Exponential Time Complexity of Weighted Counting of Independent Sets

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^Ω(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^Ω(n) and weighted counting of independent sets needs time $2^{\Omega(n/\log^3 n)}$ for all rational weights x ≠ 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 Wahlén which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.