New results for the detection of bicliques
George Manoussakis
TL;DR
This work tackles the enumeration and detection of bicliques under degree constraints. It introduces an $\mathcal{O}(\Delta^2)$ output-sensitive algorithm to enumerate all maximal non-induced bicliques in general graphs via a bipartite transformation and a space-efficient duplicate-elimination scheme. It further develops a framework for detecting and counting maximum bicliques: given a base $\mathcal{O}(c^n n^{O(1)})$ subroutine, it achieves $\mathcal{O}(n c^{\Delta} \Delta^{O(1)})$ time, and by leveraging existing maximum-biclique algorithms, attains $\mathcal{O}(n \cdot 1.2109^{\Delta})$ for finding a maximum biclique and $\mathcal{O}(n \cdot 1.2377^{\Delta})$ for counting all maximum bicliques (extendable to size-$k$). The paper also clarifies the relationship between non-induced bicliques in general graphs and induced bicliques in bipartite graphs, offering a practical path for degree-bounded graph mining with implications for data mining and network analysis.
Abstract
Building on existing algorithms and results, we offer new insights and algorithms for various problems related to detecting maximal and maximum bicliques. Most of these results focus on graphs with small maximum degree, providing improved complexities when this parameter is constant; a common characteristic in real-world graphs.