AI 中文总结
该研究针对几何二分图的加权双团覆盖问题，开发了首个具可证明保证的近似算法，在真实数据集上实现了比基准更小的双团表示，可扩展至大型图。
AI 中文摘要
二分图是推荐系统、社交网络和通信图中关系数据的基础表示形式。这些场景中的一个关键挑战是在保留精确结构和路径信息的同时，紧凑地存储和传输大型二分图。我们研究二分图G=(V,U,E)的基于双团的表示，其中边集使用一组完全二分图进行编码。我们关注加权双团覆盖问题，该问题最小化所有双团使用的顶点总数，并引入了一个广义变体，该变体还会惩罚双团的数量，以捕捉存储、传输和模型复杂度方面的实际开销。虽然已知加权双团覆盖问题是NP完全问题，但我们证明该广义变体也是NP完全问题。尽管存在这种难解性，但许多现实世界的二分图具有低维几何嵌入，或者可以通过几何嵌入很好地近似。利用这一观察结果，我们开发了针对几何二分图上（广义）加权双团覆盖问题的首个具有可证明保证的近似算法。具体而言，对于低维ℓ_∞^d空间中的δ-圆盘二分图，我们设计了一种多项式时间算法，该算法结合贪心集合覆盖、几何范围搜索和 densest 子图优化的思想，实现了O(log|U|·log^d|V|)的近似比。我们还展示了我们的算法如何扩展到任意α≥1的ℓ_α^d度量。最后，我们在现实世界的二分图数据集上评估了我们的算法，结果表明它们能够高效地计算出比自然基准小得多的基于双团的表示，同时可扩展到大型图。
英文摘要
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 $δ$-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 $\ell_α^d$ metrics for any $α\geq 1$. 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.
