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.
翻译:图形颜色是一个古典和尖锐的 NP- 硬性问题, 是将连接节点尽可能不同颜色指定为本地方法的问题。 然而, 我们观察到, 最先进的 GNN 点在图形颜色问题中不太成功。 我们从两个角度分析原因。 首先, 大多数 GNN 点没有将任务以单调的方式概括到不同颜色, 也就是说, 将连接节点分配为不同颜色的图形。 第二, GNN 受网络深度的约束, 使得它们有可能成为本地方法, 在最大独立赛( MIS) 问题中, 显示它们不是最优化的 。 在本文中, 我们侧重于 GNNNN( AC- GNNN) 的集合- GNNN (AC- G- GNNN) (AC- G- GNNNN), 这是GNNN( G- G- G) 最受欢迎的类别规则, 我们首先将颜色规则定义与直观模型的不同。 我们发现A- G- G- G- G- 直观 的模型的先前模式在假设中, 直观 直观 直观 直观 直观规则中, 我们用直观的直观的直观的直观规则, 我们用直观的直观规则 展示的直交的直交的直方的直方的直方规则, 我们用直方的直方的直方的直方方法展示的直方的直方的直方规则, 我们用直方方法展示的直方的直方的直方, 直方的直方的直方的直方, 直方的直方的直方的直方的直方的直方的直方的直方的直方的直方的直方, 。