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.
@InProceedings{nakajima_et_al:LIPIcs.ICALP.2025.121, author = {Nakajima, Tamio-Vesa and \v{Z}ivn\'{y}, Stanislav}, title = {{Maximum Bipartite vs. Triangle-Free Subgraph}}, booktitle = {52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)}, pages = {121:1--121:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-372-0}, ISSN = {1868-8969}, year = {2025}, volume = {334}, editor = {Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.121}, URN = {urn:nbn:de:0030-drops-234987}, doi = {10.4230/LIPIcs.ICALP.2025.121}, annote = {Keywords: approximation, promise constraint satisfaction, triangle-free subgraphs} }
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