Given an undirected graph, the Maximum Clique Problem (MCP) is to find a largest complete subgraph of the graph. MCP is NP-hard and has found many practical applications. In this paper, we propose a parallel Branch-and- Bound (BnB) algorithm to tackle this NP-hard problem, which carries out multiple bounded searches in parallel. Each search has its upper bound and shares a lower bound with the rest of the searches. The potential benefit of the proposed approach is that an active search terminates as soon as the best lower bound found so far reaches or exceeds its upper bound. We describe the implementation of our highly scalable and efficient parallel MCP algorithm, called PBS, which is based on a state-of-the-art sequential MCP algorithm. The proposed algorithm PBS is evaluated on hard DIMACS and BHOSLIB instances. The results show that PBS achieves a near-linear speedup on most DIMACS instances and a super-linear speedup on most BHOSLIB instances. Finally, we give a detailed analysis that explains the good speedups achieved for the tested instances.

给定一个无向图，最大团问题（MCP）是找到该图的最大完全子图。MCP 是 NP 难的，并且已经找到了许多实际应用。在本文中，我们提出了一种并行分支定界（BnB）算法来解决这个 NP 难题，该算法并行执行多个有界搜索。每个搜索都有其上限，并与其余搜索共享一个下限。所提出方法的潜在好处是，一旦迄今为止找到的最佳下界达到或超过其上限，主动搜索就会终止。我们描述了高度可扩展且高效的并行 MCP 算法（称为 PBS）的实现，该算法基于最先进的顺序 MCP 算法。所提出的算法 PBS 在硬 DIMACS 和 BHOSLIB 实例上进行了评估。结果表明，PBS 在大多数 DIMACS 实例上实现了近线性加速，在大多数 BHOSLIB 实例上实现了超线性加速。最后，我们给出了详细的分析，解释了测试实例所实现的良好加速。