Block Interpolation: A Framework for Tight Exponential-Time Counting Complexity

TL;DR

This paper introduces a framework that converts classical #P-hardness results into tight exponential-time lower bounds under #ETH, significantly advancing the understanding of computational limits for counting problems.

Contribution

The authors develop a novel framework for establishing tight lower bounds under #ETH, enabling the conversion of #P-hardness results into exponential-time complexity bounds.

Findings

Established tight lower bounds for zero-one permanent evaluation.

Derived bounds for the matching polynomial on all non-easy points.

Extended bounds to the Tutte polynomial, excluding one recently settled line.

Abstract

We devise a framework for proving tight lower bounds under the counting exponential-time hypothesis #ETH introduced by Dell et al. (ACM Transactions on Algorithms, 2014). Our framework allows us to convert classical #P-hardness results for counting problems into tight lower bounds under #ETH, thus ruling out algorithms with running time $2^{o(n)}$ on graphs with $n$ vertices and $O(n)$ edges. As exemplary applications of this framework, we obtain tight lower bounds under #ETH for the evaluation of the zero-one permanent, the matching polynomial, and the Tutte polynomial on all non-easy points except for one line. This remaining line was settled very recently by Brand et al. (IPEC 2016).