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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].

Keywords

Group Algebra Packing Problem Decision Version Polynomial Space Counting Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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