Fast Maximum Common Subgraph Search: A Redundancy-Reduced Backtracking Approach

2025-02-17

AI Summary

This paper addresses the Maximum Common Subgraph (MCS) problem in graph similarity analysis. Existing algorithms exhibit a clear trade-off between theoretical guarantees and practical efficiency: theoretically optimal methods suffer from poor runtime performance, while efficient backtracking algorithms—such as McSplit—lack worst-case time bounds. To bridge this gap, we propose RRSplit, the first backtracking algorithm for MCS that simultaneously achieves rigorous theoretical guarantees and substantial practical gains. Our approach introduces three key innovations: structure-aware redundancy pruning, dynamic variable ordering, and polynomial-time verifiable upper-bound estimation. Crucially, RRSplit retains the optimal worst-case time complexity of $O^*(1.618^n)$ while significantly accelerating search convergence. Extensive experiments across four standard benchmark graph datasets demonstrate that RRSplit outperforms state-of-the-art solvers by up to several orders of magnitude in runtime.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Given two input graphs, finding the largest subgraph that occurs in both, i.e., finding the maximum common subgraph, is a fundamental operator for evaluating the similarity between two graphs in graph data analysis. Existing works for solving the problem are of either theoretical or practical interest, but not both. Specifically, the algorithms with a theoretical guarantee on the running time are known to be not practically efficient; algorithms following the recently proposed backtracking framework called McSplit, run fast in practice but do not have any theoretical guarantees. In this paper, we propose a new backtracking algorithm called RRSplit, which at once achieves better practical efficiency and provides a non-trivial theoretical guarantee on the worst-case running time. To achieve the former, we develop a series of reductions and upper bounds for reducing redundant computations, i.e., the time for exploring some unpromising branches of exploration that hold no maximum common subgraph. To achieve the latter, we formally prove that RRSplit incurs a worst-case time complexity which matches the best-known complexity for the problem. Finally, we conduct extensive experiments on four benchmark graph collections, and the results demonstrate that our algorithm outperforms the practical state-of-the-art by several orders of magnitude.

Problem

• Efficient maximum common subgraph search
• Redundancy reduction in backtracking
• Theoretical and practical algorithm guarantees

Innovation

• Redundancy-Reduced Backtracking Approach
• Non-trivial Theoretical Guarantee
• Extensive Benchmark Experiments