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 $k\ge 2$. 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 $k\ge 4$, it is known that an approximation ratio of $1-\frac{1}{2^{k-1}}$, obtained by choosing a random assignment, is optimal, assuming $P\ne NP$. For every $k\ge 2$, an approximation ratio of at least $\frac{7}{8}$ can be obtained for MAX NAE-$\{k\}$-SAT. There was some hope, therefore, that there is also a $\frac{7}{8}$-approximation algorithm for MAX NAE-SAT, where clauses of all sizes are allowed simultaneously. Our main result is that there is no $\frac{7}{8}$-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 $\frac{3(\sqrt{21}-4)}{2}\approx 0.8739$. 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 $\approx 0.9089$. 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.

翻译：MAX NAE-SAT是一个自然的优化问题，与其更广为人知的近亲MAX SAT密切相关。当所有子句具有相同规模$k$（$k\ge 2$）时，MAX NAE-SAT的可近似性状态已几乎被完全理解。我们将该问题称为MAX NAE-$\{k\}$-SAT。对于$k=2$，它本质上是著名的MAX CUT问题。对于$k=3$，它与可在分数意义上被三角形覆盖的图中的MAX CUT问题相关。对于$k\ge 4$，已知在$P\ne NP$假设下，通过随机赋值获得的$1-\frac{1}{2^{k-1}}$近似比是最优的。对于任意$k\ge 2$，MAX NAE-$\{k\}$-SAT可获得至少$\frac{7}{8}$的近似比。因此人们曾期望，对于允许所有规模子句同时存在的MAX NAE-SAT，也可能存在$\frac{7}{8}$近似算法。我们的主要结果表明：在唯一游戏猜想（UGC）假设下，MAX NAE-SAT不存在$\frac{7}{8}$近似算法。事实上，即使对于几乎可满足的MAX NAE-$\{3,5\}$-SAT实例（即所有子句规模为$3$或$5$的MAX NAE-SAT），在UGC假设下可达到的最佳近似比至多为$\frac{3(\sqrt{21}-4)}{2}\approx 0.8739$。通过变分法，我们将O'Donnell和Wu对MAX CUT的分析推广至MAX NAE-$\{3\}$-SAT。我们在UGC假设下得到了MAX NAE-$\{3\}$-SAT的最优算法，较先前算法略有改进。新算法的近似比约为$0.9089$。我们通过实验数据补充理论结果：提出了一个针对几乎可满足MAX NAE-$\{3,5\}$-SAT实例的近似算法（推测近似比为0.8728），以及一个针对几乎可满足MAX NAE-SAT实例的近似算法（推测近似比为0.8698）。