Walking Through Waypoints

  • Saeed Akhoondian Amiri
  • Klaus-Tycho Foerster
  • Stefan Schmid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

We initiate the study of a fundamental combinatorial problem: Given a capacitated graph \(G=(V,E)\), find a shortest walk (“route”) from a source \(s\in V\) to a destination \(t\in V\) that includes all vertices specified by a set \(\mathscr {W}\subseteq V\): the waypoints. This waypoint routing problem finds immediate applications in the context of modern networked distributed systems. Our main contribution is an exact polynomial-time algorithm for graphs of bounded treewidth. We also show that if the number of waypoints is logarithmically bounded, exact polynomial-time algorithms exist even for general graphs. Our two algorithms provide an almost complete characterization of what can be solved exactly in polynomial-time: we show that more general problems (e.g., on grid graphs of maximum degree 3, with slightly more waypoints) are computationally intractable.

Notes

Acknowledgments

The authors would like to thank Riko Jacob for helpful discussions and feedback. Klaus-Tycho Foerster’s and Stefan Schmid’s research was partly supported by the Villum project ReNet and by Aalborg University’s PreLytics project. Saeed Amiri’s research was partly supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648527).

References

  1. 1.
    Akiyama, T., Nishizeki, T., Saito, N.: NP-completeness of the Hamiltonian cycle problem for bipartite graphs. J. Inf. Process. 3(2), 73–76 (1980)MathSciNetMATHGoogle Scholar
  2. 2.
    Amiri, S.A., Foerster, K.-T., Jacob, R., Schmid, S.: Charting the complexity landscape of waypoint routing. arXiv preprint arXiv:1705.00055 (2017)
  3. 3.
    Amiri, S.A., Foerster, K.-T., Schmid, S.: Walking through waypoints. arXiv preprint arXiv:1708.09827 (2017)
  4. 4.
    Akhoondian Amiri, S., Golshani, A., Kreutzer, S., Siebertz, S.: Vertex disjoint paths in upward planar graphs. In: Hirsch, E.A., Kuznetsov, S.O., Pin, J.É., Vereshchagin, N.K. (eds.) CSR 2014. LNCS, vol. 8476, pp. 52–64. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-06686-8_5 Google Scholar
  5. 5.
    Arkin, E.M., Fekete, S.P., Islam, K., Meijer, H., Mitchell, J.S.B., Rodríguez, Y.N., Polishchuk, V., Rappaport, D., Xiao, H.: Not being (super) thin or solid is hard: a study of grid hamiltonicity. Comput. Geom. 42(6–7), 582–605 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math. 23(1), 11–24 (1989)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Björklund, A., Husfeld, T., Taslaman, N.: Shortest cycle through specified elements. In: Proceedings of SODA (2012)Google Scholar
  8. 8.
    Björklund, A., Husfeldt, T.: Shortest two disjoint paths in polynomial time. In: Proceedings of ICALP (2014)Google Scholar
  9. 9.
    Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: An approximation algorithm for treewidth. In: Proceedings of FOCS (2013)Google Scholar
  10. 10.
    Bodlaender, H.L.: Dynamic programming on graphs with bounded treewidth. In: Lepistö, T., Salomaa, A. (eds.) ICALP 1988. LNCS, vol. 317, pp. 105–118. Springer, Heidelberg (1988).  https://doi.org/10.1007/3-540-19488-6_110 CrossRefGoogle Scholar
  11. 11.
    Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybern. 11(1–2), 1–21 (1993)MathSciNetMATHGoogle Scholar
  12. 12.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bodlaender, H.L., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput. 243, 86–111 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Borradaile, G., Demaine, E.D., Tazari, S.: Polynomial-time approximation schemes for subset-connectivity problems in bounded-genus graphs. Algorithmica 68(2), 287–311 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Buro, M.: Simple Amazons endgames and their connection to Hamilton circuits in cubic subgrid graphs. In: Marsland, T., Frank, I. (eds.) CG 2000. LNCS, vol. 2063, pp. 250–261. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-45579-5_17 CrossRefGoogle Scholar
