Compact Representations of Geometric Bipartite Graphs via Weighted Biclique Covers

Abstract

Bipartite graphs are a fundamental representation for relational data arising in recommendation systems, social networks, and communication graphs. A key challenge in these settings is to store and transmit large bipartite graphs compactly while preserving exact structural and path information. We study biclique-based representations of bipartite graphs $\boldsymbol{G}=(\boldsymbol{V},\boldsymbol{U},\boldsymbol{E})$, where the edge set is encoded using a collection of complete bipartite subgraphs. We focus on the Weighted Biclique Covering problem, which minimizes the total number of vertices used across all bicliques, and introduce a generalized variant that additionally penalizes the number of bicliques, capturing practical overheads in storage, transmission, and model complexity. While the weighted biclique covering problem is known to be $\mathsf{NP}$-Complete, we show that the generalized variant is also $\mathsf{NP}$-Complete. Despite this hardness, many real-world bipartite graphs admit low-dimensional geometric embeddings or can be well approximated by them. Leveraging this observation, we develop the first approximation algorithms with provable guarantees for the (generalized) weighted biclique covering problem on geometric bipartite graphs. Specifically, for $\delta$-disk bipartite graphs in low-dimensional $\ell_\infty^d$ spaces, we design a polynomial-time algorithm that achieves an $O(\log |\boldsymbol{U}| \cdot \log^d |\boldsymbol{V}|)$-approximation, combining ideas from greedy set cover, geometric range searching, and densest subgraph optimization. We also show how our algorithms extend to $\ell_\alpha^d$ metrics for any $\alpha\geq 1$. Finally, we evaluate our algorithms on real-world bipartite datasets and show that they efficiently compute significantly smaller biclique-based representations than natural baselines, while scaling to large graphs.