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 近似算法所达到的近似比 $\beta\approx 0.94016567$ 在唯一游戏猜想（UGC）和简化猜想（即算法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，从而仅基于唯一游戏猜想就确立了 LLZ 近似算法的最优性。我们的证明结合了解析与计算工具。我们还为 MAX 2-SAT 问题的两个限制版本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$(\{x\lor y,x,\bar{x}\})$（即当 2-子句不允许包含否定文字时），我们获得了 $0.95397990$ 的近似比。通过借鉴 Austrin 和我们对 MAX 2-SAT 问题的论证方法，我们证明这两个近似比在仅依赖 UGC 猜想下也是紧的。这完成了 MAX 2-SAT 问题及其限制版本的可近似性的完整刻画。