Efficient maximum clique computation and enumeration over large sparse graphs

This paper studies the problem of maximum clique computation (MCC) over sparse graphs, as large real-world graphs are usually sparse. In the literature, the problem of MCC over sparse graphs has been studied separately and less extensively than its dense counterpart—MCC over dense graphs—and advanced algorithmic techniques that are developed for MCC over dense graphs have not been utilized in the existing MCC solvers for sparse graphs. In this paper, we design an algorithm MC-BRB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {MC\text {-}BRB}$$\end{document} for sparse graphs which transforms an instance of MCC over a large sparse graph G to instances of k-clique finding (KCF) over dense subgraphs of G, each of which can be computed by the existing MCC solvers for dense graphs. To further improve the efficiency, we then develop a new branch-reduce-&-bound framework for KCF over dense graphs by proposing light-weight reducing techniques and leveraging the advanced branching and bounding techniques that are used in the existing MCC solvers for dense graphs. In addition, we also design an ego-centric algorithm MC-EGO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {MC\text {-}EGO}$$\end{document} for heuristically computing a near-maximum clique in near-linear time, and we extend our MC-BRB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {MC\text {-}BRB}$$\end{document} algorithm to enumerate all maximum cliques. Finally, we parallelize our algorithms to exploit multiple CPU cores. We conduct extensive empirical studies on large real graphs and demonstrate the efficiency and effectiveness of our techniques.