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.

Summary In this paper, a real‐time traffic index model of expressways is proposed by using a traffic index to evaluate the actual conditions of expressways. The model considers the actual situation of floating and nonfloating vehicles on expressways, in the context of massive floating car data. Included is the realization of the complete calculation model of real‐time traffic index estimation, including highway section division, spatial topology map matching, driving route calculation, and road congestion status judgment. For roads without floating car coverage, the weighted‐moving‐average time‐series prediction method is used to predict the traffic index, so that the running condition of all roads in the network can be analyzed completely. The simulation results show that the proposed highway traffic index can reflect not only the overall high‐speed operation but also real‐time congestion at specific high‐speed or specific highway intervals, providing an effective reference for travel.