On Maximising the Vertex Coverage for ${\text{Top}}-k$ t-Bicliques in Bipartite Graphs

Enumeration of all maximal bicliques in bipartite graphs is a well-studied fundamental problem. However, a wide range of applications need less overlapping bicliques with specific size constraints instead of all the maximal bicliques. In this paper, we study a new biclique problem, called the top-k t-biclique coverage problem. A t-biclique is a biclique with a size constraint $t$ for one vertex set and the problem aims to find $k$ t-bicliques maximising the coverage on the other vertex set. The top-k t-biclique coverage problem has novel applications such as finding top-k courses while maximising student engagement. We prove that this problem is NP-hard. A straightforward way to address the problem first needs to enumerate and store all t-bicliques and then greedily select $k$ promising t-bicliques, leading an approximate guarantee on the coverage. However, it takes exponential space, which is impractical. We then apply a fast approximation scheme to solve this problem, which shaves the exponential space consumption by progressively updating top-k results during the t-biclique enumeration. Observing that the fast approximation algorithm takes too much time on updating the results due to the coverage is computed from scratch for each update, an online index is devised to address the drawback. Due the hardness of the problem, even the fast approximation algorithm cannot scale to large dataset. To devise a scalable solution, we then propose a heuristic algorithm running in polynomial time. Thanks for four carefully designed heuristic rules, the heuristic algorithm can find large coverage top-k t-bicliques extremely fast for large datasets. Apart from that, the heuristic result with large coverage can effectively prune unpromising enumerations in the fast greedy algorithm, which improves the efficiency of the fast approximation algorithm without compromising the approximation ratio. Extensive experiments are conducted on real datasets to justify the effectiveness and efficiency of the proposed algorithms.