Kazuhiro Kurita

Kenta Komoto, Kazuhiro Kurita, Hirotaka Ono

Frequent tree mining asks us to enumerate tree patterns that occur frequently in a database of rooted trees. This problem is motivated by tree-structured data in bioinformatics, such as glycans and pseudoknot-free RNA secondary structures. A direct enumeration of all frequent trees is often highly redundant, because every subtree of a frequent tree is again frequent. Closed and maximal frequent trees are standard ways to reduce this redundancy, but their enumeration can still be computationally hard. In this paper, we study the effect of bounding the height of the input trees. This is a natural restriction for rooted trees, since the height is the depth of the hierarchy. We ask whether closed/maximal frequent tree mining remains hard when every input tree has a small height. Our results show that the answer depends sharply on the model. For rooted unordered trees of height at most 2, we give a polynomial-delay algorithm for enumerating closed frequent trees. On the other hand, for rooted ordered trees of height at most 2, we show that an output-polynomial time algorithm for enumerating closed frequent trees would imply an output-polynomial time algorithm for Dualization. For maximal frequent tree enumeration, we prove that no output-polynomial time algorithm exists unless P = NP already for rooted ordered trees of height at most 2 and for rooted unordered trees of height at most 3. Thus, even very small height bounds do not make the enumeration problems easy in general. At the same time, the unordered closed case of height at most 2 admits polynomial-delay enumeration. These results give a height-based classification of the complexity of closed and maximal frequent tree mining on shallow rooted trees.

Nicolás Honorato-Droguett, Kazuhiro Kurita, Tesshu Hanaka et al.

In well-studied graph modification problems, adding and deleting vertices and edges are used as graph editing operations. We propose a model for graph modification on geometric intersection graphs called Geometric Graph Edit Distance that moves objects as an edit operation. Our results are mainly focused on interval graphs. In particular, we give a linear-time algorithm to find the minimum total moving distance to render an interval graph complete. The approach of this algorithm can be applied for: (i) rendering a unit square graph complete over the $L_1$ distance and (ii) attaining the existence of a $k$-clique on unit interval graphs. In addition, we provide LP-formulations to achieve several properties in the associated graph of unit intervals.

Nicolás Honorato-Droguett, Kazuhiro Kurita, Tesshu Hanaka et al.

Removing overlaps is a central task in domains such as scheduling, visibility, and map labelling. This can be modelled using graphs, where overlap removals correspond to enforcing a certain sparsity constraint on the graph structure. We continue the study of the problem Geometric Graph Edit Distance (GGED), where the aim is to minimise the total cost of editing a geometric intersection graph to obtain a graph contained in a specific graph class. For us, the edit operation is the movement of objects, and the cost is the movement distance. We present an algorithm for rendering the intersection graph of a set of unit circular arcs edgeless and $k$-clique-free in $O(n\log n)$ time, where $n$ is the number of arcs. The algorithm can be also used to solve an open case of the points-spreading problem on cyclic domains [Li \& Wang, CGT 2025]. We also show that GGED remains strongly NP-hard on unweighted interval graphs, solving an open problem of Honorato-Droguett et al. [WADS 2025]. We complement this result by showing that GGED is strongly NP-hard on sets of $d$-balls and $d$-cubes, for any $d\ge 2$. Finally, we present an XP algorithm (parameterised by the number of maximal cliques) that removes all edges from the intersection graph of a set of weighted unit intervals.

Tesshu Hanaka, Yasuaki Kobayashi, Kazuhiro Kurita et al.

Finding diverse solutions in combinatorial problems recently has received considerable attention (Baste et al. 2020; Fomin et al. 2020; Hanaka et al. 2021). In this paper we study the following type of problems: given an integer $k$, the problem asks for $k$ solutions such that the sum of pairwise (weighted) Hamming distances between these solutions is maximized. Such solutions are called diverse solutions. We present a polynomial-time algorithm for finding diverse shortest $st$-paths in weighted directed graphs. Moreover, we study the diverse version of other classical combinatorial problems such as diverse weighted matroid bases, diverse weighted arborescences, and diverse bipartite matchings. We show that these problems can be solved in polynomial time as well. To evaluate the practical performance of our algorithm for finding diverse shortest $st$-paths, we conduct a computational experiment with synthetic and real-world instances.The experiment shows that our algorithm successfully computes diverse solutions within reasonable computational time.