Goemans and Williamson designed a 0.878-approximation algorithm for Max-Cut in undirected graphs [JACM'95]. Khot, Kindler, Mosel, and O'Donnel showed that the approximation ratio of the Goemans-Williamson algorithm is optimal assuming Khot's Unique Games Conjecture [SICOMP'07]. In the problem of maximum cuts in directed graphs (Max-DiCut), in which we seek as many edges going from one particular side of the cut to the other, the situation is more complicated but the recent work of Brakensiek, Huang, Potechin, and Zwick showed that their 0.874-approximation algorithm is tight under the Unique Games Conjecture (up to a small delta)[FOCS'23]. We consider a promise version of the problem and design an SDP-based algorithm which, if given a directed graph G that has a directed cut of value rho, finds an undirected cut in G (ignoring edge directions) with value at least \rho.

翻译：Goemans和Williamson为无向图中的最大割问题设计了0.878近似算法[JACM'95]。Khot、Kindler、Mosel和O'Donnel证明，假设Khot的唯一游戏猜想，Goemans-Williamson算法的近似比是最优的[SICOMP'07]。在有向图最大割问题（Max-DiCut）中，我们寻求尽可能多的从割的一侧指向另一侧的边，该问题更为复杂，但Brakensiek、Huang、Potechin和Zwick的最新工作表明，在唯一游戏猜想下，他们的0.874近似算法是紧的（至多相差一个小的delta）[FOCS'23]。我们考虑该问题的承诺版本，并设计了一个基于半定规划的算法：若给定一个有向图G，其存在值为rho的有向割，则该算法能在G中找到值至少为\rho的无向割（忽略边的方向）。