Absorbing Random Walks and the NAE2SAT Problem

Abstract

In this paper, we propose a simple, randomized algorithm for the NAE2SAT problem; the analysis of the algorithm uses the theory of symmetric, absorbing random walks. NAESAT (Not-All-Equal SAT) is the variant of the Satisfiability problem (SAT), in which we are interested in an assignment that satisfies all the clauses, but falsifies at least one literal in each clause. We show that the NAE2SAT problem admits an extremely simple literal-flipping algorithm, in precisely the same way that 2SAT does. On a satisfiable instance involving n variables, our algorithm finds a satisfying assignment using at most $\frac{9}{4}n^{2}$ verification calls with probability at least $\frac{5}{6}$. The randomized algorithm takes O(1) extra space, in the presence of a verifier and provides an interesting insight into checking whether a graph is bipartite. It must be noted that the bounds we derive are much sharper than the ones in [1].