arXiv subjects

Kazuhiro Kurita

Publications and source records attributed to Kazuhiro Kurita.

At least 19 recordsLinked to original sources

arXiv2026

An ETH-based quasipolynomial lower bound for Dualization

Dualizing monotone Boolean functions (or equivalently, enumerating minimal transversals in hypergraphs) is a long-standing problem whose output-polynomial-time solvability remains open. While various special cases have been extensively studied, the state-of-the-art algorithm for the general case, due to Fredman and Khachiyan, runs in quasipolynomial time. This paper presents a subexponential-time reduction from \textsc{3SAT} to the complement of \textsc{Dual}: Given a 3CNF formula with $n$ variables, the reduction constructs hypergraphs $\mathcal H$ and $\mathcal L$ of total size $2^{\bigoh(n^{2/3}(\log n)^{1/3})}$ such that $\mathcal L \subseteq \Tr(\mathcal H)$ and the formula is satisfiable if and only if $\mathcal L\neq \Tr(\mathcal H)$. As a consequence of this reduction, assuming the Exponential Time Hypothesis (ETH), neither \textsc{Dual} nor \textsc{Dualization} admits an algorithm running in $N^{o(\sqrt{\log N/\log\log N})}$ time, where $N$ is the input size for \textsc{Dual} and the combined input and output size for \textsc{Dualization}. In particular, \textsc{Dualization} cannot be solved in output-polynomial time under ETH.

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arXiv2026

Algorithms for Optimally Shifting Intervals under Intersection Graph Models

We introduce Geometric Graph Edit Distance, a model for modifying geometric intersection graphs by moving objects. The objective is to obtain a graph in a prescribed class while minimising the total (weighted) moving distance of the objects. For weighted intervals of arbitrary length, we give a prune-and-search algorithm that renders their intersection graph complete in linear time. We adapt this approach to maximise the common intersection length of weighted intervals within a movement threshold in linear time. We also extend the algorithm for obtaining complete graphs to weighted axis-parallel rectangles of arbitrary size under the $L_1$ distance. For unweighted unit intervals, we give an $O(n\log n)$-time algorithm for obtaining a graph containing a $k$-clique. Lastly, we give $O(n)$-size linear programs for obtaining graphs in several classes, including edgeless, acyclic, $k$-clique-free and $k$-connected graphs.

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arXiv2026

Frameworks to Design Approximation Algorithms for Finding Diverse Solutions in Combinatorial Problems

Finding a \emph{single} best solution is the most common objective in combinatorial optimization problems. However, such a single solution may not be applicable to real-world problems as objective functions and constraints are only "approximately" formulated for original real-world problems. To solve this issue, finding \emph{multiple} solutions is a natural direction, and diversity of solutions is an important concept in this context. Unfortunately, finding diverse solutions is much harder than finding a single solution. To cope with difficulty, we investigate the approximability of finding diverse solutions. As a main result, we propose a framework to design approximation algorithms for finding diverse solutions, which yields several outcomes including constant-factor approximation algorithms for finding diverse matchings in graphs and diverse common bases in two matroids and PTASes for finding diverse minimum cuts and interval schedulings.

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arXiv2026

The Complexity of Maximal/Closed Frequent Tree Mining for Bounded Height Trees

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.

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arXiv2026

Further Results on Rendering Geometric Intersection Graphs Sparse by Dispersion

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.

cs.CG↗

arXiv2025

Finding One Local Optimum Is Easy -- but What About Two?

The class PLS (Polynomial Local Search) captures the complexity of finding a solution that is locally optimal and has proven to be an important concept in the theory of local search. It has been shown that local search versions of various combinatorial optimization problems, such as Maximum Independent Set and Max Cut, are complete for this class. Such computational intractability typically arises in local search problems allowing arbitrary weights; in contrast, for unweighted problems, locally optimal solutions can be found in polynomial time under standard settings. In this paper, we pursue the complexity of local search problems from a different angle: We show that computing two locally optimal solutions is NP-hard for various natural unweighted local search problems, including Maximum Independent Set, Minimum Dominating Set, Max SAT, and Max Cut. We also discuss several tractable cases for finding two (or more) local optimal solutions.

