Enumeration of Maximum Common Subtree Isomorphisms with Polynomial-Delay

Abstract

The maximum common subgraph problem asks for the maximum size of a common subgraph of two given graphs. The problem is $\mathsf{NP}$-hard, but can be solved in polynomial time if both, the input graphs and the common subgraph are restricted to trees. Since the optimal solution of the maximum common subtree problem is not unique, the problem of enumerating all solutions, i.e., the isomorphisms between the two subtrees, is of interest. We present the first polynomial-delay algorithm for the problem of enumerating all maximum common subtree isomorphisms between a given pair of trees. Our approach is based on the algorithm of Edmonds for solving the maximum common subtree problem using a dynamic programming approach in combination with bipartite matching problems. As a side result, we obtain a polynomial-delay algorithm for enumerating all maximum weight matchings in a complete bipartite graph. We show how to extend the new approach in order to enumerate all solutions of the maximum weighted common subtree problem and to the maximal common subtree problem. Our experimental evaluation on both, randomly generated as well as real-world instances, demonstrates the practical usefulness of our algorithm.

This work was supported by the German Research Foundation (DFG), priority programme “Algorithms for Big Data” (SPP 1736).