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Fast Sub-exponential Algorithms and Compactness in Planar Graphs

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Algorithms – ESA 2011 (ESA 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6942))

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Abstract

We provide a new theory, alternative to bidimensionality, of sub-exponential parameterized algorithms on planar graphs, which is based on the notion of compactness. Roughly speaking, a parameterized problem is (r,q)-compact when all the faces and vertices of its YES-instances are “r-radially dominated” by some vertex set whose size is at most q times the parameter. We prove that if a parameterized problem can be solved in \(c^{\mbox{\scriptsize branchwidth}(G)}n^{O(1)}\) steps and is (r,q)-compact, then it can be solved by a \(c^{\, r\cdot 2.122\cdot \sqrt{q\cdot k} }n^{O(1)}\) step algorithm (where k is the parameter). Our framework is general enough to unify the analysis of almost all known sub-exponential parameterized algorithms on planar graphs and improves or matches their running times. Our results are based on an improved combinatorial bound on the branchwidth of planar graphs that bypasses the grid-minor exclusion theorem. That way, our approach encompasses new problems where bidimensionality theory do not directly provide sub-exponential parameterized algorithms.

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References

  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, 461–493 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon, N., Lokshtanov, D., Saurabh, S.: Fast FAST. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 49–58. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  3. Alon, N., Seymour, P., Thomas, R.: Planar separators. SIAM J. Discrete Math. 7, 184–193 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) Kernelization. In: 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), pp. 629–638. IEEE, Los Alamitos (2009)

    Chapter  Google Scholar 

  5. Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. Journal of Computer and System Sciences 67, 789–807 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, J., Kanj, I., Perkovic, L., Sedgwick, E., Xia, G.: Genus characterizes the complexity of graph problems: some tight results. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 845–856. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  7. Demaine, E., Hajiaghayi, M.: The bidimensionality theory and its algorithmic applications. The Computer Journal 51, 292–302 (2007)

    Article  Google Scholar 

  8. Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Bidimensional parameters and local treewidth. SIAM J. Discrete Math. 18, 501–511 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Fixed-parameter algorithms for (k,r)-center in planar graphs and map graphs. ACM Trans. Algorithms 1, 33–47 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. Assoc. Comput. Mach. 52, 866–893 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Demaine, E.D., Hajiaghayi, M.: Bidimensionality: new connections between FPT algorithms and PTASs. In: Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 590–601. ACM, New York (2005) (electronic)

    Google Scholar 

  12. Demaine, E.D., Hajiaghayi, M., Thilikos, D.M.: Exponential speedup of fixed-parameter algorithms for classes of graphs excluding single-crossing graphs as minors. Algorithmica 41, 245–267 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dorn, F.: Dynamic Programming and Fast Matrix Multiplication. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 280–291. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  14. Dorn, F., Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S.: Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs. In: 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010). LIPIcs, vol. 5, pp. 251–262. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2010)

    Google Scholar 

  15. Dorn, F., Fomin, F.V., Thilikos, D.M.: Subexponential parameterized algorithms. Computer Science Review 2, 29–39 (2008)

    Article  MATH  Google Scholar 

  16. Fernau, H.: Graph Separator Algorithms: A Refined Analysis. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 186–197. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  17. Fernau, H., Juedes, D.: A Geometric Approach to Parameterized Algorithms for Domination Problems on Planar Graphs. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 488–499. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  18. Fomin, F.V., Golovach, P., Thilikos, D.M.: Contraction bidimensionality: The accurate picture. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 706–717. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  19. Fomin, F.V., Lokshtanov, D., Raman, V., Saurabh, S.: Bidimensionality and EPTAS. In: 22st ACM–SIAM Symposium on Discrete Algorithms (SODA 2011), pp. 748–759. ACM-SIAM, San Francisco (2011)

    Google Scholar 

  20. Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2010), Austin, Texas, pp. 503–510. ACM-SIAM (2010)

    Google Scholar 

  21. Fomin, F.V., Thilikos, D.M.: ominating sets in planar graphs: branch-width and exponential speed-up. SIAM J. Comput. 36, 281–309 (2006) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fomin, F.V., Thilikos, D.M.: New upper bounds on the decomposability of planar graphs. Journal of Graph Theory 51, 53–81 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fomin, F.V., Villanger, Y.: Subexponential parameterized algorithm for minimum fill-in. CoRR, abs/1104.2230 (2011)

    Google Scholar 

  24. Gu, Q.-P., Tamaki, H.: Optimal branch decomposition of planar graphs in O(n 3) time. ACM Trans. Algorithms 4, 1–13 (2008)

    Article  MathSciNet  Google Scholar 

  25. Gu, Q.-P., Tamaki, H.: Improved Bounds on the Planar Branchwidth with Respect to the Largest Grid Minor Size. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part II. LNCS, vol. 6507, pp. 85–96. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  26. Guo, J., Niedermeier, R.: Linear Problem Kernels for NP-Hard Problems on Planar Graphs. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 375–386. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  27. Kanj, I., Perković, L.: Improved Parameterized Algorithms for Planar Dominating Set. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 399–410. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  28. 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: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 282–295. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  29. Koutsonas, A., Thilikos, D.: Planar feedback vertex set and face cover: Combinatorial bounds and subexponential algorithms. Algorithmica, 1–17 (2010)

    Google Scholar 

  30. Lokshtanov, D., Marx, D., Saurabh, S.: Known algorithms on graphs of bounded treewidth are probably optimal. In: 22st ACM–SIAM Symposium on Discrete Algorithms (SODA 2011), pp. 777–789. ACM-SIAM, San Francisco, California (2011)

    Google Scholar 

  31. Mazoit, F., Thomassé, S.: Branchwidth of graphic matroids. In: Surveys in combinatorics 2007. London Math. Soc. Lecture Note Ser., vol. 346, pp. 275–286. Cambridge Univ. Press, Cambridge (2007)

    Google Scholar 

  32. Pilipczuk, M.: Problems parameterized by treewidth tractable in single exponential time: a logical approach. Tech. Rep. arXiv:1104.3057, Cornell University (April 2011)

    Google Scholar 

  33. Robertson, N., Seymour, P., Thomas, R.: Quickly excluding a planar graph. J. Combin. Theory Ser. B 62, 323–348 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Combin. Theory Ser. B 52, 153–190 (1991)

    Google Scholar 

  35. Seymour, P.D., Thomas, R.: Call routing and the ratcatcher. Combinatorica 14, 217–241 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tazari, S.: Faster approximation schemes and parameterized algorithms on H-minor-free and odd-minor-free graphs. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 641–652. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  37. van Rooij, J.M.M., Bodlaender, H.L., Rossmanith, P.: Dynamic Programming on Tree Decompositions Using Generalised Fast Subset Convolution. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 566–577. Springer, Heidelberg (2009)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, 15784, Athens, Greece

    Dimitrios M. Thilikos

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  1. Dimitrios M. Thilikos
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Editors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, University of Rome " La Sapienza", via Ariosto 25, 00185, Roma, Italy

    Camil Demetrescu

  2. School of Computer Science, Reykjavik University, Menntavegur 1, 101, Reykjavik, Iceland

    Magnús M. Halldórsson

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Thilikos, D.M. (2011). Fast Sub-exponential Algorithms and Compactness in Planar Graphs. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_31

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  • DOI: https://doi.org/10.1007/978-3-642-23719-5_31

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