Cops and Robbers is a pursuit evasion game played on a graph, first introduced independently by Quilliot \cite{quilliot1978jeux} and Nowakowski and Winkler \cite{NOWAKOWSKI1983235} over four decades ago. A main interest in recent the literature is identifying the cop number of graph families. The cop number of a graph, $c(G)$, is defined as the minimum number of cops required to guarantee capture of the robber. Determining the cop number is computationally difficult and exact algorithms for this are typically restricted to small graph families. This paper investigates whether classical machine learning methods and graph neural networks can accurately predict a graph's cop number from its structural properties and identify which properties most strongly influence this prediction. Of the classical machine learning models, tree-based models achieve high accuracy in prediction despite class imbalance, whereas graph neural networks achieve comparable results without explicit feature engineering. The interpretability analysis shows that the most predictive features are related to node connectivity, clustering, clique structure, and width parameters, which aligns with known theoretical results. Our findings suggest that machine learning approaches can be used in complement with existing cop number algorithms by offering scalable approximations where computation is infeasible.

翻译：警察与强盗是一种在图（graph）上进行的追逃游戏，最早由 Quilliot \cite{quilliot1978jeux} 以及 Nowakowski 和 Winkler \cite{NOWAKOWSKI1983235} 在四十多年前独立提出。近年来文献中的一个主要关注点是确定图族的警察数。一个图 $G$ 的警察数 $c(G)$ 定义为保证能够捕获强盗所需的最少警察数量。确定警察数在计算上是困难的，其精确算法通常仅限于小的图族。本文研究了经典机器学习方法和图神经网络（graph neural networks）是否能够根据图的结构特性准确预测其警察数，并识别出哪些特性对此预测的影响最为显著。在经典机器学习模型中，基于树的模型尽管存在类别不平衡问题，仍能实现较高的预测准确率；而图神经网络则无需显式的特征工程即可达到可比的结果。可解释性分析表明，最具预测性的特征与节点连通性、聚类性、团（clique）结构以及宽度参数相关，这与已知的理论结果一致。我们的研究结果表明，机器学习方法可以作为现有警察数算法的补充，在计算不可行时提供可扩展的近似解。