TY - GEN

T1 - Efficient computation of representative weight functions with applications to parameterized counting (extended version)

AU - Lokshtanov, Daniel

AU - Saurabh, Saket

AU - Zehavi, Meirav

N1 - Publisher Copyright:
Copyright © 2021 by SIAM

PY - 2021/1/1

Y1 - 2021/1/1

N2 - 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((2.619k + no(1)) · ε 12 · (n+m)). This improves over an earlier algorithm of Brand et al. [STOC 2018] that runs in time O(4k·kO(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((3.841k + |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 4.075k

AB - 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((2.619k + no(1)) · ε 12 · (n+m)). This improves over an earlier algorithm of Brand et al. [STOC 2018] that runs in time O(4k·kO(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((3.841k + |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 4.075k

UR - http://www.scopus.com/inward/record.url?scp=85105274816&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85105274816

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 179

EP - 198

BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

A2 - Marx, Daniel

PB - Association for Computing Machinery

T2 - 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

Y2 - 10 January 2021 through 13 January 2021

ER -