Abstract
We discuss the problem of finding a longest tour for a set of points in a geometric space. In particular, we show that a longest tour for a set of n points in the plane can be computed in time O(n) if distances are determined by the Manhattan metric, while the same problem is NP-hard for points on a sphere under Euclidean distances.
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References
Applegate, D.L., Bixby, R.E., Chvátal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study (Princeton Series in Applied Mathematics). Princeton University Press, Princeton (2007)
Arkin, E.M., Chiang, Y.J., Mitchell, J.S.B., Skiena, S., Yang, T.-C.: On the maximum scatter TSP. In: Proceedings of the 8th ACM-SIAM Symposium on Discrete Algorithms (SODA 97), pp. 211–220 (1997)
Arora, S.: Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. J. ACM 45(5), 753–782 (1998)
Barvinok, A.I.: Two algorithmic results for the Traveling Salesman Problem. Math. Oper. Res. 21(1), 65–84 (1996)
Barvinok, A., Johnson, D.S., Woeginger, G.J., Woodroofe, R.: The maximum Traveling Salesman Problem under polyhedral norms. In: Proceedings of the 6th International Integer Programming and Combinatorial Optimization Conference (IPCO VI). Lecture Notes in Computer Science, vol. 1412, pp. 195–201. Springer (1998)
Barvinok, A.I., Fekete, S.P., Johnson, D.S., Tamir, A., Woeginger, G.J., Wodroofe, R.: The geometric maximum Traveling Salesman Problem. J. ACM 50, 641–664 (2003)
Cook, W.J.: In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation. Princeton University Press, Princeton (2011)
Demaine, E.D., Mitchell, J.S.B., O’Rourke, J.: The Open Problems Project (2001). http://cs.smith.edu/~orourke/TOPP/
Fekete, S.P.: Simplicity and hardness of the maximum Traveling Salesman Problem under geometric distances. In: Proceedings of the 10th ACM-SIAM Symposium Discrete Algorithms (SODA 99), pp. 337–345 (1999)
Fekete, S.P., Meijer, H.: On minimum stars, minimum Steiner stars, and maximum matchings. In: Proceedings of the 15th Annual ACM Symposium on Computational Geometry (SoCG 99), pp. 217–226. ACM (1999)
Fekete, S.P., Meijer, H.: On minimum stars and maximum matchings. Discrete Comput. Geom. 23(3), 389–407 (2000)
Fekete, S.P., Meijer, H., Rohe, A., Tietze, W.: Solving a “hard” problem to approximate an “easy” one: Heuristics for maximum matchings and maximum Traveling Salesman Problems. In: Proceedings of the 3rd International Workshop on Algorithms Engineering and Experiments (ALENEX 2001). Lecture Notes in Computer Science, vol. 2153, pp. 1–16. Springer (2001)
Fekete, S.P., Meijer, H., Rohe, A., Tietze, W.: Solving a “hard” problem to approximate an “easy” one: Heuristics for maximum matchings and maximum Traveling Salesman Problems. J. Exp. Algorithms 7, 21 (2002)
Grötschel, M., Padberg, M.W.: Ulysses 2000: In search of optimal solutions to hard combinatorial problems. SC 93-34, Zuse Institute Berlin, November 1993
Grötschel, M., Padberg, M.W.: Die optimierte Odyssee. Spektrum der Wiss. Dig. 2, 32–41 (1999)
Grötschel, M., Padberg, M.W.: The optimized Odyssey. AIROnews VI(2), 1–7 (2001)
Itai, A., Papadimitriou, C.H., Swarcfiter, J.L.: Hamilton paths in grid graphs. SIAM J. Comput. 11, 676–686 (1982)
Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.).: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, Chichester (1985)
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, 1298–1309 (1999)
Papadimitriou, C.H.: The Euclidean Traveling Salesman Problem is NP-complete. Theor. Comput. Sci. 4, 237–244 (1977)
Reinelt, G.: TSPlib—A Traveling Salesman Problem library. ORSA J. Comput. 3(4), 376–384 (1991)
Serdyukov, A.I.: An asymptotically exact algorithm for the Traveling Salesman Problem for a maximum in Euclidean space (Russian). Upravlyaemye Sistemy 27, 79–87 (1987)
Serdyukov, A.I.: Asymptotic properties of optimal solutions of extremal permutation problems in finite-dimensional normed spaces (Russian). Metody Diskret Analiz 51, 105–111 (1991)
Serdyukov, A.I.: The Traveling Salesman Problem for a maximum in finite-dimensional real spaces (Russian). Diskretnyi Analiz i Issledovanie Operatsii 2(1), 50–56 (1995)
Tamir, A., Mitchell, J.S.B.: A maximum \(b\)-matching problem arising from median location models with applications to the roommates problem. Math. Program. 80(2), 171–194 (1998)
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Fekete, S.P. (2015). Finding Longest Geometric Tours. In: Schulz, A., Skutella, M., Stiller, S., Wagner, D. (eds) Gems of Combinatorial Optimization and Graph Algorithms . Springer, Cham. https://doi.org/10.1007/978-3-319-24971-1_3
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