Exponential Time Complexity of Weighted Counting of Independent Sets

TL;DR

This paper establishes exponential lower bounds for the weighted counting of independent sets in graphs, linking their complexity to the hardness of #3SAT, and introduces novel graph transformations for reductions.

Contribution

It provides the first conditional lower bounds for weighted independent set counting, using a new reduction and graph transformation techniques.

Findings

Counting independent sets of size n/3 requires 2^{Ω(n)} time.

Weighted counting of independent sets requires 2^{Ω(n/log^3 n)} time.

Reductions from #3SAT preserve solutions and increase instance size minimally.

Abstract

We consider weighted counting of independent sets using a rational weight x: Given a graph with n vertices, count its independent sets such that each set of size k contributes x^k. This is equivalent to computation of the partition function of the lattice gas with hard-core self-repulsion and hard-core pair interaction. We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time 2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in time 2^{cn}), counting the independent sets of size n/3 of an n-vertex graph needs time 2^{\Omega(n)} and weighted counting of independent sets needs time 2^{\Omega(n/log^3 n)} for all rational weights x\neq 0. We have two technical ingredients: The first is a reduction from 3SAT to independent sets that preserves the number of solutions and increases the instance size only by a constant factor. Second, we devise a combination of vertex cloning and path addition. This graph transformation allows us to adapt a recent technique by Dell, Husfeldt, and Wahlen which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.