Deterministic counting Lovász local lemma beyond linear programming

Kun He, Chunyang Wang, Yitong Yin

Abstract: We give a simple combinatorial algorithm to deterministically approximately count the number of satisfying assignments of general constraint satisfaction problems (CSPs). Suppose that the CSP has domain size q = O(1), each constraint contains at most k = O(1) variables, shares variables with at most Δ = O(1) constraints, and is violated with probability at most p by a uniform random assignment. The algorithm returns in polynomial time in an improved local lemma regime:q2 · κ · p · Δ 5 ≤ C0 for a suitably small absolute constant C0.Here the key term Δ5 improves the previously best known Δ7 for general CSPs [21] and Δ5.714 for the special case of k-CNF [20, 16].Our deterministic counting algorithm is a derandomization of the very recent fast sampling algorithm in [17]. It departs substantially from all previous deterministic counting Lovasz local lemma algorithms which relied on linear programming, and gives a deterministic approximate counting algorithm that straightforwardly derandomizes a fast sampling algorithm, hence unifying the fast sampling and deterministic approximate counting in the same algorithmic framework.To obtain the improved regime, in our analysis we develop a refinement of the {2, 3}-trees that were used in the previous analyses of counting/sampling LLL. Similar techniques can be applied to the previous LP-based algorithms to obtain the same improved regime and may be of independent interests.