### Exact Parameterized Multilinear Monomial Counting via k-Layer Subset Convolution and k-Disjoint Sum [chapter]

Dongxiao Yu, Yuexuan Wang, Qiang-Sheng Hua, Francis C. M. Lau
2011 Lecture Notes in Computer Science
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
more » ... * ( 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 * (2 k 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 . Definition 1. (Exact multilinear k-monomial counting  ) Given a (commutative) arithmetic circuit C describing an n-variate polynomial P over a ring R, compute the sum of the coefficients of degree-k multilinear monomials in P . An arithmetic circuit C over a ring R and the set of variables x 1 , . . . , x n is a directed acyclic graph. Every node with in-degree zero is called an input gate and is labeled by either a variable x i or an element in R. Every other gate is labeled by either + or ×, for an addition gate or a product gate respectively. The size of the circuit C, denoted by |C|, is defined as the number of gates in C. If there is no confusion, for each operator gate, we call its in-neighbors input gates. As will be shown later in this paper (see Section 5.1), the #k-path problem can be reduced to the multilinear k-monomial counting problem. Hence, the #W -hardness of the #k-path problem with respect to the parameter k  implies that it is unlikely that an O(f (k)poly(n)) time algorithm for the exact multilinear k-monomial counting problem can be found. By making use of an algebraic method, Koutis and Williams in  gave an algorithm with running time O * (2 d n d +n k/2 ) for a special case where the polynomial P is a product of two polynomials P 1 , P 2 with degrees at most d. Their algorithm performs much worse for the general case, with a time complexity of O * (2 k n k ). However, some recent results indicate that one could do better. For example, Björklund et al. in  gave an O * ( n k/2 ) algorithm for the #k-path problem. In this paper, we affirmatively show that an improvement can be achieved, via the following result.