The Shortest Separating Cycle Problem

  • Esther M. Arkin
  • Jie Gao
  • Adam Hesterberg
  • Joseph S. B. Mitchell
  • Jiemin Zeng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10138)

Abstract

Given a set of pairs of points in the plane, the goal of the shortest separating cycle problem is to find a simple tour of minimum length that separates the two points of each pair to different sides. In this article we prove hardness of the problem and provide approximation algorithms under various settings. Assuming the Unique Games Conjecture, the problem cannot be approximated within a factor of 2. We provide a polynomial algorithm when all pairs are unit length apart with horizontal orientation inside a square board of size \(2-\varepsilon \). We provide constant approximation algorithms for unit length horizontal or vertical pairs or constant length pairs on points laying on a grid. For pairs with no restriction we have an \(O(\sqrt{n})\)-approximation algorithm and an \(O(\log n)\)-approximation algorithm for the shortest separating planar graph.

Keywords

Shortest separating cycle Traveling salesman problem 

References

  1. 1.
    Arkin, E.M., Hassin, R.: Approximation algorithms for the geometric covering salesman problem. Discrete Appl. Math. 55(3), 197–218 (1994)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Arora, S., Chang, K.: Approximation schemes for degree-restricted MST and red-blue separation problems. Algorithmica 40(3), 189–210 (2004)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Chan, T.H., Jiang, S.H.: Reducing curse of dimensionality: improved PTAS for TSP (with neighborhoods) in doubling metrics. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, 10–12 January 2016, Arlington, VA, USA, pp. 754–765 (2016)Google Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    Dumitrescu, A., Mitchell, J.S.: Approximation algorithms for TSP with neighborhoods in the plane. J. Algorithms 48(1), 135–159 (2003). Twelfth Annual ACM-SIAM Symposium on Discrete AlgorithmsCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: Proceedings of the Eighth Annual ACM Symposium on Theory of Computing, STOC 1976, pp. 10–22. ACM, New York (1976)Google Scholar
  8. 8.
    Gudmundsson, J., Levcopoulos, C.: Hardness result for TSP with neighborhoods. Technical report, Technical Report LU-CS-TR: 2000–216, Department of Computer Science, Lund University, Sweden (2000)Google Scholar
  9. 9.
    Karloff, H.J.: How long can a Euclidean traveling salesman tour be? SIAM J. Discrete Math. 2(1), 91–99 (1989)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Mata, C., Mitchell, J.S.B.: Approximation algorithms for geometric tour and network design problems. In: Proceedings of the 11th Annual ACM Symposium on Computational Geometry, pp. 360–369 (1995)Google Scholar
  11. 11.
    Mata, C.S., Mitchell, J.S.B.: Approximation algorithms for geometric tour and network design problems (extended abstract). In: Proceedings of the Eleventh Annual Symposium on Computational Geometry, SCG 1995, pp. 360–369. ACM, New York (1995)Google Scholar
  12. 12.
    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)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Mitchell, J.S.B.: A PTAS for TSP with neighborhoods among fat regions in the plane. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pp. 11–18 (2007)Google Scholar
  14. 14.
    Mitchell, J.S.B.: A constant-factor approximation algorithm for TSP with pairwise-disjoint connected neighborhoods in the plane. In: Proceedings of the 26th Annual ACM Symposium on Computational Geometry, pp. 183–191 (2010)Google Scholar
  15. 15.
    Papadimitriou, C.H.: The Euclidean travelling salesman problem is NP-complete. Theor. Comput. Sci. 4(3), 237–244 (1977)CrossRefMATHGoogle Scholar
  16. 16.
    Ratnasamy, S., Karp, B., Yin, L., Yu, F., Estrin, D., Govindan, R., Shenker, S.: GHT: a geographic hash table for data-centric storage in sensornets. In: Proceedings 1st ACM Workshop on Wireless Sensor Networks ands Applications, pp. 78–87 (2002)Google Scholar
  17. 17.
    Safra, S., Schwartz, O.: On the complexity of approximating TSP with neighborhoods and related problems. Comput. Complex. 14(4), 281–307 (2006)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Sarkar, R., Zhu, X., Gao, J.: Double rulings for information brokerage in sensor networks. IEEE/ACM Trans. Netw. 17(6), 1902–1915 (2009)CrossRefGoogle Scholar
  19. 19.
    Slavík, P.: The errand scheduling problem. Technical report 97-2, Department of Computer Science, SUNY, Buffalo (1997)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Esther M. Arkin
    • 1
  • Jie Gao
    • 1
  • Adam Hesterberg
    • 2
  • Joseph S. B. Mitchell
    • 1
  • Jiemin Zeng
    • 1
  1. 1.Stony Brook UniversityStony BrookUSA
  2. 2.Massachusetts Institute of TechnologyBostonUSA

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