On the Mysteries of MAX NAE-SAT

Abstract. MAX NAE-SAT is a natural optimization problem, closely related to its better-known relative MAX SAT. The approximability status of MAX NAE-SAT is almost completely understood if all clauses have the same size [Formula: see text] for some [Formula: see text]. We refer to this problem as MAX NAE-[Formula: see text]-SAT. For [Formula: see text], it is a slight extension of the celebrated MAX CUT problem. For [Formula: see text], it is related to the MAX CUT problem in graphs that can be fractionally covered by triangles. For [Formula: see text], it is known that an approximation ratio of [Formula: see text], obtained by choosing a random assignment, is optimal, assuming [Formula: see text]. For every [Formula: see text], an approximation ratio of at least [Formula: see text] can be obtained for MAX NAE-[Formula: see text]-SAT. There was some hope, therefore, that there is also a [Formula: see text]-approximation algorithm for MAX NAE-SAT, where clauses of all sizes are allowed simultaneously. Our main result is that there is no [Formula: see text]-approximation algorithm for MAX NAE-SAT, assuming the Unique Games Conjecture (UGC). In fact, even for almost satisfiable instances of MAX NAE-[Formula: see text]-SAT (i.e., MAX NAE-SAT where all clauses have size 3 or 5), the best approximation ratio that can be achieved, assuming UGC, is at most [Formula: see text]. Using calculus of variations, we extend the analysis of O’Donnell and Wu for MAX CUT to MAX NAE-[Formula: see text]-SAT. We obtain an optimal algorithm, assuming UGC, for MAX NAE-[Formula: see text]-SAT, slightly improving on previous algorithms. The approximation ratio of the new algorithm is about 0.9089. This gives a full understanding of MAX NAE-[Formula: see text]-SAT for every [Formula: see text]. Interestingly, the rounding function used by this optimal algorithm is the solution of an integral equation. We complement our theoretical results with some experimental results. We describe an approximation algorithm for almost satisfiable instances of MAX NAE-[Formula: see text]-SAT with a conjectured approximation ratio of 0.8728, and an approximation algorithm for almost satisfiable instances of MAX NAE-SAT with a conjectured approximation ratio of 0.8698. We further conjecture that these are essentially the best approximation ratios that can be achieved for these problems, assuming the UGC. Somewhat surprisingly, the rounding functions used by these approximation algorithms are nonmonotone step functions that assume only the values [Formula: see text].