arXiv subjects

Tomoki Nakamigawa

Publications and source records attributed to Tomoki Nakamigawa.

6 recordsLinked to original sources

arXiv2026

Structural Origins of Cubic Complexity in Pebble Motion

The pebble motion problem (PMP) asks whether one configuration of labeled pebbles on a graph can be transformed into another by moving pebbles to adjacent unoccupied vertices. It is a fundamental model of graph reconfiguration and is closely related to multi-agent path finding (MAPF). A central open problem since Kornhauser, Miller, and Spirakis (FOCS 1984) is to understand the origin of the classical $Θ(N^3)$ worst-case behavior. While it is known that every feasible instance on an $N$-vertex graph admits a solution sequence of length $\Ord(N^3)$, it has remained unclear which instances actually require cubic complexity. In this paper, we resolve the long-standing complexity of the pebble motion problem on trees. We show that every feasible instance on an $N$-vertex tree admits a solution sequence of length $\Ord(N^2 \log N)$, computable by an output-sensitive algorithm. Since a lower bound of $Ω(N^2)$ is known, this establishes that the $Θ(N^3)$ phenomenon does not occur on trees and nearly closes the gap $Ω(N^2)\le \OPT(N)\le \Ord(N^3)$ up to a logarithmic factor. Building on this result, we extend our approach to general graphs by applying the tree algorithm to breadth-first spanning trees. This yields an efficient framework that produces $o(N^3)$-length solution sequences for a broad class of instances, including the classical square-grid example, where we recover the $\Ord(N^{3/2})$ bound observed by Kornhauser, Miller, and Spirakis. Finally, by analyzing the behavior of this algorithm, we obtain strong structural restrictions governing when $Θ(N^3)$ complexity can arise. We show that such behavior is possible only under highly constrained conditions, specifically when $Θ(N)$ degree-two vertices lie on cycles of length $Θ(N)$, with each cycle being the shortest containing the corresponding vertex.

math.CO↗

arXiv2026

Star Decompositions of a Cyclic Polygon

Let $V$ be a set of vertices on a circumference in the plane. Let $E$ be a set of directed line segments linking two vertices of $V$. If $E$ forms a set of closed cycles and for all two adjacent edges $uv$ and $vw$, the vertices $u$, $v$, $w$ are arranged in anti-clockwise order, we call $P(V,E)$ a cyclic polygon. A star decomposition $\mathcal{S}$ of a cyclic polygon $P$ is a set of star polygons partitioning the region of $P$ with some additional diagonals. A star decomposition $\mathcal{S}$ is called maximal if there is no other star decomposition $\mathcal{S}'$ such that a set of diagonals of $\mathcal{S}$ is a proper subset of that of $\mathcal{S}'$. In this paper, it is shown that for any two maximal star decompositions $\mathcal{S}_1$ and $\mathcal{S}_2$ of a common cyclic polygon, $\mathcal{S}_1$ can be transformed into $\mathcal{S}_2$ by a finite sequence of diagonal flips. It is also shown that if a cyclic polygon $P$ admits a star decomposition, the number of diagonals contained in a maximal star decomposition of $P$ is $p - (n-2r)(n-2r-1)/2$, where $p$ is the number of all possible diagonals of $P$, $n$ is the number of vertices of $P$, and $r$ is the rotation number of $P$.

math.CO↗

arXiv2023

A Correspondence between Chord Diagrams and Families of 0-1 Young Diagrams

A chord diagram is a set of chords in which no pair of chords has a common endvertex. For a chord diagram $E$ having a crossing $S = \{ ac, bd \}$, by the chord expansion of $E$ with respect to $S$, we have two chord diagrams $E_1 = (E\setminus S) \cup \{ ab, cd \}$ and $E_2 = (E\setminus S) \cup \{ da, bc \}$. Starting from a chord diagram $E$, by iterating expansions, we have a binary tree $T$ such that $E$ is a root of $T$ and a multiset of nonintersecting chord diagrams appear in the set of leaves of $T$. The number of leaves, which is not depending on the choice of expansions, is called the chord expansion number of $E$. A $0$-$1$ Young diagram is a Young diagram having a value of $0$ or $1$ for all boxes. This paper shows that the chord expansion number of some type counts the number of $0$-$1$ Young diagrams under some conditions. In particular, it is shown that the chord expansion number of an $n$-crossing, which corresponds to the Euler number, equals the number of $0$-$1$ Young diagrams of shape $(n,n-1,\ldots,1)$ such that each column has at most one $1$ and each row has an even number of $1$'s.

