[论文解读] Rethinking Graph Neural Networks for the Graph Coloring Problem

Wei Li, Ruxuan Li+4 more|arXiv (Cornell University)|Aug 15, 2022

Advanced Graph Neural Networks被引用 6

一句话总结

该论文通过识别标准AC-GNN因局部归纳偏差和无法处理异质性而失败，重新思考了图神经网络（GNN）在图着色问题中的应用。作者提出了一种简单且深度增强的AC-GNN变体——GDN，通过提升区分能力和颜色等变性，显著优于最先进的启发式算法，在大规模工业图上的解决方案质量和速度方面最高提升10倍。

ABSTRACT

Graph coloring, a classical and critical NP-hard problem, is the problem of assigning connected nodes as different colors as possible. However, we observe that state-of-the-art GNNs are less successful in the graph coloring problem. We analyze the reasons from two perspectives. First, most GNNs fail to generalize the task under homophily to heterophily, i.e., graphs where connected nodes are assigned different colors. Second, GNNs are bounded by the network depth, making them possible to be a local method, which has been demonstrated to be non-optimal in Maximum Independent Set (MIS) problem. In this paper, we focus on the aggregation-combine GNNs (AC-GNNs), a popular class of GNNs. We first define the power of AC-GNNs in the coloring problem as the capability to assign nodes different colors. The definition is different with previous one that is based on the assumption of homophily. We identify node pairs that AC-GNNs fail to discriminate. Furthermore, we show that any AC-GNN is a local coloring method, and any local coloring method is non-optimal by exploring the limits of local methods over sparse random graphs, thereby demonstrating the non-optimality of AC-GNNs due to its local property. We then prove the positive correlation between model depth and its coloring power. Moreover, we discuss the color equivariance of graphs to tackle some practical constraints such as the pre-fixing constraints. Following the discussions above, we summarize a series of rules a series of rules that make a GNN color equivariant and powerful in the coloring problem. Then, we propose a simple AC-GNN variation satisfying these rules. We empirically validate our theoretical findings and demonstrate that our simple model substantially outperforms state-of-the-art heuristic algorithms in both quality and runtime.

研究动机与目标

• 调查为何最先进的GNN在图着色问题上表现不佳，尤其是在异质性设置下。
• 分析AC-GNN在着色任务中的局限性，特别是其局部归纳偏差和深度限制。
• 建立理论规则，用于设计在异质性设置下具备强大区分能力和颜色等变性的GNN。
• 开发并验证一种简单有效的AC-GNN变体（GDN），其在性能上超越现有GNN和启发式算法。

提出的方法

• 作者重新定义了GNN在着色任务中的能力，不再以将局部等价节点映射到相同嵌入为目标，而是强调为必须不同的节点（如相连节点）分配不同颜色的能力。
• 他们证明了模型深度与着色能力之间存在正相关关系，表明更深的GNN在全局区分能力上更占优势。
• 他们推导出对抗同质性常见GNN实践的设计规则，例如避免过度平滑并偏好更深架构。
• 他们提出了GDN，一种简单的AC-GNN变体，通过架构改进增强区分能力并确保颜色等变性。
• 该方法采用无监督学习预测节点颜色，无需真实标签。
• 在三个数据集（layout、COLOR和Queen）上进行了实证验证，与GNN-GCP、Tabucol和HybridEA进行比较。

实验结果

研究问题

• RQ1 为何标准AC-GNN在图着色任务中失败，尽管其在同质性任务中表现成功？
• RQ2 模型深度如何影响AC-GNN的着色能力，尤其是在异质性设置下？
• RQ3 哪些架构规则能够增强GNN对必须分配不同颜色的节点的区分能力？
• RQ4 一个简单的AC-GNN变体能否在解决方案质量与效率两方面均超越最先进的启发式算法？

主要发现

• 所提出的GDN模型在解决方案质量和运行时间上均优于最先进的启发式算法（Tabucol和HybridEA），在大规模图上最高实现10倍的速度提升。
• 更深的AC-GNN显示出与着色能力的正相关关系，这与“深度导致同质性任务中过度平滑”的普遍认知相矛盾。
• 广泛使用的集成AC-GNN（GCN）在着色任务中的表现显著劣于其他GNN，验证了作者提出的规则：集成聚合在异质性场景下会损害性能。
• GDN相较于其他AC-GNN，实现了更高的固定颜色比例（颜色等变性），表明其对预着色约束具有更强的鲁棒性。
• 在Queen13_13数据集上，GDN实现了33 ± 2个冲突数，运行时间为2.38 ± 0.32秒，其在准确性和速度上均优于HybridEA和Tabucol。
• 该方法在复杂且大规模的图（如COLOR和Queen数据集）上表现出色，这些图在VLSI设计等工业应用中十分常见。