Estimating Biclique Counts with Accuracy Guarantees

Efficiently counting bicliques in large bipartite graphs is a fundamental problem with applications in network analysis, bioinformatics, and social sciences. However, existing exact algorithms do not scale well to large graphs, and current approximation algorithms lack formal accuracy guarantees. In this paper, we present the first approximation algorithm for the (p,q)-biclique counting problem that offers formal accuracy guarantees. Our approach introduces a novel sampling framework, termed BC-Shadow, which refines the sample space using edge-oriented techniques to strategically balance computational costs across algorithmic stages. This refinement increases the density of bicliques in the sample space, reducing the number of samples required for accurate estimation. Our algorithm adaptively determines the number of successful samples that are needed to satisfy predefined error and failure probability, enabling real-time adjustment to graph properties. We further enhance sampling efficiency with a new sampling structure, named zstar, which establishes a one-to-one correspondence with (p,q)-bicliques, eliminating redundancies and improving accuracy. Comprehensive theoretical analyses confirm the algorithm's accuracy and running time guarantees, while extensive experiments on large real-world datasets demonstrate its scalability and effectiveness.