Exponential time complexity of the permanent and the Tutte polynomial

Abstract

The Exponential Time Hypothesis (ETH) says that deciding the satisfiability of n-variable 3-CNF formulas requires time exp (Ω(n)). We relax this hypothesis by introducing its counting version #ETH, namely that every algorithm that counts the satisfying assignments requires time exp (Ω(n)). We transfer the sparsification lemma for d-CNF formulas to the counting setting, which makes #ETH robust. Under this hypothesis, we show lower bounds for well-studied #P-hard problems: Computing the permanent of an n×n matrix with m nonzero entries requires time exp (Ω(n)). Restricted to 01-matrices, the bound is exp(Ω(m/log m) . Computing the Tutte polynomial of a multigraph with n vertices and m edges requires time exp (Ω(n)) at points (x,y) with (x-1)(y-1)≠1 and y ∈ {0,±1}. At points (x,0) with x ∉ {0, ± 1} it requires time exp(Ω(n)), and if x=-2,-3,..., it requires time exp(Ω(m)). For simple graphs, the bound is exp(Ω(m/log3m)). © 2010 Springer-Verlag Berlin Heidelberg.