A Refined Upper Bound and Inprocessing for the Maximum K-plex Problem

A k-plex of a graph G is an induced subgraph in which every vertex has at most k-1 nonadjacent vertices. The Maximum k-plex Problem (MKP) consists in finding a k-plex of the largest size, which is NP-hard and finds many applications. Existing exact algorithms mainly implement a branch-and-bound approach and improve performance by integrating effective upper bounds and graph reduction rules. In this paper, we propose a refined upper bound, which can derive a tighter upper bound than existing methods, and an inprocessing strategy, which performs graph reduction incrementally. We implement a new BnB algorithm for MKP that employs the two components to reduce the search space. Extensive experiments show that both the refined upper bound and the inprocessing strategy are very efficient in the reduction of search space. The new algorithm outperforms the state-of-the-art algorithms on the tested benchmarks significantly.