming solution of a generalized traveling salesman problem. RAIRO B-2, 43–49 (1969)
Hoogeveen, J.A.: Analysis of Christofides’ heuristic: some paths are more difficult than cycles. Oper. Res. Lett. 10(5), 291–295 (1991). https://doi.org/10.1016/0167-6377(91)90016-I
Jonker, R., Volgenant, T.: Transforming asymmetric into symmetric traveling salesman problems. Oper. Res. Lett. 2(4), 161–163 (1983)
Jonker, R., Volgenant, T.: Transforming asymmetric into symmetric traveling salesman problems: erratum. Oper. Res. Lett. 5(4), 215–216 (1986)
Karlin, A., Klein, N., Gharan, S.O.: A (slightly) improved approximation algorithm for metric TSP. In: Khuller, S., Williams, V.V. (eds.) Proceedings of the STOC 2021, pp. 32–45. ACM (2021). https://doi.org/10.1145/3406325.3451009
Karpinski, M., Lampis, M., Schmied, R.: New inapproximability bounds for TSP. J. Comput. Syst. Sci. 81(8), 1665–1677 (2015). https://doi.org/10.1016/j.jcss.2015.06.003
Klasing, R., Mömke, T.: A modern view on stability of approximation. In: Böckenhauer, H.-J., Komm, D., Unger, W. (eds.) Adventures Between Lower Bounds and Higher Altitudes. LNCS, vol. 11011, pp. 393–408. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98355-4_22
Kowalik, Ł, Mucha, M.: Two approximation algorithms for ATSP with strengthened triangle inequality. In: Dehne, F., Gavrilova, M., Sack, J.-R., Tóth, C.D. (eds.) WADS 2009. LNCS, vol. 5664, pp. 471–482. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03367-4_41
Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008). https://doi.org/10.1093/comjnl/bxm048
Marx, D., Salmasi, A., Sidiropoulos, A.: Constant-factor approximations for asymmetric TSP on nearly-embeddable graphs. In: Proceedings of APPROX/RANDOM 2016, pp. 16:1–16:54. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016). https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.16
Mori, J.C.M., Samaranayake, S.: Bounded asymmetry in road networks. Sci. Rep. 9(11951), 1–9 (2019). https://doi.org/10.1038/s41598-019-48463-z
Reinelt, G.: TSPLIB–a traveling salesman problem library. INFORMS J. Comput. 3(4), 376–384 (1991). https://doi.org/10.1287/ijoc.3.4.376
Rodríguez, A., Ruiz, R.: The effect of the asymmetry of road transportation networks on the traveling salesman problem. Comput. Oper. Res. 39(7), 1566–1576 (2012). https://doi.org/10.1016/j.cor.2011.09.005
Saksena, J.P.: Mathematical model of scheduling clients through welfare agencies. Comput. Oper. Res. J. 8, 185–200 (1970)
Svensson, O., Tarnawski, J., Végh, L.A.: A constant-factor approximation algorithm for the asymmetric traveling salesman problem. J. ACM 67(6), 37:1-37:53 (2020). https://doi.org/10.1145/3424306
Traub, V., Vygen, J.: An improved approximation algorithm for ATSP. In: Proceedings of STOC 2020, pp. 1–13. ACM (2020). https://doi.org/10.1145/3357713.3384233
Zhang, T., Li, W., Li, J.: An improved approximation algorithm for the ATSP with parameterized triangle inequality. J. Algorithms 64(2–3), 74–78 (2009). https://doi.org/10.1016/j.jalgor.2008.10.002
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Behrendt, L., Casel, K., Friedrich, T., Gregor Lagodzinski, J.A., Löser, A., Wilhelm, M. (2021). From Symmetry to Asymmetry: Generalizing TSP Approximations by Parametrization. In: Bampis, E., Pagourtzis, A. (eds) Fundamentals of Computation Theory. FCT 2021. Lecture Notes in Computer Science(), vol 12867. Springer, Cham. https://doi.org/10.1007/978-3-030-86593-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-86593-1_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-86592-4
Online ISBN: 978-3-030-86593-1
eBook Packages: Computer ScienceComputer Science (R0)