Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

Abstract

Jaeger, Vertigan, and Welsh (1990) proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén (2010) and Husfeldt and Taslaman (2010), in combination with the results of Curticapean (2015), extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line y=1, which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time exp(o(n)) unless #ETH fails.

Another dichotomy theorem we strengthen is the one of Creignou and Hermann (1996) for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases cannot be solved in time exp(o(n)) unless #ETH fails. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time exp(o(n)) unless #ETH fails.

In order to prove our results, we use the block interpolation idea by Curticapean (2015) and transfer it to systems of linear equations that might not directly correspond to interpolation.