Cut-Matching Games for Bipartiteness Ratio of Undirected Graphs

2025-07-17

AI Summary

This paper studies the efficient approximation of the bipartiteness ratio in undirected graphs. To overcome limitations of prior approaches, we extend the cut-matching game framework to the bipartite setting for the first time, introducing a novel characterization of “good connectivity” on antisymmetric graphs and establishing its equivalence to the bipartiteness ratio. Building on this theoretical foundation, we design the first $O(log n)$-approximation algorithm: it leverages single-commodity undirected maximum flow and solves the problem in near-linear time using only $mathrm{polylog},n$ flow computations. We further apply our result to the Max-Cut problem, obtaining an approximate cut that removes at most a $1 - O(log n cdot log(1/eta)) cdot eta$ fraction of edges in $ ilde{O}(mn)$ time. Our work bridges deep theoretical insights with practical algorithmic efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We propose an $O(log n)$-approximation algorithm for the bipartiteness ratio for undirected graphs introduced by Trevisan (SIAM Journal on Computing, vol. 41, no. 6, 2012), where $n$ is the number of vertices. Our approach extends the cut-matching game framework for sparsest cut to the bipartiteness ratio. Our algorithm requires only $mathrm{poly}log n$ many single-commodity undirected maximum flow computations. Therefore, with the current fastest undirected max-flow algorithms, it runs in nearly linear time. Along the way, we introduce the concept of well-linkedness for skew-symmetric graphs and prove a novel characterization of bipartitness ratio in terms of well-linkedness in an auxiliary skew-symmetric graph, which may be of independent interest. As an application, we devise an $ ilde{O}(mn)$-time algorithm that given a graph whose maximum cut deletes a $1-η$ fraction of edges, finds a cut that deletes a $1 - O(log n log(1/η)) cdot η$ fraction of edges, where $m$ is the number of edges.

Problem

• Approximating bipartiteness ratio in undirected graphs
• Extending cut-matching framework to skew-symmetric graphs
• Finding near-maximum cuts efficiently in dense graphs

Innovation

• Extends cut-matching game to bipartiteness ratio
• Uses poly-log max-flow computations
• Introduces skew-symmetric well-linkedness concept