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When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases

Journal of Artificial Intelligence Research

The subgraph isomorphism problem involves deciding whether a copy of a pattern graph occurs inside a larger target graph. The non-induced version allows extra edges in the target, whilst the induced version does not. Although both variants are NP-complete, algorithms inspired by constraint programming can operate comfortably on many real-world problem instances with thousands of vertices. However, they cannot handle arbitrary instances of this size. We show how to generate "really hard" random instances for subgraph isomorphism problems, which are computationally challenging with a couple of hundred vertices in the target, and only twenty pattern vertices. For the non-induced version of the problem, these instances lie on a satisfiable / unsatisfiable phase transition, whose location we can predict; for the induced variant, much richer behaviour is observed, and constrainedness gives a better measure of difficulty than does proximity to a phase transition. These results have practical consequences: we explain why the widely researched "filter / verify" indexing technique used in graph databases is founded upon a misunderstanding of the empirical hardness of NP-complete problems, and cannot be beneficial when paired with any reasonable subgraph isomorphism algorithm.


A Learning based Branch and Bound for Maximum Common Subgraph Problems

arXiv.org Machine Learning

Branch-and-bound (BnB) algorithms are widely used to solve combinatorial problems, and the performance crucially depends on its branching heuristic.In this work, we consider a typical problem of maximum common subgraph (MCS), and propose a branching heuristic inspired from reinforcement learning with a goal of reaching a tree leaf as early as possible to greatly reduce the search tree size.Extensive experiments show that our method is beneficial and outperforms current best BnB algorithm for the MCS.


A Constraint Programming Approach to Weighted Isomorphic Mapping of Fragment-based Shape Signatures

arXiv.org Artificial Intelligence

Fragment-based shape signature techniques have proven to be powerful tools for computer-aided drug design. They allow scientists to search for target molecules with some similarity to a known active compound. They do not require reference to the full underlying chemical structure, which is essential to deal with chemical databases containing millions of compounds. However, finding the optimal match of a part of the fragmented compound can be time-consuming. In this paper, we use constraint programming to solve this specific problem. It involves finding a weighted assignment of fragments subject to connectivity constraints. Our experiments demonstrate the practical relevance of our approach and open new perspectives, including generating multiple, diverse solutions. Our approach constitutes an original use of a constraint solver in a real time setting, where propagation allows to avoid an enumeration of weighted paths. The model must remain robust to the addition of constraints making some instances not tractable. This particular context requires the use of unusual criteria for the choice of the model: lightweight, standard propagation algorithms, data structures without prohibitive constant cost. The objective is not to design new, complex algorithms to solve difficult instances.


Fast Detection of Maximum Common Subgraph via Deep Q-Learning

arXiv.org Machine Learning

Detecting the Maximum Common Subgraph (MCS) between two input graphs is fundamental for applications in biomedical analysis, malware detection, cloud computing, etc. This is especially important in the task of drug design, where the successful extraction of common substructures in compounds can reduce the number of experiments needed to be conducted by humans. However, MCS computation is NP-hard, and state-of-the-art exact MCS solvers do not have worst-case time complexity guarantee and cannot handle large graphs in practice. Designing learning based models to find the MCS between two graphs in an approximate yet accurate way while utilizing as few labeled MCS instances as possible remains to be a challenging task. Here we propose RLMCS, a Graph Neural Network based model for MCS detection through reinforcement learning. Our model uses an exploration tree to extract subgraphs in two graphs one node pair at a time, and is trained to optimize subgraph extraction rewards via Deep Q-Networks. A novel graph embedding method is proposed to generate state representations for nodes and extracted subgraphs jointly at each step. Experiments on real graph datasets demonstrate that our model performs favorably to exact MCS solvers and supervised neural graph matching network models in terms of accuracy and efficiency.


Subgraph Matching Kernels for Attributed Graphs

arXiv.org Machine Learning

We propose graph kernels based on subgraph matchings, i.e. structure-preserving bijections between subgraphs. While recently proposed kernels based on common subgraphs (Wale et al., 2008; Shervashidze et al., 2009) in general can not be applied to attributed graphs, our approach allows to rate mappings of subgraphs by a flexible scoring scheme comparing vertex and edge attributes by kernels. We show that subgraph matching kernels generalize several known kernels. To compute the kernel we propose a graph-theoretical algorithm inspired by a classical relation between common subgraphs of two graphs and cliques in their product graph observed by Levi (1973). Encouraging experimental results on a classification task of real-world graphs are presented.