随机图着色中的相变
Phase Transitions in the Coloring of Random Graphs
摘要：
本文考虑了一个大型稀疏随机图的顶点着色问题，即用给定的颜色数对图中的顶点进行着色，使得相邻的顶点都不具有相同的颜色。使用腔方法，我们提出了一个详细的和系统的分析研究空间的正常染色（解决方案）。我们表明，对于一个固定数量的颜色和平均顶点度（约束数）的增加，一组的解决方案经历了几个相变中观察到的玻璃的平均场论。首先，在聚类过渡，熵占主导地位的部分相空间分解成指数数量的纯态，使超出此过渡的解决方案的均匀采样变得困难。之后，溶液的空间在有限数量的最大状态上凝聚，因此溶液的总熵变得比退火的小。当在所有熵占优势的状态中，有一小部分节点冻结时，就会发生另一种转变，这样，在状态内部的所有解中，每个节点都可以有一种颜色。最终，在着色阈值以上，没有更多的解决方案可用。我们计算了Erdos-Renyi和正则随机图的所有临界连通度，并确定了它们在大量颜色下的渐近值。最后，我们讨论了我们的研究结果的算法后果。我们认为，计算硬度的发病与聚类过渡无关，我们建议，而不是冻结过渡可能是相关的现象。我们还讨论了一个简单的本地Walk-COL算法和信念传播算法的性能，根据我们的研究结果。
英文摘要：
We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions). We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdos-Renyi and regular random graphs and determine their asymptotic values for large number of colors. Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.
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