Fine-grained dichotomies for the Tutte plane and Boolean #CSP

TL;DR

This paper completes the complexity classification of evaluating the Tutte polynomial and counting solutions to Boolean CSPs under the exponential time hypothesis, establishing tight lower bounds for these problems.

Contribution

It extends existing dichotomy theorems by proving tight lower bounds under #ETH for counting acyclic subgraphs and solutions to Boolean CSPs, using block interpolation techniques.

Findings

Counting all acyclic subgraphs cannot be done in sub-exponential time unless #ETH fails.

All #P-hard cases for Boolean #CSP are also hard under #ETH.

The methods transfer block interpolation to systems of linear equations.

Abstract

Jaeger, Vertigan, and Welsh [15] 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 Wahl\'en [9] and Husfeldt and Taslaman [12], in combination with Curticapean [7], 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 [6] for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases are also hard under #ETH. 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 [7] and transfer it to systems of linear equations that might not directly correspond to interpolation.