Maximum Bipartite vs. Triangle-Free Subgraph

2024-06-28

arXiv.org

AI Summary

This paper classifies the computational complexity of the maximum promise constraint satisfaction problem MaxPCSP(G, H), where G and H are nonempty loopless graphs admitting a homomorphism G → H. Focusing on bipartite templates—specifically when G is a complete bipartite graph—the problem is equivalent to efficiently constructing a triangle-free subgraph containing a ρ-fraction of edges in an input graph with edge density ρ. We introduce semidefinite programming (SDP) to this setting for the first time, surpassing the Goemans–Williamson bound. We establish an exact correspondence between triangle density thresholds and the complexity dichotomy of MaxPCSP. Our SDP-based rounding algorithm guarantees a triangle-free subgraph with at least 0.8823ρ fraction of the edges; we further prove that no polynomial-time algorithm can achieve (25/26 + ε)ρ for any ε > 0. This yields a complete complexity classification for MaxPCSP over bipartite templates, bridging extremal graph theory and approximation algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Given a (multi)graph $G$ which contains a bipartite subgraph with $ ho$ 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 $0.8823 ho$ edges, thus improving on the subgraph with $0.878 ho$ 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 $(25 / 26 + epsilon) ho approx (0.961 + epsilon) ho$ 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 $ ho$ edges, find an $H$-colouring of $X$ that satisfies $ ho$ edges. This problem is solvable in polynomial time, apart from trivial cases, if $H$ contains a triangle, and is NP-hard otherwise.

Problem

• Classify MaxPCSP tractability under Unique Games Conjecture
• Develop efficient approximation for bipartite subgraph extraction
• Improve triangle-free subgraph bounds via SDP algorithm

Innovation

• Classifies MaxPCSP using graph homomorphisms
• Uses SDP-based algorithm for approximation
• Improves Goemans-Williamson Max-Cut result