2 research outputs found

### Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

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

### Largest Weight Common Subtree Embeddings with Distance Penalties

The largest common embeddable subtree problem asks for the largest possible tree embeddable into two input trees and generalizes the classical maximum common subtree problem. Several variants of the problem in labeled and unlabeled rooted trees have been studied, e.g., for the comparison of evolutionary trees. We consider a generalization, where the sought embedding is maximal with regard to a weight function on pairs of labels. We support rooted and unrooted trees with vertex and edge labels as well as distance penalties for skipping vertices. This variant is important for many applications such as the comparison of chemical structures and evolutionary trees. Our algorithm computes the solution from a series of bipartite matching instances, which are solved efficiently by exploiting their structural relation and imbalance. Our analysis shows that our approach improves or matches the running time of the formally best algorithms for several problem variants. Specifically, we obtain a running time of O(|T| |T\u27|Delta) for two rooted or unrooted trees T and T\u27, where Delta=min{Delta(T),Delta(T\u27)} with Delta(X) the maximum degree of X. If the weights are integral and at most C, we obtain a running time of O(|T| |T\u27|sqrt Delta log (C min{|T|,|T\u27|})) for rooted trees