NuQClq: An Effective Local Search Algorithm for Maximum Quasi-Clique Problem

Abstract

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.

A local search algorithm with hybrid strategies for the maximum weighted quasi‐clique problem

“…If the solution is infeasible for a certain rounds (MaxRound = 4000), the algorithm restarts with a new solution (line 11), and if it achieves the limited time it returns the best weighted γ -quasi-clique (line 12). Moreover, the bounded-based config- uration check strategy [11] is also employed to restrict some moves of vertices so that avoid searching cycles. Experiment: All experiments were carried out on a computer with Cen-tOS 7.6.1810, configured with Intel(R) Xeon(R) E5-2630v4 CPUs (2.20 GHz) and 32GB RAM.…”

“…Three γ values are considered: γ = 0.999, 0.95, 0.90. We compare our algorithm with NuQClq, which is the state-of-the-art algorithm for the maximum γ -quasi-clique problem and outperforms the existing methods as reported [11]. We ran our algorithm and NuQClq 10 times for each instance with 1800 s. The average results over 10 runs were counted for each instance and each value of the density threshold γ .…”

“…Recently, algorithms for finding the maximum quasi-cliques have been well studied. A series of algorithms have been proposed including exact algorithms and heuristic methods [4][5][6][7][8][9][10][11]. As an early work, Pajouh et al proposed a branch-and-bound algorithm for the maximum γ -quasi-clique problem [4].…”

“…Then, based on the hybrid of a biased random-key genetic algorithm and an exact local search strategy, Pinto et al proposed a method to solve the maximal cardinality quasi-clique problem [10]. Very recently, Chen et al have devised an efficient local search algorithm, called NuQClq, and proposed two novel strategies incorporated into their algorithm [11]. Experimental results show that NuQClq has achieved an excellent performance in a large number of benchmark instances, and outperforms the previous methods.…”

“…It counts the number of critical edges. Given a vertex v, its cumulative measurement is calculated as the total number of critical edges abstracts the number of steps that v is in S since its last move to or out of S. For more details, we refer to [11].…”