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 $\exp(\Omega(n))$. We relax this hypothesis by introducing its counting version #ETH, namely that every algorithm that counts the satisfying assignments requires time $\exp(\Omega(n))$. 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 $\exp(\Omega(m))$. Restricted to 01-matrices, the bound is $\exp(\Omega(m/\log m))$. Computing the Tutte polynomial of a multigraph with n vertices and m edges requires time $\exp(\Omega(n))$ at points (x,y) with (x − 1)(y − 1) ≠ 1 and y ∉ {0,±1}. At points (x,0) with $x \not \in \{0,\pm 1\}$ it requires time $\exp(\Omega(n))$, and if x = − 2, − 3,..., it requires time $\exp(\Omega(m))$. For simple graphs, the bound is $\exp(\Omega(m/\log^3 m))$.