Finding a Maximum Clique is a classic property test from graph theory; find any one of the largest complete subgraphs in an Erd\"os-R\'enyi G(N, p) random graph. We use Maximum Clique to explore the structure of the problem as a function of N, the graph size, and K, the clique size sought. It displays a complex phase boundary, a staircase of steps at each of which 2log2 N and Kmax, the maximum size of a clique that can be found, increases by 1. Each of its boundaries has a finite width, and these widths allow local algorithms to find cliques beyond the limits defined by the study of infinite systems. We explore the performance of a number of extensions of traditional fast local algorithms, and find that much of the "hard" space remains accessible at finite N. The "hidden clique" problem embeds a clique somewhat larger than those which occur naturally in a G(N, p) random graph. Since such a clique is unique, we find that local searches which stop early, once evidence for the hidden clique is found, may outperform the best message passing or spectral algorithms.

翻译：寻找最大团是图论中的一个经典性质测试问题：在Erdős–Rényi G(N, p)随机图中找到任意一个最大的完全子图。我们利用最大团问题来探究该问题随图规模N和所求团规模K变化的结构特性。该问题展现出复杂的相边界——一个阶梯状结构，其中每级台阶上2log2 N与可找到的最大团规模Kmax增加1。每个边界均具有有限宽度，这些宽度使得局部算法能够超越无限系统研究定义的极限来找到团。我们探索了多种传统快速局部算法扩展的性能，发现大量"困难"空间在有限N下仍可触及。"隐藏团"问题在G(N, p)随机图中嵌入一个略大于自然生成规模的团。由于此类团具有唯一性，我们发现在找到隐藏团证据后提前终止的局部搜索算法，其表现可能优于最佳的消息传递算法或谱算法。