arXiv subjects

Yu Nakahata

Publications and source records attributed to Yu Nakahata.

11 recordsLinked to original sources

arXiv2025

Reconfiguring Multiple Connected Components with Size Multiset Constraints

We propose a novel generalization of Independent Set Reconfiguration (ISR): Connected Components Reconfiguration (CCR). In CCR, we are given a graph $G$, two vertex subsets $A$ and $B$, and a multiset $\mathcal{M}$ of positive integers. The question is whether $A$ and $B$ are reconfigurable under a certain rule, while ensuring that each vertex subset induces connected components whose sizes match the multiset $\mathcal{M}$. ISR is a special case of CCR where $\mathcal{M}$ only contains 1. We also propose new reconfiguration rules: component jumping (CJ) and component sliding (CS), which regard connected components as tokens. Since CCR generalizes ISR, the problem is PSPACE-complete. In contrast, we show three positive results: First, CCR-CS and CCR-CJ are solvable in linear and quadratic time, respectively, when $G$ is a path. Second, we show that CCR-CS is solvable in linear time for cographs. Third, when $\mathcal{M}$ contains only the same elements (i.e., all connected components have the same size), we show that CCR-CJ is solvable in linear time if $G$ is chordal. The second and third results generalize known results for ISR and exhibit an interesting difference between the reconfiguration rules.

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arXiv2025

Computing Time-varying Network Reliability using Binary Decision Diagrams

Computing the reliability of a time-varying network, taking into account its dynamic nature, is crucial for networks that change over time, such as space networks, vehicular ad-hoc networks, and drone networks. These networks are modeled using temporal graphs, in which each edge is labeled with a time indicating its existence at a specific point in time. The time-varying network reliability is defined as the probability that a data packet from the source vertex can reach the terminal vertex, following links with increasing time labels (i.e., a journey), while taking into account the possibility of network link failures. Currently, the existing method for calculating this reliability involves explicitly enumerating all possible journeys between the source and terminal vertices and then calculating the reliability using the sum of disjoint products method. However, this method has high computational complexity. In contrast, there is an efficient algorithm that uses binary decision diagrams (BDDs) to evaluate the reliability of a network whose topology does not change over time. This paper presents an efficient exact algorithm that utilizes BDDs for computing the time-varying network reliability. Experimental results show that the proposed method runs faster than the existing method up to four orders of magnitude.

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arXiv2023

Reconfiguring (non-spanning) arborescences

In this paper, we investigate the computational complexity of subgraph reconfiguration problems in directed graphs. More specifically, we focus on the problem of reconfiguring arborescences in a digraph, where an arborescence is a directed graph such that its underlying undirected graph forms a tree and all vertices have in-degree at most 1. Given two arborescences in a digraph, the goal of the problem is to determine whether there is a (reconfiguration) sequence of arborescences between the given arborescences such that each arborescence in the sequence can be obtained from the previous one by removing an arc and then adding another arc. We show that this problem can be solved in polynomial time, whereas the problem is PSPACE-complete when we restrict arborescences in a reconfiguration sequence to directed paths or relax to directed acyclic graphs. We also show that there is a polynomial-time algorithm for finding a shortest reconfiguration sequence between two spanning arborescences.

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arXiv2022

ZDD-Based Algorithmic Framework for Solving Shortest Reconfiguration Problems

This paper proposes an algorithmic framework for various reconfiguration problems using zero-suppressed binary decision diagrams (ZDDs), a data structure for families of sets. In general, a reconfiguration problem checks if there is a step-by-step transformation between two given feasible solutions (e.g., independent sets of an input graph) of a fixed search problem such that all intermediate results are also feasible and each step obeys a fixed reconfiguration rule (e.g., adding/removing a single vertex to/from an independent set). The solution space formed by all feasible solutions can be exponential in the input size, and indeed many reconfiguration problems are known to be PSPACE-complete. This paper shows that an algorithm in the proposed framework efficiently conducts the breadth-first search by compressing the solution space using ZDDs, and finds a shortest transformation between two given feasible solutions if exists. Moreover, the proposed framework provides rich information on the solution space, such as the connectivity of the solution space and all feasible solutions reachable from a specified one. We demonstrate that the proposed framework can be applied to various reconfiguration problems, and experimentally evaluate their performances.

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arXiv2022

Independent set reconfiguration on directed graphs

\textsc{Directed Token Sliding} asks, given a directed graph and two sets of pairwise nonadjacent vertices, whether one can reach from one set to the other by repeatedly applying a local operation that exchanges a vertex in the current set with one of its out-neighbors, while keeping the nonadjacency. It can be seen as a reconfiguration process where a token is placed on each vertex in the current set, and the local operation slides a token along an arc respecting its direction. Previously, such a problem was extensively studied on undirected graphs, where the edges have no directions and thus the local operation is symmetric. \textsc{Directed Token Sliding} is a generalization of its undirected variant since an undirected edge can be simulated by two arcs of opposite directions. In this paper, we initiate the algorithmic study of \textsc{Directed Token Sliding}. We first observe that the problem is PSPACE-complete even if we forbid parallel arcs in opposite directions and that the problem on directed acyclic graphs is NP-complete and W[1]-hard parameterized by the size of the sets in consideration. We then show our main result: a linear-time algorithm for the problem on directed graphs whose underlying undirected graphs are trees, which are called polytrees. Such a result is also known for the undirected variant of the problem on trees~[Demaine et al.~TCS 2015], but the techniques used here are quite different because of the asymmetric nature of the directed problem. We present a characterization of yes-instances based on the existence of a certain set of directed paths, and then derive simple equivalent conditions from it by some observations, which admits an efficient algorithm. For the polytree case, we also present a quadratic-time algorithm that outputs, if the input is a yes-instance, one of the shortest reconfiguration sequences.

