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

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 ) = ∑d i=0 ( 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 kpacking 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].