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 \(k\geq2\). 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 \(k\geq4\), it is known that an approximation ratio of \(1-{1}/{2}^{k-1}\), obtained by choosing a random assignment, is optimal, assuming P ≠ NP. For every \(k\geq2\), an approximation ratio of at least ^{7}⁄_{8} can be obtained for MAX NAE-\({k}\)-SAT. There was some hope, therefore, that there is also a ^{7}⁄_{8}-approximation algorithm for MAX NAE-SAT, where clauses of all sizes are allowed simultaneously.

In this talk, we prove that there is no ^{7}⁄_{8}-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 0.8739.

Joint work with Joshua Brakensiek, Aaron Potechin and Uri Zwick.