Automated Reencoding Meets Graph Theory

2026-03-29

AI Summary

This work investigates the capabilities and limitations of Bounded Variable Addition (BVA) in the re-encoding of 2-CNF formulas. By characterizing the idealized encodings producible by BVA through graph-theoretic constructs, it establishes the first formal connection between BVA and graph theory. The study derives tight bounds on clause compression: for general 2-CNF formulas over $n$ variables, BVA can achieve up to approximately $0.396 \cdot n^2 / \log n$ clauses, while for at-most-one constraints, it proves a lower bound of $3n - 6$, demonstrating that BVA cannot generate certain highly efficient encodings such as product encodings. A new implementation incorporating lightweight preprocessing significantly improves computational efficiency while preserving comparable compression performance.

Abstract

Bounded Variable Addition (BVA) is a central preprocessing method in modern state-of-the-art SAT solvers. We provide a graph-theoretic characterization of which 2-CNF encodings can be constructed by an idealized BVA algorithm. Based on this insight, we prove new results about the behavior and limitations of BVA and its interaction with other preprocessing techniques. We show that idealized BVA, plus some minor additional preprocessing (e.g., equivalent literal substitution), can reencode any 2-CNF formula with $n$ variables into an equivalent 2-CNF formula with $(\tfrac{\lg(3)}{4}+o(1))\,\tfrac{n^2}{\lg n}$ clauses. Furthermore, we show that without the additional preprocessing the constant factor worsens from $\tfrac{\lg(3)}{4} \approx 0.396$ to $1$, and that no reencoding method can achieve a constant below $0.25$. On the other hand, for the at-most-one constraint on $n$ variables, we prove that idealized BVA cannot reencode this constraint using fewer than $3n-6$ clauses, a bound that we prove is achieved by actual implementations. In particular, this shows that the product encoding for at-most-one, which uses $2n+o(n)$ clauses, cannot be constructed by BVA regardless of the heuristics used. Finally, our graph-theoretic characterization of BVA allows us to leverage recent work in algorithmic graph theory to develop a drastically more efficient implementation of BVA that achieves a comparable clause reduction on random monotone 2-CNF formulas.

Problem

• Bounded Variable Addition
• 2-CNF encoding
• SAT preprocessing
• graph-theoretic characterization
• clause complexity

Innovation

• Bounded Variable Addition
• 2-CNF reencoding
• graph-theoretic characterization
• at-most-one constraint
• SAT preprocessing