Algorithmica

, Volume 58, Issue 3, pp 790–810 | Cite as

Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions

  • Frederic Dorn
  • Eelko Penninkx
  • Hans L. Bodlaender
  • Fedor V. Fomin
Article
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Abstract

We present a general framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour and Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. To exemplify our approach we show how to obtain an  \(O(2^{6.903\sqrt{n}})\) time algorithm solving weighted Hamiltonian Cycle on an n-vertex planar graph. Similar technique solves Planar Graph Travelling Salesman Problem with n cities in time \(O(2^{9.8594\sqrt{n}})\) . Our approach can be used to design parameterized algorithms as well. For example, we give an algorithm that for a given k decides if a planar graph on n vertices has a cycle of length at least k in time \(O(2^{13.6\sqrt{k}}n+n^{3})\) .

Keywords

Exact and parameterized algorithms Planar graphs Treewidth Branchwidth Traveling salesman problem Hamiltonian cycle 
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References

  1. 1.
    Alber, J., Bodlaender, H.L., Fernau, H., Kloks, T., Niedermeier, R.: Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica 33(4), 461–493 (2002) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alber, J., Fernau, H., Niedermeier, R.: Graph separators: a parameterized view. J. Comput. Syst. Sci. 67(4), 808–832 (2003) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alber, J., Fernau, H., Niedermeier, R.: Parameterized complexity: exponential speed-up for planar graph problems. J. Algorithms 52(1), 26–56 (2004) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Arora, S., Grigni, M., Karger, D., Klein, P.N., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1998), pp. 33–41. ACM, New York (1998) Google Scholar
  5. 5.
    Bian, Z., Gu, Q.-P., Marzban, M., Tamaki, H., Yoshitake, Y.: Study on branchwidth and branch decomposition of planar graphs. In: Proceedings of the 10th Workshop on Algorithm Engineering and Experiments (ALENEX 2008), pp. 152–165. ACM, New York (2008) Google Scholar
  6. 6.
    Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybern. 11, 1–21 (1993) MATHMathSciNetGoogle Scholar
  7. 7.
    Cook, W., Seymour, P.: Tour merging via branch-decomposition. INFORMS J. Comput. 15, 233–248 (2003) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Deĭneko, V.G., Klinz, B., Woeginger, G.J.: Exact algorithms for the Hamiltonian cycle problem in planar graphs. Oper. Res. Lett. 34(2), 269–274 (2006) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M.: Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs. J. Assoc. Comput. Math. 52(6), 866–893 (2005) MathSciNetGoogle Scholar
  10. 10.
    Demaine, E.D., Hajiaghayi, M.T.: Bidimensionality: new connections between FPT algorithms and PTASs. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), pp. 590–601. ACM, New York (2005) Google Scholar
  11. 11.
    Demaine, E.D., Hajiaghayi, M.T.: The bidimensionality theory and its algorithmic applications. Comput. J. 51(3), 292–302 (2008) CrossRefGoogle Scholar
  12. 12.
    Dorn, F., Fomin, F.V., Thilikos, D.M.: Fast subexponential algorithm for non-local problems on graphs of bounded genus. In: Proceedings of the 10th Scandinavian Workshop on Algorithm Theory (SWAT 2006). LNCS, vol. 4059, pp. 172–183. Springer, Berlin (2006) Google Scholar
  13. 13.
    Dorn, F., Fomin, F.V., Thilikos, D.M.: Catalan structures and dynamic programming on H-minor-free graphs. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 631–640. ACM, New York (2008) Google Scholar
  14. 14.
    Dorn, F., Fomin, F.V., Thilikos, D.M.: Subexponential parameterized algorithms. Comput. Sci. Rev. 2(1), 29–39 (2008) CrossRefGoogle Scholar
  15. 15.
    Dorn, F., Penninkx, E., Bodlaender, H.L., Fomin, F.V.: Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions. In: Proceedings of the 13th Annual European Symposium on Algorithms (ESA 2005). LNCS, vol. 3669, pp. 95–106. Springer, Berlin (2005) Google Scholar
  16. 16.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999) Google Scholar
  17. 17.
    Eppstein, D.: Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms Appl. 3, 1–27 (1999) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Fomin, F.V., Thilikos, D.M.: New upper bounds on the decomposability of planar graphs. J. Graph Theory 51(1), 53–81 (2006) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979) MATHGoogle Scholar
  20. 20.
    Gu, Q.-P., Tamaki, H.: Optimal branch-decomposition of planar graphs in O(n 3) time. ACM Trans. Algorithms 4(3) (2008). Article 30 Google Scholar
  21. 21.
    Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. SIAM 10, 196–210 (1962) MATHMathSciNetGoogle Scholar
  22. 22.
    Hwang, R.Z., Chang, R.C., Lee, R.C.T.: The searching over separators strategy to solve some NP-hard problems in subexponential time. Algorithmica 9(4), 398–423 (1993) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kloks, T., Lee, C.M., Liu, J.: New algorithms for k-face cover, k-feedback vertex set, and k-disjoint cycles on plane and planar graphs. In: Proceedings of the 28th International Workshop on Graph-theoretic Concepts in Computer Science (WG 2002). Lecture Notes in Comput. Sci., vol. 2573, pp. 282–295. Springer, Berlin (2002) CrossRefGoogle Scholar
  24. 24.
    Kreweras, G.: Sur les partitions non croisées d’un circle. Discrete Math. 1(4), 333–350 (1972) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G. (eds.): The Traveling Salesman Problem. Wiley, New York (1985) MATHGoogle Scholar
  26. 26.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36(2), 177–189 (1979) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9(3), 615–627 (1980) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Marzban, M., Gu, Q.-P., Jia, X.: Computational study on dominating set problem of planar graphs. In: Proceedings of the Second International Conference on Combinatorial Optimization and Applications (COCOA 2008). Lecture Notes in Comput. Sci., vol. 5165, pp. 89–102. Springer, Berlin (2008) Google Scholar
  29. 29.
    Robertson, N., Seymour, P., Thomas, R.: Quickly excluding a planar graph. J. Comb. Theory, Ser. B 62, 323–348 (1994) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Robertson, N., Seymour, P.D.: Graph minors X. Obstructions to tree-decomposition. J. Comb. Theory, Ser. B 52(2), 153–190 (1991) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Seymour, P., Thomas, R.: Call routing and the ratcatcher. Combinatorica 15, 217–241 (1994) CrossRefMathSciNetGoogle Scholar
  32. 32.
    Smith, W.D., Wormald, N.C.: Geometric separator theorems and applications. In: Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS 1998), pp. 232–243. IEEE Comput. Soc., Los Alamitos (1998) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Frederic Dorn
    • 1
  • Eelko Penninkx
    • 2
  • Hans L. Bodlaender
    • 2
  • Fedor V. Fomin
    • 1
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands

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