Austrin showed that the approximation ratio $\beta\approx 0.94016567$ obtained by the MAX 2-SAT approximation algorithm of Lewin, Livnat and Zwick (LLZ) is optimal modulo the Unique Games Conjecture (UGC) and modulo a Simplicity Conjecture that states that the worst performance of the algorithm is obtained on so called simple configurations. We prove Austrin's conjecture, thereby showing the optimality of the LLZ approximation algorithm, relying only on the Unique Games Conjecture. Our proof uses a combination of analytic and computational tools. We also present new approximation algorithms for two restrictions of the MAX 2-SAT problem. For MAX HORN-$\{1,2\}$-SAT, i.e., MAX CSP$(\{x\lor y,\bar{x}\lor y,x,\bar{x}\})$, in which clauses are not allowed to contain two negated literals, we obtain an approximation ratio of $0.94615981$. For MAX CSP$(\{x\lor y,x,\bar{x}\})$, i.e., when 2-clauses are not allowed to contain negated literals, we obtain an approximation ratio of $0.95397990$. By adapting Austrin's and our arguments for the MAX 2-SAT problem we show that these two approximation ratios are also tight, modulo only the UGC conjecture. This completes a full characterization of the approximability of the MAX 2-SAT problem and its restrictions.

翻译：Austrin证明了Lewin、Livnat和Zwick（LLZ）提出的MAX 2-SAT近似算法所获得的近似比$\beta\approx 0.94016567$在唯一博弈猜想（UGC）和简单性猜想（即算法的最差性能出现在所谓简单配置上）下是最优的。我们证明了Austrin的猜想，从而仅依赖唯一博弈猜想就证明了LLZ近似算法的最优性。我们的证明结合了解析和计算工具。此外，我们还针对MAX 2-SAT问题的两个限制变体提出了新的近似算法。对于MAX HORN-$\{1,2\}$-SAT（即MAX CSP$(\{x\lor y,\bar{x}\lor y,x,\bar{x}\})$，其中子句不允许包含两个否定文字），我们获得了$0.94615981$的近似比。对于MAX CSP$(\{x\lor y,x,\bar{x}\})$（即禁止2-子句包含否定文字的情况），我们获得了$0.95397990$的近似比。通过将Austrin及我们针对MAX 2-SAT问题的论证进行改编，我们证明了这两个近似比在仅依赖UGC猜想的情况下也是紧的。这完整刻画了MAX 2-SAT问题及其限制变体的近似性。