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Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

Deterministic counting Lovász 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 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 5C0 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.

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cover image Proceedings
Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)
Pages: 3388 - 3425
Editors: Nikhil Bansal, University of Michigan, Ann Arbor, Michigan, USA and Viswanath Nagarajan, University of Michigan, Ann Arbor, Michigan, USA
ISBN (Online): 978-1-61197-755-4

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Published online: 16 January 2023

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