  16. 16.
    Chekuri, C., Khanna, S., Shepherd, F.B.: A note on multiflows and treewidth. Algorithmica 54(3), 400–412 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21275-3 CrossRefMATHGoogle Scholar
  18. 18.
    Cygan, M., Marx, D., Pilipczuk, M., Pilipczuk, M.: The planar directed k-vertex-disjoint paths problem is fixed-parameter tractable. In: Proceedings of FOCS (2013)Google Scholar
  19. 19.
    de Verdière, E.C., Schrijver, A.: Shortest vertex-disjoint two-face paths in planar graphs. ACM Trans. Algorithms (TALG) 7(2), 19 (2011)MathSciNetMATHGoogle Scholar
  20. 20.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999).  https://doi.org/10.1007/978-1-4612-0515-9 CrossRefMATHGoogle Scholar
  21. 21.
    Eilam-Tzoreff, T.: The disjoint shortest paths problem. Discrete Appl. Math. 85(2), 113–138 (1998)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ene, A., Mnich, M., Pilipczuk, M., Risteski, A.: On routing disjoint paths in bounded treewidth graphs. In: Proceedings of SWAT (2016)Google Scholar
  23. 23.
    ETSI: Network functions virtualisation. White Paper, October 2013Google Scholar
  24. 24.
    ETSI: Network functions virtualisation (NFV); use cases. http://www.etsi.org/deliver/etsi_gs/NFV/001_099/001/01.01.01_60/gs_NFV001v010101p.pdf (2014)
  25. 25.
    Even, G., Medina, M., Patt-Shamir, B.: On-line path computation and function placement in SDNs. In: Bonakdarpour, B., Petit, F. (eds.) SSS 2016. LNCS, vol. 10083, pp. 131–147. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-49259-9_11 CrossRefGoogle Scholar
  26. 26.
    Even, G., Rost, M., Schmid, S.: An approximation algorithm for path computation and function placement in SDNs. In: Suomela, J. (ed.) SIROCCO 2016. LNCS, vol. 9988, pp. 374–390. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-48314-6_24 CrossRefGoogle Scholar
  27. 27.
    Feamster, N., Rexford, J., Zegura, E.: The road to SDN. Queue 11(12), 1–21 (2013)CrossRefGoogle Scholar
  28. 28.
    Fellows, M., Fomin, F.V., Lokshtanov, D., Rosamond, F., Saurabh, S., Szeider, S., Thomassen, C.: On the complexity of some colorful problems parameterized by treewidth. In: Dress, A., Xu, Y., Zhu, B. (eds.) COCOA 2007. LNCS, vol. 4616, pp. 366–377. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-73556-4_38 CrossRefGoogle Scholar
  29. 29.
    Fenner, T., Lachish, O., Popa, A.: Min-sum 2-paths problems. Theor. Comp. Sys. 58(1), 94–110 (2016)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Fleischner, H., Woeginger, G.J.: Detecting cycles through three fixed vertices in a graph. Inf. Process. Lett. 42(1), 29–33 (1992)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Foerster, K.-T., Parham, M., Schmid, S.: A walk in the clouds: routing through VNFs on bidirected networks. In: Proceedings of ALGOCLOUD (2017)Google Scholar
  32. 32.
    Fortune, S., Hopcroft, J.E., Wyllie, J.: The directed subgraph homeomorphism problem. Theor. Comput. Sci. 10, 111–121 (1980)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Itai, A., Perl, Y., Shiloach, Y.: The complexity of finding maximum disjoint paths with length constraints. Networks 12(3), 277–286 (1982)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Karp, R.M.: On the computational complexity of combinatorial problems. Networks 5(1), 45–68 (1975)CrossRefMATHGoogle Scholar
  35. 35.
    Kawarabayashi, K.: An improved algorithm for finding cycles through elements. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 374–384. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-68891-4_26 CrossRefGoogle Scholar
  36. 36.
