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 \(\mathsf {MC\text {-}BRB}\) 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 \(\mathsf {MC\text {-}EGO}\) for heuristically computing a near-maximum clique in near-linear time, and we extend our \(\mathsf {MC\text {-}BRB}\) 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.

中文翻译：

---

大型稀疏图的有效最大集团计算和枚举

本文研究稀疏图的最大集团计算（MCC）问题，因为大型现实世界图通常是稀疏的。在文献中，稀疏图上的MCC问题已被单独研究，没有比其稠密对应物（稠密图上的MCC）研究范围更广，并且针对现有的MCC求解器尚未使用针对稠密图上的MCC开发的高级算法技术。稀疏图。在本文中，我们为稀疏图设计了一种算法\（\ mathsf {MC \ text {-} BRB} \），该算法将大型稀疏图G上的MCC实例转换为稠密的k -clique发现（KCF）实例G的子图，每个都可以由现有的MCC求解器针对密集图进行计算。为了进一步提高效率，我们然后通过提出轻量减少技术并利用现有MCC求解器中用于稠密图的先进分支和定界技术，为稠密图上的KCF开发了一个新的分支减少与约束框架。 。此外，我们还设计了一个以自我为中心的算法\（\ mathsf {MC \ text {-} EGO} \），用于在近似线性时间内启发式计算近最大集团，并扩展\（\ mathsf {MC \ text {-} BRB} \）枚举所有最大集团的算法。最后，我们并行化算法以利用多个CPU内核。我们对大型实图进行了广泛的实证研究，并证明了我们技术的效率和有效性。