Pp-Gnn: Pretraining Position-Aware Graph Neural Networks with the Np-Hard Metric Dimension Problem

Abstract

On a graph G=(V, E), we call S ⊂ V resolving if ∀ u, v ∈ V with u ≠ v, ∃ w ⊂ V such that d(u, w) ≠ d(v, w). The smallest possible cardinality of S is called the metric dimension, computing which is known to be NP-hard. Solving the metric dimension problem (MDP) and the associated minimal resolving set has many important applications across science and engineering. In this paper, we introduce MaskGNN, a method using a graph neural network (GNN) model to learn the minimal resolving set in a self-supervised manner by optimizing a novel surrogate objective. We provide a construction showing the global minimum of this objective coincides with the solution to the MDP. MaskGNN attains 51-72% improvement over the best baseline and up to 98% the reward of integer programming in 0.72% of the running time. On this foundation, we introduce Pretraining Position-aware GNNs (PP-GNN) and evaluate on popular benchmark position-based tasks on graphs. PP-GNN's strong results challenge the currently popular paradigm -- heuristic-driven anchor-selection -- with a new learning-based paradigm -- simultaneously learning the metric basis of the graph and pretraining position-aware representations for transferring to downstream position-based tasks.

Keywords: Graph, Metric dimension, NP-hard, Graph neural network, Graph representation learning