Exact Parameterized Multilinear Monomial Counting via k-Layer Subset Convolution and k-Disjoint Sum

  • Dongxiao Yu
  • Yuexuan Wang
  • Qiang-Sheng Hua
  • Francis C. M. Lau
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6842)


We present new algorithms for exact multilinear k-monomial counting which is to compute the sum of coefficients of all degree-k multilinear monomials in a given polynomial P over a ring R described by an arithmetic circuit C. If the polynomial can be represented as a product of two polynomials with degree at most d < k, our algorithm can solve this problem in \(O^{*}(\binom{n}{\downarrow d})\) time, where \(\binom{n}{\downarrow d}=\sum_{i=0}^d\binom{n}{i}\). O * omits a polynomial factor in n. For the general case, the proposed algorithm takes time \(O^{*}(\binom{n}{\downarrow k})\). In both cases, our results are superior to previous approaches presented in [Koutis, I. and Williams, R.: Limits and applications of group algebras for parameterized problems. ICALP, pages 653-664 (2009)]. We also present a polynomial space algorithm with time bound \(O^{*}(2^k\binom{n}{k})\). By reducing the #k-path problem and the #m-set k-packing problem to the exact multilinear k-monomial counting problem, we give algorithms for these two problems that match the fastest known results presented in [2].


Group Algebra Packing Problem Decision Version Polynomial Space Counting Problem 
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  1. 1.
    Alon, N., Yuster, R., Zwick, U.: Color coding. Journal of the ACM 42(4), 844–856 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Counting paths and packings in halves. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 578–586. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets möbius: fast subset convolution. In: STOC, pp. 67–74 (2007)Google Scholar
  4. 4.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Trimmed moebius inversion and graphs of bounded degree. Theory Comput. Syst. 47(3), 637–654 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, J., Lu, S., Sze, S.-H., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: SODA, pp. 298–307 (2007)Google Scholar
  6. 6.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symbolic Computation 9(3), 251–280 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33, 892–922 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jia, W., Zhang, C., Chen, J.: An efficient parameterized algorithm for m-set packing. J. Algorithms 50, 106–117 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kennes, R.: Computational aspects of the Moebius transform of a graph. IEEE Transactions on Systems, Man, and Cybernetics 22, 201–223 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Koutis, I.: Faster algebraic algorithms for path and packing problems. In: ICALP, pp. 575–586 (2009)Google Scholar
  13. 13.
    Koutis, I., Williams, R.: Limits and applications of group algebras for parameterized problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 653–664. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Lokshtanov, D., Nederlof, J.: Saving space by algebraization. In: STOC, pp. 321–330 (2010)Google Scholar
  15. 15.
    Vassilevska, V., Williams, R.: Finding, minimizing, and counting weighted subgraphs. In: STOC, pp. 455–464 (2009)Google Scholar
  16. 16.
    Williams, R.: Finding paths of length k in O *(2k) time. Inf. Process. Lett. 109(6), 315–318 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yates, F.: The design and analysis of factorial experiments, Technical Communication No. 35, Commonwealth Bureau of Soil Science, Harpenden, UK (1937)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dongxiao Yu
    • 1
  • Yuexuan Wang
    • 2
  • Qiang-Sheng Hua
    • 2
    • 1
  • Francis C. M. Lau
    • 1
  1. 1.Department of Computer ScienceThe University of Hong KongPokfulamHong Kong
  2. 2.Institute for Interdisciplinary Information SciencesTsinghua UniversityBeijingP.R. China

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