Assuming the Unique Games Conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the MAX CUT problem is $\alpha_{\text{CUT}}\simeq 0.87856$, obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. The currently best approximation algorithm for MAX DI-CUT, i.e., the MAX CUT problem in directed graphs, achieves a ratio of about $0.87401$, leaving open the question whether MAX DI-CUT can be approximated as well as MAX CUT. We obtain a slightly improved algorithm for MAX DI-CUT and a new UGC-hardness result for it, showing that $0.87446\le \alpha_{\text{DI-CUT}}\le 0.87461$, where $\alpha_{\text{DI-CUT}}$ is the best approximation ratio that can be obtained in polynomial time for MAX DI-CUT under UGC. The new upper bound separates MAX DI-CUT from MAX CUT, resolving a question raised by Feige and Goemans. A natural generalization of MAX DI-CUT is the MAX 2-AND problem in which each constraint is of the form $z_1\land z_2$, where $z_1$ and $z_2$ are literals, i.e., variables or their negations (In MAX DI-CUT each constraint is of the form $\bar{x}_1\land x_2$, where $x_1$ and $x_2$ are variables.) Austrin separated MAX 2-AND from MAX CUT by showing that $\alpha_{\text{2AND}} < 0.87435$ and conjectured that MAX 2-AND and MAX DI-CUT have the same approximation ratio. Our new lower bound on MAX DI-CUT refutes this conjecture, completing the separation of the three problems MAX 2-AND, MAX DI-CUT and MAX CUT. We also obtain a new lower bound for MAX 2-AND, showing that $0.87414\le \alpha_{\text{2AND}}\le 0.87435$. Our upper bound on MAX DI-CUT is achieved via a simple, analytical proof. The lower bounds on MAX DI-CUT and MAX 2-AND (the new approximation algorithms) use experimentally-discovered distributions of rounding functions which are then verified via computer-assisted proofs.

翻译：假设唯一博弈猜想（UGC）成立，MAX CUT 问题在多项式时间内可达到的最佳近似比为 $\alpha_{\text{CUT}}\simeq 0.87856$，该结果由 Goemans 和 Williamson 基于半定规划（SDP）的著名近似算法获得。当前 MAX DI-CUT（即有向图中的 MAX CUT 问题）的最佳近似算法达到约 $0.87401$ 的比率，使得 MAX DI-CUT 能否被近似到与 MAX CUT 相同的程度仍是一个开放问题。我们为 MAX DI-CUT 提出一个略有改进的算法，并给出其新的 UGC-困难性结果，证明 $0.87446\le \alpha_{\text{DI-CUT}}\le 0.87461$，其中 $\alpha_{\text{DI-CUT}}$ 是在 UGC 假设下 MAX DI-CUT 在多项式时间内可达到的最佳近似比。这一新的上界将 MAX DI-CUT 与 MAX CUT 分离，解决了 Feige 和 Goemans 提出的问题。MAX DI-CUT 的一个自然推广是 MAX 2-AND 问题，其中每个约束的形式为 $z_1\land z_2$，$z_1$ 和 $z_2$ 为文字（即变量或其否定）（在 MAX DI-CUT 中，每个约束的形式为 $\bar{x}_1\land x_2$，其中 $x_1$ 和 $x_2$ 是变量）。Austrin 通过证明 $\alpha_{\text{2AND}} < 0.87435$ 将 MAX 2-AND 与 MAX CUT 分离，并猜想 MAX 2-AND 与 MAX DI-CUT 具有相同的近似比。我们关于 MAX DI-CUT 的新下界反驳了这一猜想，完成了对 MAX 2-AND、MAX DI-CUT 和 MAX CUT 三个问题的分离。我们还获得了 MAX 2-AND 的新下界，证明 $0.87414\le \alpha_{\text{2AND}}\le 0.87435$。我们对 MAX DI-CUT 的上界通过简洁的分析证明实现。而 MAX DI-CUT 和 MAX 2-AND 的下界（新的近似算法）则使用了通过实验发现的舍入函数分布，并通过计算机辅助证明进行验证。