On the Mysteries of MAX NAE-SAT

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 k, for some k≥ 2. We refer to this problem as MAX NAE-k-SAT. For k=2, it is essentially the celebrated MAX CUT problem. For k=3, it is related to the MAX CUT problem in graphs that can be fractionally covered by triangles. For k≥ 4, it is known that an approximation ratio of 1-frac12k-1, obtained by choosing a random assignment, is optimal, assuming Pne NP. For every k≥ 2, an approximation ratio of at least frac78 can be obtained for MAX NAE-k-SAT. There was some hope, therefore, that there is also a frac78-approximation algorithm for MAX NAE-SAT, where clauses of all sizes are allowed simultaneously. Our main result is that there is no frac78-approximation algorithm for MAX NAE-SAT, assuming the unique games conjecture (UGC). In fact, even for almost satisfiable instances of MAX NAE-3,5-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 frac3(√21-4)2≈ 0.8739. Using calculus of variations, we extend the analysis of O'Donnell and Wu for MAX CUT to MAX NAE-3-SAT. We obtain an optimal algorithm, assuming UGC, for MAX NAE-3-SAT, slightly improving on previous algorithms. The approximation ratio of the new algorithm is ≈ 0.9089. We complement our theoretical results with some experimental results. We describe an approximation algorithm for almost satisfiable instances of MAX NAE-3,5-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.