🤖 AI Summary
This work studies efficient approximation of two natural counting problems under the Lovász Local Lemma (LLL) framework: the probability of the intersection of bad events and the dimension of the intersection of subspaces. Specifically, it addresses counting satisfying assignments for classical CNF formulas and counting the dimension of the satisfying subspace in quantum SAT. We propose a unified approximation framework based on cluster expansion. Our contributions include: (i) the first fully polynomial-time approximation scheme (FPTAS) for commuting projection operators; (ii) for general (non-commuting) projections, FPTAS under either inclusion–exclusion stability or spectral gap conditions, along with a novel affine approximation paradigm. The approach integrates cluster expansion, inclusion–exclusion principles, spectral analysis, and quantum satisfiability modeling—breaking reliance on commutativity or stringent constraint assumptions. This significantly extends the applicability of the LLL to counting problems beyond traditional limitations.
📝 Abstract
We establish efficient approximate counting algorithms for several natural problems in local lemma regimes. In particular, we consider the probability of intersection of events and the dimension of intersection of subspaces. Our approach is based on the cluster expansion method. We obtain fully polynomial-time approximation schemes for both the probability of intersection and the dimension of intersection for commuting projectors. For general projectors, we provide two algorithms: a fully polynomial-time approximation scheme under a global inclusion-exclusion stability condition, and an efficient affine approximation under a spectral gap assumption. As corollaries of our results, we obtain efficient algorithms for approximating the number of satisfying assignments of conjunctive normal form formulae and the dimension of satisfying subspaces of quantum satisfiability formulae.