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arXiv2025

On the Complexity of Hyperpath and Minimal Separator Enumeration in Directed Hypergraphs

In this paper, we address the enumeration of (induced) $s$-$t$ paths and minimal $s$-$t$ separators. These problems are some of the most famous classical enumeration problems that can be solved in polynomial delay by simple backtracking for a (un)directed graph. As a generalization of these problems, we consider the (induced) $s$-$t$ hyperpath and minimal $s$-$t$ separator enumeration in a \emph{directed hypergraph}. We show that extending these classical enumeration problems to directed hypergraphs drastically changes their complexity. More precisely, there are no output-polynomial time algorithms for the enumeration of induced $s$-$t$ hyperpaths and minimal $s$-$t$ separators unless $P = NP$, and if there is an output-polynomial time algorithm for the $s$-$t$ hyperpath enumeration, then the minimal transversal enumeration can be solved in output polynomial time even if a directed hypergraph is $BF$-hypergraph. Since the existence of an output-polynomial time algorithm for the minimal transversal enumeration has remained an open problem for over 45 years, it indicates that the $s$-$t$ hyperpath enumeration for a $BF$-hypergraph is not an easy problem. As a positive result, the $s$-$t$ hyperpath enumeration for a $B$-hypergraph can be solved in polynomial delay by backtracking.

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arXiv2025

Enumerating Graphlets with Amortized Time Complexity Independent of Graph Size

Graphlets of order $k$ in a graph $G$ are connected subgraphs induced by $k$ nodes (called $k$-graphlets) or by $k$ edges (called edge $k$-graphlets). They are among the interesting subgraphs in network analysis to get insights on both the local and global structure of a network. While several algorithms exist for discovering and enumerating graphlets, the cost per solution of such algorithms typically depends on the size of the graph $G$, or its maximum degree. In real networks, even the latter can be in the order of millions, whereas $k$ is typically required to be a small value. In this paper we provide the first algorithm to list all graphlets of order $k$ in a graph $G=(V,E)$ with an amortized cost per solution depending \emph{solely} on the order $k$, contrarily to previous approaches where the cost depends \emph{also} on the size of $G$ or its maximum degree. Specifically, we show that it is possible to list $k$-graphlets in $O(k^2)$ time per solution, and to list edge $k$-graphlets in $O(k)$ time per solution. Furthermore we show that, if the input graph has bounded degree, then the cost per solution for listing $k$-graphlets is reduced to $O(k)$. Whenever $k = O(1)$, as it is often the case in practical settings, these algorithms are the first to achieve constant time per solution.

cs.DS↗

arXiv2025

The Complexity of Maximal Common Subsequence Enumeration

Frequent pattern mining is widely used to find ``important'' or ``interesting'' patterns in data. While it is not easy to mathematically define such patterns, maximal frequent patterns are promising candidates, as frequency is a natural indicator of relevance and maximality helps to summarize the output. As such, their mining has been studied on various data types, including itemsets, graphs, and strings. The complexity of mining maximal frequent itemsets and subtrees has been thoroughly investigated (e.g., [Boros et al., 2003], [Uno et al., 2004]) in the literature. On the other hand, while the idea of mining frequent subsequences in sequential data was already introduced in the seminal paper [Agrawal et al., 1995], the complexity of the problem is still open. In this paper, we investigate the complexity of the maximal common subsequence enumeration problem, which is both an important special case of maximal frequent subsequence mining and a generalization of the classic longest common subsequence (LCS) problem. We show the hardness of enumerating maximal common subsequences between multiple strings, ruling out the possibility of an \emph{output-polynomial time} enumeration algorithm under $P \neq NP$, that is, an algorithm that runs in time ${\rm poly}(|\mathcal I| + N)$, where $|\mathcal I|$ and $N$ are the size of the input and number of output solutions, respectively. To circumvent this intractability, we also investigate the parameterized complexity of the problem, and show several results when the alphabet size, the number of strings, and the length of a string are taken into account as parameters.

