Equivariant Neural Primal-Dual Assignment for Maximum Common Edge Subgraphs

2026-09-26

AI Summary

This study addresses the prohibitive query overhead in molecular similarity search, where maximum common edge subgraph (MCES) matching necessitates pairwise network training. We propose ENPDA, an equivariant neural primal-dual assignment method that learns a shared matching strategy across graph pairs. By integrating Hungarian projection with exact marginal computation, ENPDA generalizes to unseen graph pairs without retraining while providing parameter-free pairwise guarantees and global optimality certificates. Experimental results demonstrate that the proposed approach accelerates query speed by three orders of magnitude, improves accuracy by 7.4–8.6 percentage points over baselines, and achieves cross-domain transfer gains of up to 17.6 percentage points, thereby enabling efficient and generalizable graph matching.

Abstract

Maximum common edge subgraph (MCES) matching finds a partial vertex correspondence between two labeled graphs that preserves as many labeled edges as possible. Molecular similarity search requires matching many graph pairs, making the cost of repeated queries important. The strongest baseline attains accurate MCES solutions but trains a separate network for each pair. We introduce Equivariant Neural Primal-Dual Assignment (ENPDA), which learns a shared matching policy and applies it to new pairs without further training, answering queries roughly three orders of magnitude faster and recovering its training cost after a few dozen queries. The policy recomputes exact objective marginals for candidate matches and learns corrections and step sizes that update their scores. Target prices respond to competition when several source vertices favor the same target. Four update rounds and a Hungarian projection produce a partial one-to-one matching. We prove per-pair guarantees that hold for any network parameters. In exact arithmetic, reordering either graph permutes the assignment and price states, the projected matching is one-to-one, and repaired prices give a valid MCES upper bound. Subtracting the preserved-edge count bounds the optimality gap; combined with structural caps, these certificates prove global optimality for 60 of 291 native test pairs. On three molecular benchmarks with disjoint train/validation/test splits, ENPDA improves over an analytic counterpart with the same update and projection budget by 7.4-8.6 accuracy points; after one second of refinement search, 2.5-3 points of the gain remain. Transferred without fine-tuning to edge-deletion tasks from social and protein graphs, the policy gains 9.1-17.6 points over the analytic counterpart. When output matchings must keep aromatic rings intact, ENPDA recovers more reference bonds than the baselines on all three datasets.

Problem

• Maximum Common Edge Subgraph
• Graph Matching
• Molecular Similarity Search
• Equivariant Neural Network

Innovation

• Equivariant Neural Primal-Dual Assignment
• Maximum Common Edge Subgraph
• Graph Matching
• Molecular Similarity
• Hungarian Projection