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)

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 \(O^{*}(\binom{n}{\downarrow d})\) 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].

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Copyright information

© 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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