מירב זהבי

In this paper we prove an analogue of the classic Bollobás lemma for approximate counting. In fact, we match an analogous result of Fomin et al. [JACM 2016] for decision. This immediately yields, for a number of fundamental problems, parameterized approximate counting algorithms with the same running times as what is obtained for the decision variant using the representative family technique of Fomin et al. [JACM 2016]. For example, we devise an algorithm for approximately counting (a factor (1 ± ε) approximation algorithm) k-paths in an n-vertex directed graph (#k-Path) running in time O((2.619^k + n^o(1)) · ε ¹₂ · (n+m)). This improves over an earlier algorithm of Brand et al. [STOC 2018] that runs in time O(4^k·k^O(1)· ε ¹₂ ·(n+m)). Additionally, we obtain an approximate counting analogue of the efficient computation of representative families for product families of Fomin et al. [TALG 2017], again essentially matching the running time for decision. This results in an algorithm with running time O((3.841^k + |I|^o(1)) · ε ¹₆ · |I|) for computing a (1 + ε) approximation of the sum of the coefficients of the multilinear monomials in a degree-k homogeneous n-variate polynomial encoded by a monotone circuit (#Multilinear Monomial Detection). When restricted to monotone circuits (rather than polynomials of non-negative coefficients), this improves upon an earlier algorithm of Pratt [FOCS 2019] that runs in time 4.075^k