Combining the k-CNF and XOR Phase-Transitions

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 phasetransition for k-CNF-XOR formulas exists for k = 2 and (when the number of k-CNF constraints is small) for k > 2.