Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

TL;DR

This paper presents a faster algorithm for the maximum common subtree isomorphism problem in trees, achieving improved theoretical running times and practical performance, especially for trees with bounded degrees.

Contribution

The authors develop a new algorithm with improved running time for the maximum common subtree isomorphism problem in unrooted, unordered trees, outperforming previous approaches.

Findings

Achieves a running time of O(n^2 Δ) for trees of order n and maximum degree Δ.

Outperforms previous algorithms in both theoretical analysis and experiments.

Improves enumeration of all maximum common subtree isomorphisms with a faster algorithm.

Abstract

The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is ${\sf NP}$-hard in general graphs. Confining to trees renders polynomial time algorithms possible and is of fundamental importance for approaches on more general graph classes. Various variants of this problem in trees have been intensively studied. We consider the general case, where trees are neither rooted nor ordered and the isomorphism is maximum w.r.t. a weight function on the mapped vertices and edges. For trees of order $n$ and maximum degree $\Delta$ our algorithm achieves a running time of $\mathcal{O}(n^2\Delta)$ by exploiting the structure of the matching instances arising as subproblems. Thus our algorithm outperforms the best previously known approaches. No faster algorithm is possible for trees of bounded degree and for trees of unbounded degree we show that a further reduction of the running time would directly improve the best known approach to the assignment problem. Combining a polynomial-delay algorithm for the enumeration of all maximum common subtree isomorphisms with central ideas of our new algorithm leads to an improvement of its running time from $\mathcal{O}(n^6+Tn^2)$ to $\mathcal{O}(n^3+Tn\Delta)$, where $n$ is the order of the larger tree, $T$ is the number of different solutions, and $\Delta$ is the minimum of the maximum degrees of the input trees. Our theoretical results are supplemented by an experimental evaluation on synthetic and real-world instances.