Skip to main content
SpringerLink
Log in
Book cover

Latin American Symposium on Theoretical Informatics

LATIN 2018: LATIN 2018: Theoretical Informatics pp 37–51Cite as

Walking Through Waypoints

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    A preliminary full version is provided at [3].

References

  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)

    MathSciNet  MATH  Google Scholar 

  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. Amiri, S.A., Foerster, K.-T., Schmid, S.: Walking through waypoints. arXiv preprint arXiv:1708.09827 (2017)

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Björklund, A., Husfeld, T., Taslaman, N.: Shortest cycle through specified elements. In: Proceedings of SODA (2012)

    Google Scholar 

  8. Björklund, A., Husfeldt, T.: Shortest two disjoint paths in polynomial time. In: Proceedings of ICALP (2014)

    Google Scholar 

  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. 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

    Chapter  Google Scholar 

  11. Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybern. 11(1–2), 1–21 (1993)

    MathSciNet  MATH  Google Scholar 

  12. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  16. Chekuri, C., Khanna, S., Shepherd, F.B.: A note on multiflows and treewidth. Algorithmica 54(3), 400–412 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Book  MATH  Google Scholar 

  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. de Verdière, E.C., Schrijver, A.: Shortest vertex-disjoint two-face paths in planar graphs. ACM Trans. Algorithms (TALG) 7(2), 19 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999). https://doi.org/10.1007/978-1-4612-0515-9

    Book  MATH  Google Scholar 

  21. Eilam-Tzoreff, T.: The disjoint shortest paths problem. Discrete Appl. Math. 85(2), 113–138 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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. ETSI: Network functions virtualisation. White Paper, October 2013

    Google Scholar 

  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. 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

    Chapter  Google Scholar 

  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

    Chapter  Google Scholar 

  27. Feamster, N., Rexford, J., Zegura, E.: The road to SDN. Queue 11(12), 1–21 (2013)

    Article  Google Scholar 

  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

    Chapter  Google Scholar 

  29. Fenner, T., Lachish, O., Popa, A.: Min-sum 2-paths problems. Theor. Comp. Sys. 58(1), 94–110 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Fleischner, H., Woeginger, G.J.: Detecting cycles through three fixed vertices in a graph. Inf. Process. Lett. 42(1), 29–33 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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. Fortune, S., Hopcroft, J.E., Wyllie, J.: The directed subgraph homeomorphism problem. Theor. Comput. Sci. 10, 111–121 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  33. Itai, A., Perl, Y., Shiloach, Y.: The complexity of finding maximum disjoint paths with length constraints. Networks 12(3), 277–286 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  34. Karp, R.M.: On the computational complexity of combinatorial problems. Networks 5(1), 45–68 (1975)

    Article  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  38. Klein, P.N., Marx, D.: A subexponential parameterized algorithm for subset TSP on planar graphs. In: Proceedings of SODA (2014)

    Google Scholar 

  39. Kloks, T. (ed.): Treewidth, Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994). https://doi.org/10.1007/BFb0045375

    MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  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 0387526854

    MATH  Google Scholar 

  42. Marx, D.: List edge multicoloring in graphs with few cycles. Inf. Process. Lett. 89(2), 85–90 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  46. Papadimitriou, C.H., Vazirani, U.V.: On two geometric problems related to the traveling salesman problem. J. Algorithms 5(2), 231–246 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Rost, M., Schmid, S.: Service chain and virtual network embeddings: approximations using randomized rounding. arXiv preprint arXiv:1604.02180 (2016)

  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)

    Article  Google Scholar 

  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. Schrijver, A.: Finding k disjoint paths in a directed planar graph. SIAM J. Comput. 23(4), 780–788 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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. Seymour, D.P.: Disjoint paths in graphs. Discrete Math. 29(3), 293–309 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  55. Shiloach, Y.: A polynomial solution to the undirected two paths problem. J. ACM 27(3), 445–456 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  56. Srinivas, A., Modiano, E.: Finding minimum energy disjoint paths in wireless ad-hoc networks. Wireless Netw. 11(4), 401–417 (2005)

    Article  Google Scholar 

  57. Thomassen, C.: 2-linked graphs. Europ. J. Comb. 1(4), 371–378 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

  1. TU Berlin and Max Planck Institute for Informatics (Saarland), Saarbrücken, Germany

    Saeed Akhoondian Amiri

  2. University of Vienna, Vienna, Austria

    Klaus-Tycho Foerster & Stefan Schmid

Authors
  1. Saeed Akhoondian Amiri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Klaus-Tycho Foerster
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Schmid
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Klaus-Tycho Foerster .

Editor information

Editors and Affiliations

  1. Stony Brook University, Stony Brook, New York, USA

    Michael A. Bender

  2. Rutgers University, New Brunswick, New Jersey, USA

    Martín Farach-Colton

  3. Pace University, New York, New York, USA

    Miguel A. Mosteiro

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Akhoondian Amiri, S., Foerster, KT., Schmid, S. (2018). Walking Through Waypoints. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-77404-6_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-77403-9

  • Online ISBN: 978-3-319-77404-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation