Maximum Bipartite vs. Triangle-Free Subgraph
Nakajima, Tamio-Vesa, Živný, Stanislav · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2025
Given a (multi)graph G which contains a bipartite subgraph with ρ 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 ρ edges, thus improving on the subgraph with 0.878 ρ edges obtained by the classic Max-Cut algorithm of Goemans and Williamson. On the other hand, by a reduction from Håstad’s 3-bit PCP we show that it is NP-hard to find a triangle-free subgraph with (25 / 26 + ε) ρ ≈ (0.961 + ε) ρ edges. As an application, we classify the Maximum Promise Constraint Satisfaction Problem, denoted byMaxPCSP(G, H), for all bipartite G: Given an input (multi)graph X which admits a G-colouring satisfying ρ edges, find an H-colouring of X that satisfies ρ edges. This problem is solvable in polynomial time, apart from trivial cases, if H contains a triangle, and is NP-hard otherwise.