Given a (multi)graph $G$ which contains a bipartite subgraph with $\rho$ edges, what is the largest triangle-free subgraph of $G$ that can be found efficiently? We present an SDP-based algorithm that finds one with at least $0.8823 \rho$ edges, thus improving on the subgraph with $0.878 \rho$ edges obtained by the classic Max-Cut algorithm of Goemans and Williamson. On the other hand, by a reduction from Hastad's 3-bit PCP we show that it is NP-hard to find a triangle-free subgraph with $(25 / 26 + \epsilon) \rho \approx (0.961 + \epsilon) \rho$ edges. As an application, we classify the Maximum Promise Constraint Satisfaction Problem MaxPCSP($G$,$H$) for all bipartite $G$: Given an input (multi)graph $X$ which admits a $G$-colouring satisfying $\rho$ edges, find an $H$-colouring of $X$ that satisfies $\rho$ edges. This problem is solvable in polynomial time, apart from trivial cases, if $H$ contains a triangle, and is NP-hard otherwise.

翻译：给定一个包含具有 $\rho$ 条边的二分子图的（多重）图 $G$，我们能否高效地找到 $G$ 中最大的无三角形子图？本文提出一种基于半正定规划（SDP）的算法，该算法能找到至少包含 $0.8823 \rho$ 条边的无三角形子图，从而改进了由 Goemans 和 Williamson 的经典最大割算法所获得的 $0.878 \rho$ 条边的结果。另一方面，通过从 Hastad 的 3 位概率可检查证明（PCP）进行归约，我们证明了找到具有 $(25 / 26 + \epsilon) \rho \approx (0.961 + \epsilon) \rho$ 条边的无三角形子图是 NP 难的。作为应用，我们对所有二分图 $G$ 的最大承诺约束满足问题 MaxPCSP($G$,$H$) 进行了分类：给定一个允许存在满足 $\rho$ 条边的 $G$ 着色的输入（多重）图 $X$，寻找 $X$ 的一个满足 $\rho$ 条边的 $H$ 着色。除非平凡情况，当 $H$ 包含三角形时，该问题可在多项式时间内求解；否则，该问题是 NP 难的。