Fast Exact Algorithms Using Hadamard Product of Polynomials

Let C be an arithmetic circuit of size s, given as input that computes a polynomial \(f\in {\mathbb {F}}[x_1,x_2,\ldots ,x_n]\), where \({\mathbb {F}}\) is finite field or the rationals. Using Hadamard product polynomials, we obtain new algorithms for following two problems first studied by Koutis and Williams (Faster algebraic path packing problems, 2008, https://doi.org/10.1007/978-3-540-70575-8_47; ACM Trans Algorithms 12(3):31:1–31:18, 2016, https://doi.org/10.1145/2885499; Inf Process Lett 109(6):315–318, 2009, https://doi.org/10.1016/j.ipl.2008.11.004): \({{{(\textit{k,n}){-}\mathrm{M{L}\normalsize {C}}}}}\): problem computing sum coefficients all degree-k multilinear monomials in f. We deterministic algorithm running time \({n\atopwithdelims (){\downarrow k/2}}\cdot n^{O(\log k)}\cdot s^{O(1)}\). This improvement over \(O(n^k)\) brute-force search answers positively question (2016). As applications, give exact counting algorithms, faster than search, number copies tree k graph, also m-dimensional k-matchings. \({{{\textit{k}{-}\mathrm{M{M}\normalsize {D}}}}}\): checking if there monomial f with non-zero coefficient. randomized \(O(4.32^k\cdot n^{O(1)})\). Additionally, our space bounded. Other results include fast {C}}}}}\) {D}}}}}\) depth three circuits.