Exponential Time Complexity of the Permanent and the Tutte Polynomial

TL;DR

This paper establishes exponential time lower bounds for computing the permanent, Tutte polynomial, and counting solutions for #P-hard problems, based on variants of the Exponential Time Hypothesis, highlighting their computational intractability.

Contribution

It introduces conditional lower bounds for #P-hard problems using #ETH, extending the sparsification lemma to the counting setting, and demonstrates their exponential complexity.

Findings

Permanent cannot be computed in time exp(o(n))

Tutte polynomial evaluation is exponential time hard for multigraphs and simple graphs

Counting satisfying assignments of 2-CNF formulas is exponential time hard

Abstract

We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) 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/polylog 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.