$O(\log n)$-Approximation Algorithms for Bipartiteness Ratio
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| Format: | Preprint |
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2025
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| author | Soma, Tasuku Ye, Mingquan Yoshida, Yuichi |
| author_facet | Soma, Tasuku Ye, Mingquan Yoshida, Yuichi |
| contents | We propose an $O(\log n)$-approximation algorithm for the bipartiteness ratio of 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, and requires only $\mathop{\mathrm{polylog}} n$ many single-commodity undirected maximum flow computations. Therefore, with the current fastest undirected max-flow algorithms, it runs in almost linear time. Along the way, we introduce the concept of well-linkedness for skew-symmetric graphs and prove a novel characterization of bipartiteness 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 for the minimum uncut problem: given a graph whose optimal cut leaves an $η$ fraction of edges uncut, we find a cut that leaves only an $O(\log n \log(1/η)) \cdot η$ fraction of edges uncut, where $m$ is the number of edges.
Finally, we propose a directed analogue of the bipartiteness ratio, and we give a polynomial-time algorithm that achieves an $O(\log n)$ approximation for this measure via a directed Leighton--Rao-style embedding. We also propose an algorithm for the minimum directed uncut problem with a guarantee similar to that for the minimum uncut problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_12847 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $O(\log n)$-Approximation Algorithms for Bipartiteness Ratio Soma, Tasuku Ye, Mingquan Yoshida, Yuichi Data Structures and Algorithms We propose an $O(\log n)$-approximation algorithm for the bipartiteness ratio of 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, and requires only $\mathop{\mathrm{polylog}} n$ many single-commodity undirected maximum flow computations. Therefore, with the current fastest undirected max-flow algorithms, it runs in almost linear time. Along the way, we introduce the concept of well-linkedness for skew-symmetric graphs and prove a novel characterization of bipartiteness 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 for the minimum uncut problem: given a graph whose optimal cut leaves an $η$ fraction of edges uncut, we find a cut that leaves only an $O(\log n \log(1/η)) \cdot η$ fraction of edges uncut, where $m$ is the number of edges. Finally, we propose a directed analogue of the bipartiteness ratio, and we give a polynomial-time algorithm that achieves an $O(\log n)$ approximation for this measure via a directed Leighton--Rao-style embedding. We also propose an algorithm for the minimum directed uncut problem with a guarantee similar to that for the minimum uncut problem. |
| title | $O(\log n)$-Approximation Algorithms for Bipartiteness Ratio |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2507.12847 |