$O(\log n)$-Approximation Algorithms for Bipartiteness Ratio

TL;DR

This work delivers the first $O(\log n)$-approximation for the bipartiteness ratio in undirected graphs, via a generalized cut-matching game that uses a skew-symmetric auxiliary graph and a flow-based well-linkedness characterization. It leverages matrix multiplicative weight updates and fast approximate Gram decompositions to realize nearly-linear running time with polylogarithmic max-flow calls, and extends the approach to directed graphs through a directed metric embedding framework. The results yield a powerful set of consequences, including a tilde-$O(mn)$-time algorithm for minimum uncut and a family of directed counterparts with comparable guarantees. The paper broadens spectral and combinatorial methods for near-optimal cuts by linking bipartiteness to well-linkedness and directed metric embeddings, offering practical algorithms for network design and analysis. The techniques introduce novel notions (well-linkedness for skew-symmetric graphs) and a versatile framework (cut-matching with MMWU) that may be of independent interest in graph partitioning and flow-based approximation.”

Abstract

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.

Figures (3)

• Figure 1.1: \ref{['fig:ori']} the original graph $G = (V, E)$. Given a vector $\Bx = [1, 1, 1, 1, -1, -1, -1, 0, 0]^\top$, we have a corresponding partition of $V = L \cup R \cup Z$, where $L = \{1, 2, 3, 4\}$, $R = \{5, 6, 7\}$, and $Z = \{8, 9\}$. \ref{['fig:bip']} the corresponding bipartite graph $G' = (V^+ \cup V^-, E')$, where each edge $(u, v) \in E$ correponds to two edges $(u^+, v^-)$ and $(v^+, u^-)$ in $E'$. Moreover, the three subsets $L, R$, and $Z$ of $V$ correspond to the subsets $L^+, L^-$, $R^+, R^-$, and $Z^+, Z^-$ of $V'$, respectively.
• Figure 3.1: The figure of auxiliary network $N_{A, B, r}$, where $A = L^+ \cup R^-$, $B = L^- \cup R^+$, and $s^+, s^-$ are the source and sink. Additionally, $s^+$ has an edge to each vertex $u \in A$ with capacity $b(u)$, each $v \in B$ has an edge to $s^-$ with capacity $b(v)$, and each edge $e \in E'$ has capacity $w(e) / r$.
• Figure 5.1: The gadget for directed edge $e = (i, j)$. The dummy vertices are drawn as hollow circles.

Theorems & Definitions (50)

• Theorem 1.1: informal version of \ref{['thm:cut-matching']}
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Flow decomposition theorem
• Definition 2.2: demand graph
• Lemma 2.3
• Lemma 2.4: Laurent-Massart bound, Laurent2000
• Lemma 2.5
• proof