The Curse and Blessing of Not-All-Equal in k-Satisfiability

As a natural variant of the \(k\)-SAT problem, NAE-\(k\)-SAT additionally requires the literals in each clause to take not-all-equal (NAE) truth values. In this paper, we study the worst-case time complexities of solving NAE-\(k\)-SAT and MAX-NAE-\(k\)-SAT approximation, as functions of \(k\), the number of variables \(n\), and the performance ratio \(\delta\). The latter problem asks for a solution of at least \(\delta\) times the optimal. Our main results include: (1) A deterministic algorithm for NAE-\(k\)-SAT that is faster than the best deterministic algorithm for \(k\)-SAT on all \(k \ge 3\). Previously, no NAE-\(k\)-SAT algorithm is known to be faster than \(k\)-SAT algorithms. For \(k = 3\), we achieve an upper bound of \(1.326^n\). The corresponding bound for \(3\)-SAT is \(1.328^n\). (2) A randomized algorithm for MAX-NAE-\(k\)-SAT approximation, with upper bound \((2 - \epsilon_k(\delta))^n\) where \(\epsilon_k(\delta) > 0\) only depends on \(k\) and \(\delta\). Previously, no upper bound better than the trivial \(2^n\) is known for MAX-NAE-\(k\)-SAT approximation on \(k \ge 4\). For \(\delta = 0.9\) and \(k = 4\), we achieve an upper bound of \(1.947^n\). (3) A deterministic algorithm for MAX-NAE-\(k\)-SAT approximation. For \(\delta = 0.9\) and \(k = 3\), we achieve an upper bound of \(1.698^n\), which is better than the upper bound \(1.731^n\) of the exact algorithm for MAX-NAE-\(3\)-SAT. Our finding sheds new light on the following question: Is NAE-\(k\)-SAT easier than \(k\)-SAT? The answer might be affirmative at least on solving the problems exactly and deterministically, while approximately solving MAX-NAE-\(k\)-SAT might be harder than MAX-\(k\)-SAT on \(k \ge 4\).