Exponential Time Complexity of the Permanent and the Tutte Polynomial

Abstract

We show conditional lower bounds for well-studied #P-hard problems: The number of satisfying assignments of a 2-CNF formula with n variables cannot be computed in time exp( o ( n )), and the same is true for computing the number of all independent sets in an n -vertex graph. The permanent of an n × n matrix with entries 0 and 1 cannot be computed in time exp( o ( n )). The Tutte polynomial of an n -vertex multigraph cannot be computed in time exp( o ( n )) at most evaluation points ( x , y ) in the case of multigraphs, and it cannot be computed in time exp( o ( n /poly log n )) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n -variable 3-CNF formulas cannot be decided in time exp( o ( n )). We relax this hypothesis by introducing its counting version #ETH; namely, that the satisfying assignments cannot be counted in time exp( o ( n )). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d -CNF formulas to the counting setting.