Deterministic counting Lov\'{a}sz local lemma beyond linear programming

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 $\Delta=O(1)$ 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: \[ q^2\cdot k\cdot p\cdot\Delta^5\le C_0\quad\text{for a suitably small absolute constant }C_0. \] Here the key term $\Delta^5$ improves the previously best known $\Delta^7$ for general CSPs [JPV21b] and $\Delta^{5.714}$ for the special case of $k$-CNF [JPV21a, HSW21]. Our deterministic counting algorithm is a derandomization of the very recent fast sampling algorithm in [HWY22]. It departs substantially from all previous deterministic counting Lov\'{a}sz 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.