Choco Banana is one of Nikoli’s pencil puzzles. We study the computational complexity of Choco Banana. It is shown that deciding whether a given instance of the Choco Banana puzzle has a solution is NP-complete.

This paper proposes graph linear notations and an extension of them with regular expressions. Graph linear notations are a set of strings to represent labeled general graphs. They are extended with regular expressions to represent sets of graphs by specifying chosen parts for selections and repetitions of certain induced subgraphs. Methods for the conversion between graph linear notations and labeled general graphs are shown. The NP-completeness of the membership problem for graph regular expressions is proved.

Chained Block is one of Nikoli's pencil puzzles. We study the computational complexity of Chained Block puzzles. It is shown that deciding whether a given instance of the Chained Block puzzle has a solution is NP-complete.

Peg solitaire is a single-player board game. The goal of the game is to remove all but one peg from the game board. Peg solitaire on graphs is a peg solitaire played on arbitrary graphs. A graph is called solvable if there exists some vertex s such that it is possible to remove all but one peg starting with s as the initial hole. In this paper, we prove that it is NP-complete to decide if a graph is solvable or not.

Calculation is a solitaire card game with a standard 52-card deck. Initially, cards A, 2, 3, and 4 of any suit are laid out as four foundations. The remaining 48 cards are piled up as the stock, and there are four empty tableau piles. The purpose of the game is to move all cards of the stock to foundations. The foundation starting with A is to be built up in sequence from an ace to a king. The other foundations are similarly built up, but by twos, threes, and fours from 2, 3, and 4 until a king is reached. Here, a card of rank i may be used as a card of rank i + 13j for j ∈ {0, 1, 2, 3}. During the game, the player moves (i) the top card of the stock either onto a foundation or to the top of a tableau pile, or (ii) the top card of a tableau pile onto a foundation. We prove that the generalized version of Calculation Solitaire is NP-complete.

Moon-or-Sun, Nagareru, and Nurimeizu are Nikoli's pencil puzzles. We study the computational complexity of Moon-or-Sun, Nagareru, and Nurimeizu puzzles. It is shown that deciding whether a given instance of each puzzle has a solution is NP-complete.

Five Cells and Tilepaint are Nikoli's pencil puzzles. We study the computational complexity of Five Cells and Tilepaint puzzles. It is shown that deciding whether a given instance of each puzzle has a solution is NP-complete.

Some textbooks of formal languages and automata theory implicitly state the structural equality of the binary n-dimensional de Bruijn graph and the state diagram of minimum state deterministic finite automaton which accepts regular language (0+1)*1(0+1)n-1. By introducing special finite automata whose accepting states are refined with two or more colors, we extend this fact to both k-ary versions. That is, we prove that k-ary n-dimensional de Brujin graph and the state diagram for minimum state deterministic colored finite automaton which accepts the (k-1)-tuple of the regular languages (0+1+…+k-1)*1(0+1+…+k-1)n-1,...,and(0+1+…+k-1)*(k-1)(0+1+…+k-1)n-1 are isomorphic for arbitrary k more than or equal to 2. We also investigate the properties of colored finite automata themselves and give computational complexity results on three decision problems concerning color unmixedness of nondeterminisitic ones.

Nurimisaki and Sashigane are Nikoli's pencil puzzles. We study the computational complexity of Nurimisaki and Sashigane puzzles. It is shown that deciding whether a given instance of each puzzle has a solution is NP-complete.

Picross 3D is a popular single-player puzzle video game for the Nintendo DS. It presents a rectangular parallelepiped (i.e., rectangular box) made of unit cubes, some of which must be removed to construct an object in three dimensions. Each row or column has at most one integer on it, and the integer indicates how many cubes in the corresponding 1D slice remain when the object is complete. Kusano et al. showed that Picross 3D is NP-complete and Kimura et al. showed that the counting version, the another solution problem, and the fewest clues problem of Picross 3D are #P-complete, NP-complete, and Σ2P-complete, respectively, where those results are shown for the restricted input that the rectangular parallelepiped is of height four. On the other hand, Igarashi showed that Picross 3D is NP-complete even if the height of the input rectangular parallelepiped is one. Extending the result by Igarashi, we in this paper show that the counting version, the another solution problem, and the fewest clues problem of Picross 3D are #P-complete, NP-complete, and Σ2P-complete, respectively, even if the height of the input rectangular parallelepiped is one. Since the height of the rectangular parallelepiped of any instance of Picross 3D is at least one, our hardness results are best in terms of height.

Kurotto and Juosan are Nikoli's pencil puzzles. We study the computational complexity of Kurotto and Juosan puzzles. It is shown that deciding whether a given instance of each puzzle has a solution is NP-complete.

