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Maximum Bipartite vs. Triangle-Free Subgraph

Authors Tamio-Vesa Nakajima , Stanislav Živný

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Problems, reductions and completeness
Keywords
  • approximation
  • promise constraint satisfaction
  • triangle-free subgraphs

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Abstract

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.

Cite As Get BibTex

Tamio-Vesa Nakajima and Stanislav Živný. Maximum Bipartite vs. Triangle-Free Subgraph. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 121:1-121:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.121

Author Details

Tamio-Vesa Nakajima
  • Department of Computer Science, University of Oxford, UK
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK

Funding

  • Nakajima, Tamio-Vesa: UKRI EP/X024431/1, Clarendon Fund Scholarship.
  • Živný, Stanislav: UKRI EP/X024431/1.

Acknowledgements

We thank the anonymous reivewers for their feedback. We also thank Jakub Opršal, who asked us about the complexity of maximisation PCSPs for graphs beyond cliques.

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