Near-Optimal Encodings of Cardinality Constraints

2026-03-30

AI Summary

This study addresses the problem of compact CNF encodings for cardinality constraints, particularly AtMostOne and general AtMostₖ. By introducing a novel AtMostOne encoding, refining Adleman’s threshold circuit construction, and incorporating a hash-table-inspired “grid compression” technique for general AtMostₖ constraints, the work significantly reduces encoding size. The main contributions include disproving Chen’s conjecture on the optimality of product encoding by achieving an AtMostOne clause count of 2n + 2√(2n) + O(n¹/³), establishing the first unconditional lower bound of 2n + √(n+1) − 2 for such encodings, and improving the clause complexity for AtMostₖ to 4n + o(n) when k = o(n)—outperforming existing bounds of (k+1)n or 7n.

Abstract

We present several novel encodings for cardinality constraints, which use fewer clauses than previous encodings and, more importantly, introduce new generally applicable techniques for constructing compact encodings. First, we present a CNF encoding for the $\text{AtMostOne}(x_1,\dots,x_n)$ constraint using $2n + 2 \sqrt{2n} + O(\sqrt[3]{n})$ clauses, thus refuting the conjectured optimality of Chen's product encoding. Our construction also yields a smaller monotone circuit for the threshold-2 function, improving on a 50-year-old construction of Adleman and incidentally solving a long-standing open problem in circuit complexity. On the other hand, we show that any encoding for this constraint requires at least $2n + \sqrt{n+1} - 2$ clauses, which is the first nontrivial unconditional lower bound for this constraint and answers a question of Kučera, Savický, and Vorel. We then turn our attention to encodings of $\text{AtMost}_k(x_1,\dots,x_n)$, where we introduce "grid compression", a technique inspired by hash tables, to give encodings using $2n + o(n)$ clauses as long as $k = o(\sqrt[3]{n})$ and $4n + o(n)$ clauses as long as $k = o(n)$. Previously, the smallest known encodings were of size $(k+1)n + o(n)$ for $k \le 5$ and $7n - o(n)$ for $k \ge 6$.

Problem

• cardinality constraints
• CNF encoding
• AtMostOne
• AtMost_k
• clause complexity

Innovation

• cardinality constraints
• CNF encoding
• grid compression
• monotone circuits
• lower bounds