A Dichotomy for Maximum PCSPs on Graphs

Fix two non-empty loopless graphs $G$ and $H$ such that $G$ maps homomorphically to $H$. The Maximum Promise Constraint Satisfaction Problem parameterised by $G$ and $H$ is the following computational problem, denoted by MaxPCSP($G$, $H$): Given an input (multi)graph $X$ that admits a map to $G$ preserving a $ρ$-fraction of the edges, find a map from $X$ to $H$ that preserves a $ρ$-fraction of the edges. As our main result, we give a complete classification of this problem under Khot's Unique Games Conjecture: The only tractable cases are when $G$ is bipartite and $H$ contains a triangle. Along the way, we establish several results, including an efficient approximation algorithm for the following problem: Given a (multi)graph $X$ which contains a bipartite subgraph with $ρ$ edges, what is the largest triangle-free subgraph of $X$ that can be found efficiently? We present an SDP-based algorithm that finds one with at least $0.8823 ρ$ edges, thus improving on the subgraph with $0.878 ρ$ edges obtained by the classic Max-Cut algorithm of Goemans and Williamson.