Combining the $k$-CNF and XOR Phase-Transitions

Abstract

The runtime performance of modern SAT solvers on random $k$-CNF formulas is deeply connected with the 'phase-transition' phenomenon seen empirically in the satisfiability of random $k$-CNF formulas. Recent universal hashing-based approaches to sampling and counting crucially depend on the runtime performance of SAT solvers on formulas expressed as the conjunction of both $k$-CNF and XOR constraints (known as $k$-CNF-XOR formulas), but the behavior of random $k$-CNF-XOR formulas is unexplored in prior work. In this paper, we present the first study of the satisfiability of random $k$-CNF-XOR formulas. We show empirical evidence of a surprising phase-transition that follows a linear trade-off between $k$-CNF and XOR constraints. Furthermore, we prove that a phase-transition for $k$-CNF-XOR formulas exists for $k=2$ and (when the number of $k$-CNF constraints is small) for $k>2$.

Presented at The 25th International Joint Conference on Artificial Intelligence (IJCAI-16)

References in corpus (2)

• Balancing Scalability and Uniformity in SAT Witness Generator
• Aspects of Statistical Physics in Computational Complexity

Cited by in corpus (3)

• Bit-Vector Model Counting using Statistical Estimation
• FourierSAT: A Fourier Expansion-Based Algebraic Framework for Solving Hybrid Boolean Constraints
• Phase Transition Behavior of Cardinality and XOR Constraints