Yasuhiko Takenaga

TETRAVEX is NP-complete

TETRAVEX is a widely played one person computer game in which you are given n2 unit tiles, each edge of which is labelled with a number. The objective is to place each tile within a n by n square such that all neighbouring edges are labelled with an identical number. Unfortunately, playing TETRAVEX is computationally hard. More precisely, we prove that deciding if there is a tiling of the TETRAVEX board given n2 unit tiles is NP-complete. Deciding where to place the tiles is therefore NP-hard. This may help to explain why TETRAVEX is a good puzzle. This result compliments a number of similar results for one person games involving tiling. For example, NP-completeness results have been show for: the offline version of Tetris [1], KPlumber (which involves rotating tiles containing drawings of pipes to make a connected network) [2], and shortest sliding puzzle problems [3]. It raises a number of open questions. For example, is the infinite version Turing-complete? How do we generate TETRAVEX problems which are truly puzzling as random NP-complete problems are often surprising easy to solve? Can we observe phase transition behaviour? What about the complexity of the problem when it is guaranteed to have an unique solution? How do we generate puzzles with unique solutions?

Size and Variable Ordering of OBDDs Representing Treshold Functions

An ordered binary decision diagram (OBDD) is a graph representation of a Boolean function, and it is considered as a restricted branching program. According to its good properties, an OBDD is widely used in computer aided logic design. In this paper, the size of ordered binary decision diagrams representing threshold functions is discussed. First, we prove an Ω(n2cn1-e) lower bound on the OBDD size necessary to represent any threshold function when the variable ordering can be chosen adaptively to minimize the OBDD size. Next, we show that it is not possible to find a good variable ordering only from the total order of weights, that is, for any variable ordering of this kind, there exists a threshold function that requires an exponential size OBDD, but is represented in polynomial size by the optimal variable ordering.

Vertex Coloring of Comparability+ke and –ke Graphs

(mathcal{F}+k)e and (mathcal{F}-k)e graphs are classes of graphs close to graphs in a graph class (mathcal{F}). They are the classes of graphs obtained by adding or deleting at most k edges from a graph in (mathcal{F}). In this paper, we consider vertex coloring of comparability+ke and comparability–ke graphs. We show that for comparability+ke graphs, vertex coloring is solved in polynomial time for k=1 and NP-complete for k ≥2. We also show that vertex coloring of comparability–1e graphs is solved in polynomial time.

NP-Completeness of Pandemic

Pandemic is a multi-player board game which simulates the outbreak of epidemics and the human effort to prevent them. It is a characteristic of this game that all the players cooperate for a goal and they are not competitive. We show that the problem to decide if the player can win the generalized Pandemic from the given situation of the game is NP-complete.

Tree-shellability of Boolean functions

In this paper, we define tree-shellable and ordered tree-shellable Boolean functions. A tree-shellable function is a positive Boolean function such that the number of prime implicants equals the number of paths from the root node to a 1-node in its binary decision tree representation. A tree-shellable function is easy to dualize and good for a kind of reliability computation. We show their basic properties and clarify the relations between several shellable functions, i.e. shellable, tree-shellable, ordered tree-shellable, aligned and lexico-exchange functions. We also discuss on tree-shellable quadratic functions.

Hardness of identifying the minimum ordered binary decision diagram

An ordered binary decision diagram (OBDD) is a graph representation of a Boolean function. We consider minimum OBDD identification problems: given positive and negative examples of a Boolean function, identify the OBDD with minimum number of nodes (or with minimum width) that is consistent with all the examples. We prove in this paper that the problems are NP-complete. The result implies that f(n)-width OBDD and f(n)-node OBDD are not learnable for some fixed f(n) under the PAC-learning model unless NP = RP. We also show that the problems are still NP-hard even if we restrict the functions to monotone functions.

Number of Three-point Tilings with Triangle Tiles

Three-point tiling is the problem to cover all the lattice points in a triangular region of the triangular lattice with triangle tiles that connect three adjacent lattice points. All the lattice points must be used by exactly one triangle tile. In this paper, we enumerate all the solutions and rotation symmetric solutions using ordered binary decision diagrams. In addition, the number of essentially different solutions, any two of which do not become identical by rotating and turning over, is computed.

10th Workshop on Advances in Parallel and Distributed Computational Models - APDCM'08

Parallel and distributed computing offer the promise to deliver the computing power necessary to solve many important problems whose requirements exceed the capabilities of the most powerful existing computers. Aiming to fulfill this promise, recent years have seen a flurry of activity in the arena of parallel and distributed computing which evolved into novel and robust computing models. These models reflect advances in computational devices and environments such as optical interconnects, programmable logic arrays, networks of workstations, radio communications, mobile computing, DNA computing, quantum computing, sensor networks, etc. In addition, practical experience with both parallel computers and distributed data communication networks has brought about an understanding of their potential and limitations which, in turn, have fostered the development of sophisticated algorithms. It is very encouraging see that the advent of these models, combined with the availability of efficient algorithms, has led to significant advances in the resolution of various difficult problems of practical interest.

Tree-Shellability of Restricted DNFs

A tree-shellable function is a positive Boolean function which can be represented by a binary decision tree whose number of paths from the root to a leaf labeled 1 equals the number of prime implicants. In this paper, we consider the tree-shellability of DNFs with restrictions. We show that, for read-k DNFs, the number of terms in a tree-shellable function is at most k2. We also show that, for k-DNFs, recognition of ordered tree-shellable functions is NP-complete for k=4 and tree-shellable functions can be recognized in polynomial time for constant k.

Vertex coloring of comparability+ k e and - k e

(mathcal{F}+k)e and (mathcal{F}-k)e graphs are classes of graphs close to graphs in a graph class (mathcal{F}). They are the classes of graphs obtained by adding or deleting at most k edges from a graph in (mathcal{F}). In this paper, we consider vertex coloring of comparability+ke and comparability–ke graphs. We show that for comparability+ke graphs, vertex coloring is solved in polynomial time for k=1 and NP-complete for k ≥2. We also show that vertex coloring of comparability–1e graphs is solved in polynomial time.