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 $\tilde{O}(mn)$-time algorithm that given a graph whose maximum cut deletes a $1-\eta$ fraction of edges, finds a cut that deletes a $1 - O(\log n \log(1/\eta)) \cdot \eta$ fraction of edges, where $m$ is the number of edges.

翻译：我们针对Trevisan（《SIAM计算期刊》，第41卷第6期，2012年）提出的无向图二分性比率问题，提出了一种$O(\log n)$近似算法，其中$n$为顶点数。我们的方法将稀疏割的割匹配博弈框架扩展至二分性比率问题。该算法仅需$\mathrm{poly}\log n$次单商品无向最大流计算。因此，结合当前最快的无向最大流算法，其运行时间接近线性。在此过程中，我们引入了斜对称图的良连通性概念，并基于辅助斜对称图中的良连通性证明了二分性比率的新颖表征，该结果可能具有独立研究价值。作为应用，我们设计了一个$\tilde{O}(mn)$时间算法：对于最大割能删除$1-\eta$比例边的给定图，该算法可找到一个能删除$1 - O(\log n \log(1/\eta)) \cdot \eta$比例边的割，其中$m$为边数。