有向反馈顶点集问题的自旋玻璃方法
A spin glass approach to the directed feedback vertex set problem
摘要：
有向图（有向图）由顶点和从一个顶点到另一个顶点的弧（有向边）构成。反馈顶点集（FVS）是指有向图中每个有向圈至少包含一个顶点的顶点集。有向反馈顶点集问题的目标是构造一个基数最小的有向反馈顶点集。这是一个基本的循环约束组合优化问题，具有广泛的实际应用。通过将全局环约束转化为局部弧约束，构造了有向FVS问题的自旋玻璃模型，并利用统计物理中的复制对称（RS）平均场论对该模型进行了研究.然后，我们实现了一个信念传播引导抽取（BPD）算法的单有向图的实例。BPD算法在大型随机图实例上的性能略优于模拟退火算法。RS平均场论的预测值明显低于BPD的结果，这可能是由于它忽略了周期引起的长程相关性。
英文摘要：
A directed graph (digraph) is formed by vertices and arcs (directed edges) from one vertex to another. A feedback vertex set (FVS) is a set of vertices that contains at least one vertex of every directed cycle in this digraph. The directed feedback vertex set problem aims at constructing a FVS of minimum cardinality. This is a fundamental cycle-constrained hard combinatorial optimization problem with wide practical applications. In this paper we construct a spin glass model for the directed FVS problem by converting the global cycle constraints into local arc constraints, and study this model through the replica-symmetric (RS) mean field theory of statistical physics. We then implement a belief propagation-guided decimation (BPD) algorithm for single digraph instances. The BPD algorithm slightly outperforms the simulated annealing algorithm on large random graph instances. The predictions of the RS mean field theory are noticeably lower than the BPD results, possibly due to its neglect of cycle-caused long range correlations.
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