The Multi-Maximum and Quasi-Maximum Common Subgraph Problem

The Maximum Common Subgraph problem has been long proven NP-hard. Nevertheless, it has countless practical applications, and researchers are still searching for exact solutions and scalable heuristic approaches. Driven by applications in molecular science and cyber-security, we concentrate on the Maximum Common Subgraph among an indefinite number of graphs. We first extend a state-of-the-art branch-and-bound procedure working on two graphs to N graphs. Then, given the high computational cost of this approach, we trade off complexity for accuracy, and we propose a set of heuristics to approximate the exact solution for N graphs. We analyze sequential, parallel multi-core, and parallel-many core (GPU-based) approaches, exploiting several leveraging techniques to decrease the contention among threads, improve the workload balance of the different tasks, reduce the computation time, and increase the final result size. We also present several sorting heuristics to order the vertices of the graphs and the graphs themselves. We compare our algorithms with a state-of-the-art method on publicly available benchmark sets. On graph pairs, we are able to speed up the exact computation by a 2× factor, pruning the search space by more than 60%. On sets of more than two graphs, all exact solutions are extremely time-consuming and of a complex application in many real cases. On the contrary, our heuristics are far less expensive (as they show a lower-bound for the speed up of 10×), have a far better asymptotic complexity (with speed ups up to several orders of magnitude in our experiments), and obtain excellent approximations of the maximal solution with 98.5% of the nodes on average.