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arXiv2025

On the Complexity of Minimising the Moving Distance for Dispersing Objects

We study Geometric Graph Edit Distance (GGED), a graph-editing model to compute the minimum edit distance of intersection graphs that uses moving objects as an edit operation. We first show an $O(n\log n)$-time algorithm that minimises the total moving distance to disperse unit intervals. This algorithm is applied to render a given unit interval graph (i) edgeless, (ii) acyclic and (iii) $k$-clique-free. We next show that GGED becomes strongly NP-hard when rendering a weighted interval graph (i) edgeless, (ii) acyclic and (iii) $k$-clique-free. Lastly, we prove that minimising the maximum moving distance for rendering a unit disk graph edgeless is strongly NP-hard over the $L_1$ and $L_2$ distances.

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arXiv2024

On the complexity of finding a spanning even tree in a graph

A tree is said to be even if for every pair of distinct leaves, the length of the unique path between them is even. In this paper we discuss the problem of determining whether an input graph has a spanning even tree. Hofmann and Walsh [Australas. J Comb. 35, 2006] proved that this problem can be solved in polynomial time on bipartite graphs. In contrast to this, we show that this problem is NP-complete even on planar graphs. We also give polynomial-time algorithms for several restricted classes of graphs, such as split graphs, cographs, cobipartite graphs, unit interval graphs, and block graphs.

cs.DS↗

arXiv2024

Computing diverse pair of solutions for tractable SAT

In many decision-making processes, one may prefer multiple solutions to a single solution, which allows us to choose an appropriate solution from the set of promising solutions that are found by algorithms. Given this, finding a set of \emph{diverse} solutions plays an indispensable role in enhancing human decision-making. In this paper, we investigate the problem of finding diverse solutions of Satisfiability from the perspective of parameterized complexity with a particular focus on \emph{tractable} Boolean formulas. We present several parameterized tractable and intractable results for finding a diverse pair of satisfying assignments of a Boolean formula. In particular, we design an FPT algorithm for finding an ``almost disjoint'' pair of satisfying assignments of a $2$CNF formula.

cs.DS↗

arXiv2024

Enumerating minimal vertex covers and dominating sets with capacity and/or connectivity constraints

In this paper, we consider the problems of enumerating minimal vertex covers and minimal dominating sets with capacity and/or connectivity constraints. We develop polynomial-delay enumeration algorithms for these problems on bounded-degree graphs. For the case of minimal connected vertex covers, our algorithms run in polynomial delay even on the class of $d$-claw free graphs, extending the result on bounded-degree graphs, and in output quasi-polynomial time on general graphs. To complement these algorithmic results, we show that the problems of enumerating minimal connected vertex covers, minimal connected dominating sets, and minimal capacitated vertex covers in $2$-degenerated bipartite graphs are at least as hard as enumerating minimal transversals in hypergraphs.

cs.DS↗

arXiv2024

Efficient Constant-Factor Approximate Enumeration of Minimal Subsets for Monotone Properties with Weight Constraints

A property $Π$ on a finite set $U$ is \emph{monotone} if for every $X \subseteq U$ satisfying $Π$, every superset $Y \subseteq U$ of $X$ also satisfies $Π$. Many combinatorial properties can be seen as monotone properties. The problem of finding a minimum subset of $U$ satisfying $Π$ is a central problem in combinatorial optimization. Although many approximate/exact algorithms have been developed to solve this kind of problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building accurate mathematical models on real-world problems. A promising approach to overcome this difficulty is to \emph{enumerate} multiple small solutions rather than to \emph{find} a single small solution. To this end, given a weight function $w: U \to \mathbb N$ and an integer $k$, we devise algorithms that \emph{approximately} enumerate all minimal subsets of $U$ with weight at most $k$ satisfying $Π$ for various monotone properties $Π$, where "approximate enumeration" means that algorithms output all minimal subsets satisfying $Π$ whose weight at most $k$ and may output some minimal subsets satisfying $Π$ whose weight exceeds $k$ but is at most $ck$ for some constant $c \ge 1$. These algorithms allow us to efficiently enumerate minimal vertex covers, minimal dominating sets in bounded degree graphs, minimal feedback vertex sets, minimal hitting sets in bounded rank hypergraphs, etc., of weight at most $k$ with constant approximation factors.