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arXiv2022

On the Clique-Width of Unigraphs

Clique-width is a well-studied graph parameter. For graphs of bounded clique-width, many problems that are NP-hard in general can be polynomial-time solvable. The fact motivates several studies to investigate whether the clique-width of graphs in a certain class is bounded or not. We focus on unigraphs, that is, graphs that are uniquely determined by their degree sequences up to isomorphism. We show that every unigraph has clique-width at most 4. It follows that many problems that are NP-hard in general are polynomial-time solvable for unigraphs.

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arXiv2021

A Note on Exponential-Time Algorithms for Linearwidth

In this note, we give an algorithm that computes the linearwidth of input $n$-vertex graphs in time $O^*(2^n)$, which improves a trivial $O^*(2^m)$-time algorithm, where $n$ and $m$ the number of vertices and edges, respectively.

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arXiv2020

Compiling Crossing-free Geometric Graphs with Connectivity Constraint for Fast Enumeration, Random Sampling, and Optimization

Given $n$ points in the plane, we propose algorithms to compile connected crossing-free geometric graphs into directed acyclic graphs (DAGs). The DAGs allow efficient counting, enumeration, random sampling, and optimization. Our algorithms rely on Wettstein's framework to compile several crossing-free geometric graphs. One of the remarkable contributions of Wettstein is to allow dealing with geometric graphs with connectivity, since it is known to be difficult to efficiently represent geometric graphs with such global property. To achieve this, Wettstein proposed specialized techniques for crossing-free spanning trees and crossing-free spanning cycles and invented compiling algorithms running in $\mathrm{O}(7.044^n)$ time and $\mathrm{O}(5.619^n)$ time, respectively. Our first contribution is to propose a technique to deal with the connectivity constraint more simply and efficiently. It makes the design and analysis of algorithms easier, and yields improved time complexity. Our algorithms achieve $\mathrm{O}(6^n)$ time and $\mathrm{O}(4^n)$ time for compiling crossing-free spanning trees and crossing-free spanning cycles, respectively. As the second contribution, we propose an algorithm to optimize the area surrounded by crossing-free spanning cycles. To achieve this, we modify the DAG so that it has additional information. Our algorithm runs in $\mathrm{O}(4.829^n)$ time to find an area-minimized (or maximized) crossing-free spanning cycle of a given point set. Although the problem was shown to be NP-complete in 2000, as far as we know, there were no known algorithms faster than the obvious $\mathrm{O}(n!)$ time algorithm for 20 years.

cs.CG↗

arXiv2019

Implicit Enumeration of Topological-Minor-Embeddings and Its Application to Planar Subgraph Enumeration

Given graphs $G$ and $H$, we propose a method to implicitly enumerate topological-minor-embeddings of $H$ in $G$ using decision diagrams. We show a useful application of our method to enumerating subgraphs characterized by forbidden topological minors, that is, planar, outerplanar, series-parallel, and cactus subgraphs. Computational experiments show that our method can find all planar subgraphs in a given graph at most five orders of magnitude faster than a naive backtracking-based method.

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arXiv2019

Decomposing a Graph into Unigraphs

Unigraphs are graphs uniquely determined by their own degree sequence up to isomorphism. There are many subclasses of unigraphs such as threshold graphs, split matrogenic graphs, matroidal graphs, and matrogenic graphs. Unigraphs and these subclasses are well studied in the literature. Nevertheless, there are few results on superclasses of unigraphs. In this paper, we introduce two types of generalizations of unigraphs: $k$-unigraphs and $k$-strong unigraphs. We say that a graph $G$ is a $k$-unigraph if $G$ can be partitioned into $k$ unigraphs. $G$ is a $k$-strong unigraph if not only each subgraph is a unigraph but also the whole graph can be uniquely determined up to isomorphism, by using the degree sequences of all the subgraphs in the partition. We describe a relation between $k$-strong unigraphs and the subgraph isomorphism problem. We show some properties of $k$-(strong) unigraphs and algorithmic results on calculating the minimum $k$ such that a graph $G$ is a $k$-(strong) unigraph. This paper will open many other research topics.

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arXiv2018

Enumerating Graph Partitions Without Too Small Connected Components Using Zero-suppressed Binary and Ternary Decision Diagrams

Partitioning a graph into balanced components is important for several applications. For multi-objective problems, it is useful not only to find one solution but also to enumerate all the solutions with good values of objectives. However, there are a vast number of graph partitions in a graph, and thus it is difficult to enumerate desired graph partitions efficiently. In this paper, an algorithm to enumerate all the graph partitions such that all the weights of the connected components are at least a specified value is proposed. To deal with a large search space, we use zero-suppressed binary decision diagrams (ZDDs) to represent sets of graph partitions and we design a new algorithm based on frontier-based search, which is a framework to directly construct a ZDD. Our algorithm utilizes not only ZDDs but also ternary decision diagrams (TDDs) and realizes an operation which seems difficult to be designed only by ZDDs. Experimental results show that the proposed algorithm runs up to tens of times faster than an existing state-of-the-art algorithm.

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