Fill-a-Pix is a pencil-and-paper puzzle, which is popular worldwide since announced by Conceptis in 2003. It provides a rectangular grid of squares that must be filled in to create a picture. Precisely, we are given a rectangular grid of squares some of which has an integer from 0 to 9 in it, and our task is to paint some squares black so that every square with an integer has the same number of painted squares around it including the square itself. Despite its popularity, computational complexity of Fill-a-Pix has not been known. We in this paper show that the puzzle is NP-complete, ASP-complete, and #P-complete via a parsimonious reduction from the Boolean satisfiability problem. We also consider the fewest clues problem of Fill-a-Pix, where the fewest clues problem is recently introduced by Demaine et al. for analyzing computational complexity of designing “good” puzzles. We show that the fewest clues problem of Fill-a-Pix is Σ2P-complete.

In this paper, we investigate computational complexity of pipe puzzles. A pipe puzzle is a kind of tiling puzzle; the input is a set of cards, and a part of a pipe is drawn on each card. For a given set of cards, we arrange them and connect the pipes. We have to connect all pipes without creating any local loop. While ordinary tiling puzzles, like jigsaw puzzles, ask to arrange the tiles with local consistency, pipe puzzles ask to join all pipes. We first show that the pipe puzzle is NP-complete in general even if the goal shape is quite restricted. We also investigate restricted cases and show some polynomial-time algorithms.

Herugolf and Makaro are Nikoli's pencil puzzles. We study the computational complexity of Herugolf and Makaro puzzles. It is shown that deciding whether a given instance of each puzzle has a solution is NP-complete.

The Machine-to-Machine (M2M) service network platform accommodates M2M communications traffic efficiently by using tree-structured networks and the computation resources deployed on network nodes. In the M2M service network platform, program files required for controlling devices are placed on network nodes, which have different amounts of computation resources according to their position in the hierarchy. The program files must be dynamically repositioned in response to service quality requests from each device, such as computation power, link bandwidth, and latency. This paper proposes a Program File Placement (PFP) method for the M2M service network platform. First, the PFP problem is formulated in the Mixed-Integer Linear Programming (MILP) approach. We prove that the decision version of the PFP problem is NP-complete. Next, we present heuristic algorithms that attain sub-optimal but attractive solutions. Evaluations show that the heuristic algorithm based on the number of devices that share a program file reduces the total number of placed program files compared to the algorithm that moves program files based on their position.

A term is a connected acyclic graph (unrooted unordered tree) pattern with structured variables, which are ordered lists of one or more distinct vertices. A variable of a term has a variable label and can be replaced with an arbitrary tree by hyperedge replacement according to the variable label. The dimension of a term is the maximum number of vertices in the variables of it. A term is said to be linear if each variable label in it occurs exactly once. Let T be a tree and t a linear term. In this paper, we study the graph pattern matching problem (GPMP) for T and t, which decides whether or not T is obtained from t by replacing variables in t with some trees. First we show that GPMP for T and t is NP-complete if the dimension of t is greater than or equal to 4. Next we give a polynomial time algorithm for solving GPMP for a tree of bounded degree and a linear term of bounded dimension. Finally we show that GPMP for a tree of arbitrary degree and a linear term of dimension 2 is solvable in polynomial time.

Usowan is one of Nikoli's pencil puzzles. We study the computational complexity of Usowan puzzles. It is shown that deciding whether a given instance of the Usowan puzzle has a solution is NP-complete.

Given an undirected graph G, an edge dominating set is a subset F of edges such that each edge not in F is adjacent to some edge in F, and computing the minimum size of an edge dominating set is known to be NP-hard. Since the size of any edge dominating set is at least half of the maximum size µ(G) of a matching in G, we study the problem of testing whether a given graph G has an edge dominating set of size ⌈µ(G)/2⌉ or not. In this paper, we prove that the problem is NP-complete, whereas we design an O*(2.0801µ(G)/2)-time and polynomial-space algorithm to the problem.

The 3-D channel routing is a fundamental problem on the physical design of 3-D integrated circuits. The 3-D channel is a 3-D grid G and the terminals are vertices of G located in the top and bottom layers. A net is a set of terminals to be connected. The objective of the 3-D channel routing problem is to connect the terminals in each net with a Steiner tree (wire) in G using as few layers as possible and as short wires as possible in such a way that wires for distinct nets are disjoint. This paper shows that the problem is intractable. We also show that a sparse set of ν 2-terminal nets can be routed in a 3-D channel with O(√ν) layers using wires of length O(√ν).

The Building puzzle is played on an N×N grid of cells. Initially, some numbers are given around the border of the grid. The object of the puzzle is to fill out blank cells such that every row and column contains the numbers 1 through N. The number written in each cell represents the height of the building. The numbers around the border indicate the number of buildings which a person can see from that direction. A shorter building behind a taller one cannot be seen by him. It is shown that deciding whether the Building puzzle has a solution is NP-complete.