Abstract
The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. This problem is APX-hard in the general metric case but admits polynomial-time approximation schemes in the geometric setting, when the edge weights are induced by a vector norm in fixed-dimensional real space. We propose the first approximation scheme for Max TSP in an arbitrary metric space of fixed doubling dimension. The proposed algorithm implements an efficient PTAS which, for any fixed \(\varepsilon \in (0,1)\), computes a \((1-\varepsilon )\)-approximate solution of the problem in cubic time. Additionally, we suggest a cubic-time algorithm which finds asymptotically optimal solutions of the metric Max TSP in fixed and sublogarithmic doubling dimensions.
Similar content being viewed by others
References
Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)
Bartal, Y., Gottlieb, L.-A., Krauthgamer, R.: The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme. SIAM J. Comput. 45(4), 1563–1581 (2016)
Barvinok, A., Fekete, S.P., Johnson, D.S., Tamir, A., Woeginger, G.J., Woodroofe, R.: The geometric maximum traveling salesman problem. J. ACM 50(5), 641–664 (2003)
Barvinok, A.I., Gimadi, E.K., Serdyukov, A.I.: The maximum traveling salesman problem. In: Gutin, G., Punnen, A.P. (eds.) The traveling salesman problem and its applications, pp. 585–608. Kluwer Academic Publishers, Dordrecht (2002)
Bellman, R.: Dynamic programming treatment of the travelling salesman problem. J. ACM 9(1), 61–63 (1962)
Engebretsen, L., Karpinski, M.: TSP with bounded metrics. J. Comp. Syst. Sci. 72(4), 509–546 (2006)
Fekete, S.P.: Simplicity and hardness of the maximum traveling salesman problem under geometric distances. In: Proceedings 10th ACM-SIAM Symposium on Discrete Algorithms (SODA 1999), pp. 337–345 (2015)
Gabow, H.: An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems. In: Proceedings 15th ACM Symposium on Theory of Computing (STOC 1983), pp. 448–456 (1983)
Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. Soc. Indust. Appl. Math. 10(1), 196–210 (1962)
Kaplan, H., Lewenstein, M., Shafrir, N., Sviridenko, M.: Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs. J. ACM 52(4), 602–626 (2005)
Kostochka, A.V., Serdyukov, A.I.: Polynomial algorithms with the estimates \(3/4\) and \(5/6\) for the traveling salesman problem of the maximum (in Russian). Upravlyaemye Sistemy 26, 55–59 (1985)
Kowalik, L., Mucha, M.: Deterministic \(7/8\)-approximation for the metric maximum TSP. Theor. Comp. Sci. 410(47–49), 5000–5009 (2009)
Kowalik, L., Mucha, M.: \(35/44\)-approximation for asymmetric maximum TSP with triangle inequality. Algorithmica 59(2), 240–255 (2011)
Mitchell, J.S.B.: Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, \(k\)-MST, and related problems. SIAM J. Comput. 28(4), 1298–1309 (1999)
Paluch, K., Mucha, M., Ma̧dry, A.: A \(7/9\)-approximation algorithm for the maximum traveling salesman problem. In: Proceedings 12th Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX 2009), Lecture Notes in Computer Science, vol. 5687, 298–311 (2009)
Papadimitriou, C.H.: The Euclidean traveling salesman problem is NP-complete. Theor. Comp. Sci. 4(3), 237–244 (1977)
Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Math. Oper. Res. 18(1), 1–11 (1993)
Sahni, S., Gonzalez, T.: P-complete approximation problems. J. ACM 23(3), 555–565 (1976)
Serdyukov, A.I.: An asymptotically exact algorithm for the traveling salesman problem for a maximum in Euclidean space (in Russian). Upravlyaemye sistemy 27, 79–87 (1987)
Serdyukov, A.I.: Polynomial time algorithm with estimates of accuracy of solutions for one class of the maximum cost TSP (in Russian). In: Kombinatorno-Algebraicheskie Metody v Diskretnoi Optimizatsii, pp. 107–114. Nizhny Novgorod Univ., Nizhny Novgorod (1991)
Serdyukov, A.I.: The maximum-weight traveling salesman problem in finite-dimensional real spaces. In: Operations research and discrete analysis. Mathematics and its applications, vol. 391, pp. 233–239. Kluwer Academic Publishers, Dordrecht (1997)
Shenmaier, V.V.: An asymptotically exact algorithm for the maximum traveling salesman problem in a finite-dimensional normed space. J. Appl. Industr. Math. 5(2), 296–300 (2011)
Shenmaier, V.V.: Asymptotically optimal algorithms for geometric Max TSP and Max \(m\)-PSP. Discrete Appl. Math. 163(2), 214–219 (2014)
Acknowledgements
The study was carried out within the framework of the state contract of the Sobolev Institute of Mathematics (Project 0314-2019-0014).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shenmaier, V. Efficient PTAS for the maximum traveling salesman problem in a metric space of fixed doubling dimension. Optim Lett 16, 2115–2122 (2022). https://doi.org/10.1007/s11590-021-01769-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-021-01769-2