Exact parameterized multilinear monomial counting via k-layer subset convolution and k-disjoint

  • Authors:

  • Dongxiao Yu;Yuexuan Wang;Qiang-Sheng Hua;Francis C. M. Lau

  • Affiliations:

  • Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong, P.R. China;Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, P.R. China;Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, P.R. China and Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong, P.R. China;Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong, P.R. China

  • Venue:
  • COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
  • Year:
  • 2011

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Abstract

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*((n ↓d)) time, where (n ↓d) = Σi=0d (n i). O* omits a polynomial factor in n. For the general case, the proposed algorithm takes time O*((n ↓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*(2k(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].