Efficient Maximum $k$-Defective Clique Computation with Improved Time Complexity

Abstract

$k$-defective cliques relax cliques by allowing up-to $k$ missing edges from being a complete graph. This relaxation enables us to find larger near-cliques and has applications in link prediction, cluster detection, social network analysis and transportation science. The problem of finding the largest $k$-defective clique has been recently studied with several algorithms being proposed in the literature. However, the currently fastest algorithm KDBB does not improve its time complexity from being the trivial $O(2^n)$, and also, KDBB's practical performance is still not satisfactory. In this paper, we advance the state of the art for exact maximum $k$-defective clique computation, in terms of both time complexity and practical performance. Moreover, we separate the techniques required for achieving the time complexity from others purely used for practical performance consideration; this design choice may facilitate the research community to further improve the practical efficiency while not sacrificing the worst case time complexity. In specific, we first develop a general framework kDC that beats the trivial time complexity of $O(2^n)$ and achieves a better time complexity than all existing algorithms. The time complexity of kDC is solely achieved by non-fully-adjacent-first branching rule, excess-removal reduction rule and high-degree reduction rule. Then, to make kDC practically efficient, we further propose a new upper bound, two reduction rules, and an algorithm for efficiently computing a large initial solution. Extensive empirical studies on three benchmark graph collections with $290$ graphs in total demonstrate that kDC outperforms the currently fastest algorithm KDBB by several orders of magnitude.