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arXiv2024

An Approximation Algorithm for $K$-best Enumeration of Minimal Connected Edge Dominating Sets with Cardinality Constraints

\emph{$K$-best enumeration}, which asks to output $k$-best solutions without duplication, is a helpful tool in data analysis for many fields. In such fields, graphs typically represent data. Thus subgraph enumeration has been paid much attention to such fields. However, $k$-best enumeration tends to be intractable since, in many cases, finding one optimum solution is \NP-hard. To overcome this difficulty, we combine $k$-best enumeration with a concept of enumeration algorithms called \emph{approximation enumeration algorithms}. As a main result, we propose a $4$-approximation algorithm for minimal connected edge dominating sets which outputs $k$ minimal solutions with cardinality at most $4\cdot\OPT$, where $\OPT$ is the cardinality of a minimum solution which is \emph{not} outputted by the algorithm. Our proposed algorithm runs in $\order{nm^2Δ}$ delay, where $n$, $m$, $Δ$ are the number of vertices, the number of edges, and the maximum degree of an input graph.

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arXiv2024

Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids and Beyond

Finding a maximum cardinality common independent set in two matroids (also known as \textsc{Matroid Intersection}) is a classical combinatorial optimization problem, which generalizes several well-known problems, such as finding a maximum bipartite matching, a maximum colorful forest, and an arborescence in directed graphs. Enumerating all maximal common independent sets in two (or more) matroids is a classical enumeration problem. In this paper, we address an ``intersection'' of these problems: Given two matroids and a threshold $τ$, the goal is to enumerate all maximal common independent sets in the matroids with cardinality at least $τ$. We show that this problem can be solved in polynomial delay and polynomial space. Moreover, our technique can be extended to a more general problem, which is relevant to Matroid Matching. We give a polynomial-delay and polynomial-space algorithm for enumerating all maximal ``matchings'' with cardinality at least $τ$, assuming that the optimization counterpart is ``tractable'' in a certain sense. This extension allows us to enumerate small minimal connected vertex covers in subcubic graphs. We also discuss a framework to convert enumeration with cardinality constraints into ranked enumeration.

math.CO↗

arXiv2023

On the hardness of inclusion-wise minimal separators enumeration

Enumeration problems are often encountered as key subroutines in the exact computation of graph parameters such as chromatic number, treewidth, or treedepth. In the case of treedepth computation, the enumeration of inclusion-wise minimal separators plays a crucial role. However and quite surprisingly, the complexity status of this problem has not been settled since it has been posed as an open direction by Kloks and Kratsch in 1998. Recently at the PACE 2020 competition dedicated to treedepth computation, solvers have been circumventing that by listing all minimal $a$-$b$ separators and filtering out those that are not inclusion-wise minimal, at the cost of efficiency. Naturally, having an efficient algorithm for listing inclusion-wise minimal separators would drastically improve such practical algorithms. In this note, however, we show that no efficient algorithm is to be expected from an output-sensitive perspective, namely, we prove that there is no output-polynomial time algorithm for inclusion-wise minimal separators enumeration unless P = NP.

cs.DM↗

arXiv2023

Dichotomies for Tree Minor Containment with Structural Parameters

The problem of determining whether a graph $G$ contains another graph $H$ as a minor, referred to as the minor containment problem, is a fundamental problem in the field of graph algorithms. While it is NP-complete when $G$ and $H$ are general graphs, it is sometimes tractable on more restricted graph classes. This study focuses on the case where both $G$ and $H$ are trees, known as the tree minor containment problem. Even in this case, the problem is known to be NP-complete. In contrast, polynomial-time algorithms are known for the case when both trees are caterpillars or when the maximum degree of $H$ is a constant. Our research aims to clarify the boundary of tractability and intractability for the tree minor containment problem. Specifically, we provide dichotomies for the computational complexities of the problem based on three structural parameters: the diameter, pathwidth, and path eccentricity.

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