A New Upper Bound Based on Vertex Partitioning for the Maximum K-plex Problem

Abstract

Given an undirected graph, the Maximum k-plex Problem (MKP) is to find a largest induced subgraph in which each vertex has at most k−1 non-adjacent vertices. The problem arises in social network analysis and has found applications in many important areas employing graph-based data mining. Existing exact algorithms usually implement a branch-and-bound approach that requires a tight upper bound to reduce the search space. In this paper, we propose a new upper bound for MKP, which is a partitioning of the candidate vertex set with respect to the constructing solution. We implement a new branch-and-bound algorithm that employs the upper bound to reduce the number of branches. Experimental results show that the upper bound is very effective in reducing the search space. The new algorithm outperforms the state-of-the-art algorithms significantly on real-world massive graphs, DIMACS graphs and random graphs.

An Exact Algorithm with New Upper Bounds for the Maximum k-Defective Clique Problem in Massive Sparse Graphs

“…The second one is massive sparse graphs from real-world applications. Among which, 139 graphs, originally from the Network Data Repository online (Rossi and Ahmed 2015), were frequently tested in previous works (Rossi and Ahmed 2014;Cai 2015;Lin et al 2017;Jiang et al 2021) 2 . The third one is a set of massive graphs from DIMACS10 3 and SNAP 4 benchmarks, where 37 instances were tested the same as the instances used in (Chen et al 2021).…”

“…In the past decade, graphs from real-world applications have revealed some new characteristics, i.e., massive, sparse and the power-law distribution of vertex degrees. To find the maximum (relaxed) clique or enumerate maximal (relaxed) cliques in such graphs, a number of algorithms for cliques Cai and Lin 2016;Jiang, Li, and Manyà 2017), k-plexes (Xiao et al 2017;Miao and Balasundaram 2017;Conte et al 2017;Zhou et al 2021;Jiang et al 2021), andquasi-cliques (Sanei-Mehri et al 2021;Marinelli, Pizzuti, and Rossi 2021) have been proposed. However, compared with the clique and its other relaxations, existing k-defective algorithms can hardly handle massive sparse graphs.…”