Improved Approximation Algorithm for Maximum Balanced Biclique

2026-04-29

AI Summary

This work addresses the Maximum Balanced Biclique (MBB) problem in bipartite graphs, which seeks the largest complete bipartite subgraph with $n$ vertices on each side. We propose a novel polynomial-time approximation algorithm grounded in combinatorial optimization and graph theory. By employing a refined probabilistic analysis within an improved approximation framework, we enhance the best-known approximation ratio from $n/\Omega((\log n)^2)$ to $n/\widetilde{O}((\log n)^3)$. This advancement resolves an open question posed by Chalermsook et al. and aligns the approximability of MBB with Feige’s classical result for the Maximum Clique problem up to an $O(\log \log n)$ factor, thereby achieving the current best theoretical guarantee for this fundamental problem.

Abstract

We study the Maximum Balanced Biclique (MBB) problem: Given a bipartite graph $G$ with $n$ vertices on each side, find a balanced biclique in $G$ with maximum size. We give a polynomial-time $\left(\frac{n}{\widetildeΩ\left((\log n)^3\right)}\right)$-approximation algorithm for the problem, which improves upon an $\left(\frac{n}{Ω\left((\log n)^2\right)}\right)$-approximation by Chalermsook et al. (2020) and answers their open question. Furthermore, our approximation ratio matches that of the maximum clique problem by Feige (2004) up to an $O(\log \log n)$ factor.

Problem

• Maximum Balanced Biclique
• bipartite graph
• balanced biclique
• approximation algorithm
• combinatorial optimization

Innovation

• Maximum Balanced Biclique
• Approximation Algorithm
• Bipartite Graph
• Polynomial-time Algorithm
• Feige's Approximation