We show how to couple phase-oscillators on a graph so that collective dynamics “searches” for the coloring of that graph as it relaxes toward the dynamical equilibrium. This translates a combinatorial optimization problem (graph coloring) into a functional optimization problem (finding and evaluating the global minimum of dynamical non-equilibrium potential, done by the natural system’s evolution). Using a sample of graphs, we show that our method can serve as a viable alternative to the traditional combinatorial algorithms. Moreover, we show that, with the same computational cost, our method efficiently solves the harder problem of improper coloring of weighed graphs.

中文翻译：

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相斥振荡器的集体动力学解决了图形着色问题

我们展示了如何在图形上耦合相位振荡器，以使集体动力学“搜索”该图形向动态平衡松弛时的着色。这将组合优化问题（图形着色）转换为功能优化问题（通过自然系统的演化完成，找到并评估动态非平衡势的全局最小值）。使用图形样本，我们证明了我们的方法可以替代传统的组合算法。此外，我们表明，在相同的计算成本的情况下，我们的方法有效地解决了加权图着色不正确的难题。