Efficient k-Clique Listing: An Edge-Oriented Branching Strategy

Abstract

k-clique listing is a vital graph mining operator with diverse applications in various networks. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial k-clique with a vertex. These algorithms have the time complexity of O(k · m · (δ/2)k-2 ), where m is the number of edges in the graph and δ is the degeneracy of the graph. In this paper, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial k-clique with two vertices that connect each other (which correspond to an edge ). We explore various edge orderings for EBBkC such that it achieves a time complexity of O( m · δ + k · m · (τ/2)k-2 ), where τ is an integer related to the maximum truss number of the graph and we have τ < δ. The time complexity of EBBkC is better than that of VBBkC algorithms for k>3 since both O(m · δ) and O(k · m · (τ/2)k-2 ) are bounded by O(k · m · (δ/2)k-2 ). Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC based algorithms with the early termination technique consistently and largely outperform the state-of-the-art (VBBkC based) algorithms.