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

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].