    Khuller, S., Mitchell, S.G., Vazirani, V.V.: Processor efficient parallel algorithms for the two disjoint paths problem and for finding a kuratowski homeomorph. SIAM J. Comput. 21(3), 486–506 (1992)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Khuller, S., Schieber, B.: Efficient parallel algorithms for testing k-connectivity and finding disjoint s-t paths in graphs. SIAM J. Comput. 20(2), 352–375 (1991)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Klein, P.N., Marx, D.: A subexponential parameterized algorithm for subset TSP on planar graphs. In: Proceedings of SODA (2014)Google Scholar
  39. 39.
    Kloks, T. (ed.): Treewidth, Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994).  https://doi.org/10.1007/BFb0045375 MATHGoogle Scholar
  40. 40.
    Kobayashi, Y., Sommer, C.: On shortest disjoint paths in planar graphs. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 293–302. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-10631-6_31 CrossRefGoogle Scholar
  41. 41.
    Schrijver, A., Lovasz, L.: Paths, Flows, and VLSI-Layout. Springer-Verlag New York, Inc., Secaucus (1990). Korte, B., Promel, H.J., Graham, R.L. (eds.). ISBN 0387526854MATHGoogle Scholar
  42. 42.
    Marx, D.: List edge multicoloring in graphs with few cycles. Inf. Process. Lett. 89(2), 85–90 (2004)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Nishizeki, T., Vygen, J., Zhou, X.: The edge-disjoint paths problem is NP-complete for series-parallel graphs. Discrete Appl. Math. 115, 177–186 (2001)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Ogier, R.G., Rutenburg, V., Shacham, N.: Distributed algorithms for computing shortest pairs of disjoint paths. IEEE Trans. Inf. Theory 39(2), 443–455 (1993)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Ohtsuki, T.: The two disjoint path problem and wire routing design. In: Saito, N., Nishizeki, T. (eds.) Graph Theory and Algorithms. LNCS, vol. 108, pp. 207–216. Springer, Heidelberg (1981).  https://doi.org/10.1007/3-540-10704-5_18 CrossRefGoogle Scholar
  46. 46.
    Papadimitriou, C.H., Vazirani, U.V.: On two geometric problems related to the traveling salesman problem. J. Algorithms 5(2), 231–246 (1984)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Perković, L., Reed, B.A.: An improved algorithm for finding tree decompositions of small width. Int. J. Found. Comput. Sci. 11(3), 365–371 (2000)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Robertson, N., Seymour, P.D.: Graph minors .XIII. The disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995)Google Scholar
  49. 49.
    Rost, M., Schmid, S.: Service chain and virtual network embeddings: approximations using randomized rounding. arXiv preprint arXiv:1604.02180 (2016)
  50. 50.
    Saltzer, J.H., Reed, D.P., Clark, D.D.: End-to-end arguments in system design. ACM Trans. Comput. Syst. 2(4), 277–288 (1984)CrossRefGoogle Scholar
  51. 51.
    Scheffler, P.: A practical linear time algorithm for disjoint paths in graphs with bounded tree-width. Technical report, TU Berlin (1994)Google Scholar
  52. 52.
    Schrijver, A.: Finding k disjoint paths in a directed planar graph. SIAM J. Comput. 23(4), 780–788 (1994)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Sebö, A., van Zuylen, A.: The salesman’s improved paths: A 3/2+1/34 approximation. In: Proceedings of FOCS (2016)Google Scholar
  54. 54.
    Seymour, D.P.: Disjoint paths in graphs. Discrete Math. 29(3), 293–309 (1980)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Shiloach, Y.: A polynomial solution to the undirected two paths problem. J. ACM 27(3), 445–456 (1980)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Srinivas, A., Modiano, E.: Finding minimum energy disjoint paths in wireless ad-hoc networks. Wireless Netw. 11(4), 401–417 (2005)CrossRefGoogle Scholar
  57. 57.
    Thomassen, C.: 2-linked graphs. Europ. J. Comb. 1(4), 371–378 (1980)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.TU Berlin and Max Planck Institute for Informatics (Saarland)SaarbrückenGermany
  2. 2.University of ViennaViennaAustria

Personalised recommendations