The minimum dominating set (MDS) problem is a crucial NP-hard combinatorial optimization problem with wide applications in real-world scenarios. In this paper, we propose an efficient local search algorithm namely NuMDS to solve the MDS, which comprises three key ideas. First, we introduce a dominate propagation-based reduction method that fixes a portion of vertices in a given graph. Second, we develop a novel two-phase initialization method based on the decomposition method. Third, we propose a multi-stage local search procedure, which adopts three different search manners according to the current stage of the search. We conduct extensive experiments to demonstrate the outstanding effectiveness of NuMDS, and the results clearly indicate that NuMDS outperforms previous state-of-the-art algorithms on almost all instances.

The maximum k-club problem (MkCP) is an important clique relaxation problem with wide applications. Previous MkCP algorithms only work on small-scale instances and are not applicable for large-scale instances. For solving instances with different scales, this paper develops an efficient local search algorithm named NukCP for the MkCP which mainly includes two novel ideas. First, we propose a dynamic reduction strategy, which makes a good balance between the time efficiency and the precision effectiveness of the upper bound calculation. Second, a stratified threshold configuration checking strategy is designed by giving different priorities for the neighborhood in the different levels. Experiments on a broad range of different scale instances show that NukCP significantly outperforms the state-of-the-art MkCP algorithms on most instances.

The maximum quasi-clique problem (MQCP) is an important extension of maximum clique problem with wide applications. Recent heuristic MQCP algorithms can hardly solve large and hard graphs effectively. This paper develops an efficient local search algorithm named NuQClq for the MQCP, which has two main ideas. First, we propose a novel vertex selection strategy, which utilizes cumulative saturation information to be a selection criterion when the candidate vertices have equal values on the primary scoring function. Second, a variant of configuration checking named BoundedCC is designed by setting an upper bound for the threshold of forbidding strength. When the threshold value of vertex exceeds the upper bound, we reset its threshold value to increase the diversity of search process. Experiments on a broad range of classic benchmarks and sparse instances show that NuQClq significantly outperforms the state-of-the-art MQCP algorithms for most instances.

The maximum k-plex, a generalization of maximum clique, is used to cope with a great number of real-world problems. The aim of this paper is to propose a novel exact k-plex algorithm that can deal with large-scaled graphs with millions of vertices and edges. Specifically, we first propose several new graph reduction methods through a careful analyzing of structures of induced subgraphs. Afterwards, we present a preprocessing method to simplify initial graphs. Additionally, we present a branch-and-bound algorithm integrating the reduction methods as well as a new dynamic vertex selection mechanism. We perform intensive experiments to evaluate our algorithm, and show that the proposed strategies are effective and our algorithm outperforms state-of-the-art algorithms, especially for real-world massive graphs.

The minimum weight dominating set (MWDS) problem is NP-hard and also important in many applications. Recent heuristic MWDS algorithms can hardly solve massive real world graphs effectively. In this paper, we design a fast local search algorithm called FastMWDS for the MWDS problem, which aims to obtain a good solution on massive graphs within a short time. In this novel local search framework, we propose two ideas to make it effective. Firstly, we design a new fast construction procedure with four reduction rules to cut down the size of massive graphs. Secondly, we propose the three-valued two-level configuration checking strategy to improve local search, which is interestingly a variant of configuration checking (CC) with two levels and multiple values. Experiment results on a broad range of massive real world graphs show that FastMWDS finds much better solutions than state of the art MWDS algorithms.