To this day, the maximum clique problem remains a computationally challenging problem. Indeed, despite researchers’ best efforts, there exist unsolved benchmark instances with one thousand vertices. However, relatively simple algorithms solve real-life instances with millions of vertices in a few seconds. Why is this the case? Why is the problem apparently so easy in many naturally occurring networks? In this paper, we provide an explanation. First, we observe that the graph’s clique number ω is very near to the graph’s degeneracy d in most real-life instances. This observation motivates a main contribution of this paper, which is an algorithm for the maximum clique problem that runs in time polynomial in the size of the graph, but exponential in the gap g := (d+ 1)− ω between the clique number ω and its degeneracy-based upper bound d + 1. When this gap g can be treated as a constant, as is often the case for real-life graphs, the proposed algorithm runs in time O(dm) = O(m). This provides a rigorous explanation for the apparent easiness of these instances despite the intractability of the problem in the worst case. Further, our implementation of the proposed algorithm is actually practical—competitive with the best approaches from the literature.

中文翻译：

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为什么在实践中通常容易实现最大派系？

时至今日，最大派系问题仍然是计算难题。确实，尽管研究人员做出了最大的努力，但仍存在具有一千个顶点的未解决基准实例。但是，相对简单的算法可以在几秒钟内解决具有数百万个顶点的真实实例。为什么会这样呢？为什么在许多自然发生的网络中问题似乎如此容易？在本文中，我们提供了一个解释。首先，我们观察到在大多数现实情况下，图的集团数ω非常接近图的简并性d。这种观察激发了本文的主要贡献，这是一种针对最大集团问题的算法，该算法在图形大小的时间多项式中运行，但在集团数ω之间的间隙g：=（d + 1）-ω中呈指数形式以及基于退化的上限d + 1。当这个间隙g可以被视为一个常数时（对于现实生活中的图形通常如此），所提出的算法在O（dm）= O（m）的时间内运行。尽管在最坏的情况下问题难以解决，但是这为这些情况的明显易用性提供了严格的解释。此外，我们对提出的算法的实现实际上是可行的，与文献中的最佳方法相竞争。