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Counting Paths and Packings in Halves

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Algorithms - ESA 2009 (ESA 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5757))

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Abstract

We show that one can count k-edge paths in an n-vertex graph and m-set k-packings on an n-element universe, respectively, in time \({n \choose k/2}\) and \({n \choose mk/2}\), up to a factor polynomial in n, k, and m; in polynomial space, the bounds hold if multiplied by 3k/2 or 5mk/2, respectively. These are implications of a more general result: given two set families on an n-element universe, one can count the disjoint pairs of sets in the Cartesian product of the two families with O(n ℓ) basic operations, where ℓ is the number of members in the two families and their subsets.

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References

  1. Alon, N., Gutner, S.: Balanced hashing, color coding and approximate counting. Electronic Colloquium on Computational Complexity, Report TR09-12 (2009)

    Google Scholar 

  2. Alon, N., Yuster, R., Zwick, U.: Color-coding. J. Assoc. Comput. Mach. 42, 844–856 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bellman, R.: Combinatorial processes and dynamic programming. In: Bellman, R., Hall Jr., M. (eds.) Combinatorial Analysis. Proceedings of Symposia in Applied Mathematics 10, pp. 217–249. American Mathematical Society (1960)

    Google Scholar 

  4. Bellman, R.: Dynamic programming treatment of the travelling Salesman Problem. J. Assoc. Comput. Mach. 9, 61–63 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  5. Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion–exclusion. SIAM J. Comput. (to appear) Special Issue for FOCS 2006

    Google Scholar 

  6. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbius: fast subset convolution. In: 39th ACM Symposium on Theory of Computing (STOC 2007), pp. 67–74. ACM Press, New York (2007)

    Google Scholar 

  7. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Trimmed Moebius inversion and graphs of bounded degree. In: 25th International Symposium on Theoretical Aspects of Computer Science (STACS 2008). Dagstuhl Seminar Proceedings 08001, pp. 85–96. IBFI Schloss Dagstuhl (2008)

    Google Scholar 

  8. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: The fast intersection transform with applications to counting paths. CoRR, abs/0809.2489 (2008)

    Google Scholar 

  9. Chen, J., Lu, S., Sze, S.-H., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), Philadelphia, PA, USA, pp. 298–307. Society for Industrial and Applied Mathematics (2007)

    Google Scholar 

  10. Danzig, G.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)

    Book  Google Scholar 

  11. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  12. Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33, 892–922 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gurevich, Y., Shelah, S.: Expected computation time for Hamiltonian path problem. SIAM J. Comput. 16, 486–502 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. Soc. Indust. Appl. Math. 10, 196–210 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  15. Horowitz, E., Sahni, S.: Computing partitions with applications to the knapsack Problem. J. Assoc. Comput. Mach. 21, 277–292 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jia, W., Zhang, C., Chen, J.: An efficient parameterized algorithm for m-set packing. J. Algorithms 50, 106–117 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Karp, R.M.: Dynamic programming meets the principle of inclusion and exclusion. Oper. Res. Lett. 1, 49–51 (1982)

    Google Scholar 

  18. Kennes, R.: Computational aspects of the Moebius transform on a graph. IEEE Transactions on System, Man, and Cybernetics 22, 201–223 (1991)

    Google Scholar 

  19. Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Divide-and-color. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 58–67. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  20. Kohn, S., Gottlieb, A., Kohn, M.: A generating function approach to the traveling salesman problem. In: ACM Annual Conference (ACM 1977), pp. 294–300. ACM Press, New York (1977)

    Google Scholar 

  21. Koutis, I.: A faster parameterized algorithm for set packing. Information Processing Letters 94, 4–7 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Koutis, I.: Faster algebraic algorithms for path and packing problems. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 575–586. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  23. Koutis, I., Williams, R.: Limitations and applications of group algebras for parameterized problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. Part I. LNCS, vol. 5555, pp. 653–664. Springer, Heidelberg (2009)

    Google Scholar 

  24. Papadimitriou, C.H., Yannakakis, M.: On limited nondeterminism and the complexity of the V-C dimension. J. Comput. Syst. Sci. 53, 161–170 (1996)

    Google Scholar 

  25. Pohl, I.: Bi-directional and heuristic search in path problems. PhD thesis, Report SLAC-104, Stanford University (1969)

    Google Scholar 

  26. Vassilevska, V., Williams, R.: Finding, minimizing, and counting weighted subgraphs. In: 41st ACM Symposium on Theory of Computing (STOC 2009), pp. 455–464 (2009)

    Google Scholar 

  27. Williams, R.: Finding paths of length k in O*O*(2k) time. Information Processing Letters 109, 315–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Lund University, P.O. Box 118, SE-22100, Lund, Sweden

    Andreas Björklund & Thore Husfeldt

  2. IT University of Copenhagen, Rued Langgaards Vej 7, 2300, København S, Denmark

    Thore Husfeldt

  3. Helsinki Institute for Information Technology HIIT, Department of Computer Science, University of Helsinki, P.O. Box 68, FI-00014, Finland

    Petteri Kaski & Mikko Koivisto

Authors
  1. Andreas Björklund
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  2. Thore Husfeldt
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  3. Petteri Kaski
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  4. Mikko Koivisto
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Editor information

Editors and Affiliations

  1. School of Computer Science, Tel Aviv University, Tel Aviv, Israel

    Amos Fiat

  2. Fakultät für Informatik, Universität Karlsruhe (TH), Am Fasanengarten 5, 76131, Karlsruhe, Germany

    Peter Sanders

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© 2009 Springer-Verlag Berlin Heidelberg

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Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M. (2009). Counting Paths and Packings in Halves. In: Fiat, A., Sanders, P. (eds) Algorithms - ESA 2009. ESA 2009. Lecture Notes in Computer Science, vol 5757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04128-0_52

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  • DOI: https://doi.org/10.1007/978-3-642-04128-0_52

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04127-3

  • Online ISBN: 978-3-642-04128-0

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