math.CO↗

arXiv2021

On the average hitting times of the squares of cycles

The exact formula for the average hitting time (HT, as an abbreviation) of simple random walks from one vertex to any other vertex on the square $C^2_N$ of an $N$-vertex cycle graph $C_N$ was given by N. Chair [\textit{Journal of Statistical Physics}, \textbf{154} (2014) 1177-1190]. In that paper, the author gives the expression for the even $N$ case and the expression for the odd $N$ case separately. In this paper, by using an elementary method different from Chair (2014), we give a much simpler single formula for the HT's of simple random walks on $C^2_N$. Our proof is considerably short and fully combinatorial, in particular, has no-need of any spectral graph theoretical arguments. Not only the formula itself but also intermediate results through the process of our proof describe clear relations between the HT's of simple random walks on $C^2_N$ and the Fibonacci numbers.

math.CO↗

arXiv2021

Pebble Exchange Group of Graphs

A graph puzzle ${\rm Puz}(G)$ of a graph $G$ is defined as follows. A configuration of ${\rm Puz}(G)$ is a bijection from the set of vertices of a board graph to the set of vertices of a pebble graph, both graphs being isomorphic to some input graph $G$. A move of pebbles is defined as exchanging two pebbles which are adjacent on both a board graph and a pebble graph. For a pair of configurations $f$ and $g$, we say that $f$ is equivalent to $g$ if $f$ can be transformed into $g$ by a finite sequence of moves. Let ${\rm Aut}(G)$ be the automorphism group of $G$, and let ${\rm 1}_G$ be the unit element of ${\rm Aut}(G)$. The pebble exchange group of $G$, denoted by ${\rm Peb}(G)$, is defined as the set of all automorphisms $f$ of $G$ such that ${\rm 1}_G$ and $f$ are equivalent to each other. In this paper, some basic properties of ${\rm Peb}(G)$ are studied. Among other results, it is shown that for any connected graph $G$, all automorphisms of $G$ are contained in ${\rm Peb}(G^2)$, where $G^2$ is a square graph of $G$.

cs.DM↗

arXiv2012

Agent Arrangement Problem

An {\em arrangement} of an ordered pair $(G_A, G_M)$ of graphs is defined as a function $f$ from $V(G_A)$ to $V(G_M)$ such that, for each vertex $c$ of $G_M$, the vertex-set $f^{-1}(c)$ of $G_A$ either is $\emptyset$ (the case when $c \not\in f(V(G_A))$) or induces a connected subgraph of $G_A$ and that the family $\{f^{-1}(y) : y \in V(G_M), f^{-1}(y) \neq \emptyset\}$ is a partition of $V(G_A)$. Let $f$ be an arrangement of $(G_A, G_M)$, let $pq$ be an edge of $G_M$ and let $U$ be a subset of $f^{-1}(p)$ such that each of the three graphs $G_A[U]$, $G_A[f^{-1}(p)\setminus U]$ and $G_A[f^{-1}(q)\cup U]$ is ether connected or $\emptyset$ and that $\big(f^{-1}(p)\cup f^{-1}(q) \big) \setminus U \neq \emptyset$. A {\em transfer} of $U$ from $p$ to $q$ is defined as the modification $f^{\prime}$ of $f$ such that $f^{\prime}(x):=f(x)$ for every $ x \notin U$ and $f^{\prime}(u):=q$ for every $u \in U$. Two arrangements $f$ and $g$ of $(G_A, G_M)$ are called {\em t-equivalent} if they can be transformed into each other by a finite sequence of transfers. An ordered pair $(G_A, G_M)$ of graphs is called {\em almighty} if every two arrangements of the pair $(G_A, G_M)$ are t-equivalent. In this study, we consider the following two decision problems. [{\bf (P1)}]{For a given pair of arrangements $f$ and $g$ of a given ordered pair $(G_A,G_M)$ of graphs, decide whether $f$ is t-equivalent to $g$ or not.} [{\bf (P2)}]{For a given ordered pair $(G_A,G_M)$ of graphs, decide whether the pair $(G_A,G_M)$ is almighty or not.} We show an $\Od(|E(G_A)|+(|V(G_M)|+|E(G_A)|)|V(G_A)|)$-time algorithm for {\bf (P1)}, and prove the $\co\np$-completeness of {\bf (P2)}.

math.CO↗