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 − _2k¹₋1, obtained by choosing a random assignment, is optimal, assuming P 6= NP. For every k ≥ 2, an approximation ratio of at least ⁷₈ can be obtained for MAX NAE-{k}-SAT. There was some hope, therefore, that there is also a ⁷₈ -approximation algorithm for MAX NAE-SAT, where clauses of all sizes are allowed simultaneously. Our main result is that there is no ⁷₈ -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 3(√21−⁴⁾ ≈ 0.8739. 2 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. This gives a full understanding of MAX NAE-{k}-SAT for every k ≥ 2. 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-{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. 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 non-monotone step functions that assume only the values ±1.