发表机构
University of Southern California; USC Information Sciences Institute（南加州大学; 南加州大学信息科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究通过图同构测试，探讨具有全局注意力的图基础模型对混合整数线性规划的表达能力，证明一类分层图变换器受1-WL测试限制，验证多种图编码器，表明超越其表达能力源于输入编码，提供与编码器无关的诊断方法。
AI 中文摘要
具有全局注意力的图基础模型（GFMs）越来越多地用于表示混合整数线性规划（MILPs），旨在捕捉超越标准图神经网络局部性的结构。我们通过图同构测试研究它们的表达能力，询问哪些MILP实例会映射到相同的表示。我们证明，一类广泛的分层图变换器，结合全局线性注意力、边加权交叉注意力和二分消息传递，受一维Weisfeiler-Leman（1-WL）测试的限制：在任何参数设置下，1-WL等效的MILP图会接收相同的图嵌入。我们的组合证明表明，每个架构组件都是一个对称多重集函数，因此保留1-WL等效性。我们在十种不同的图编码器上验证了这一特征，包括Graphormer、GraphGPS、Set-Transformer和Gasse风格的模型。在模型容量、图规模和池化算子方面，每个测试编码器都将1-WL等效的非同构图对映射到数值相同的嵌入。因此，无法从这些表示中恢复在1-WL等效类中变化的图不变量。我们进一步表明，超越1-WL的表达能力来自输入编码而非注意力：随机游走位置编码分离了构造的对，而其他构造揭示了这种补救方法的局限性。这些结果表征了全局注意力GFMs的表达能力，并为检测1-WL诱导的表示等效性提供了一种与编码器无关的诊断方法。
英文摘要
Graph foundation models (GFMs) with global attention are increasingly used to represent mixed-integer linear programs (MILPs), aiming to capture structure beyond the locality of standard graph neural networks. We study their expressive power through graph isomorphism testing, asking which MILP instances they map to identical representations. We prove that a broad class of hierarchical graph transformers combining global linear attention, edge-weighted cross-attention, and bipartite message passing is bounded by the one-dimensional Weisfeiler-Leman (1-WL) test: under any parameter setting, 1-WL-equivalent MILP graphs receive identical graph embeddings. Our compositional proof shows that each architectural component is a symmetric multiset function and thus preserves 1-WL equivalence. We validate this characterization across ten diverse graph encoders, including Graphormer-, GraphGPS-, Set-Transformer-, and Gasse-style models. Across model capacities, graph scales, and pooling operators, every tested encoder maps 1-WL-equivalent non-isomorphic graph pairs to numerically identical embeddings. Consequently, graph invariants that vary within a 1-WL equivalence class cannot be recovered from these representations. We further show that expressiveness beyond 1-WL arises from input encoding rather than attention: random-walk positional encodings separate the constructed pairs, while additional constructions expose the limits of this remedy. These results characterize the expressive power of global-attention GFMs and provide an encoder-agnostic diagnostic for detecting 1-WL-induced representation equivalence.
CommentsAccepted to Topology, Algebra, and Geometry in Data Science (TAG-DS) 2026 (20 pages, 10 figures) [Lightning Oral Presentation]
