The Fewest Clues Problem of Picross 3D

Picross 3D is a popular single-player puzzle video game for the Nintendo DS. It is a 3D variant of Nonogram, which is a popular pencil-and-paper puzzle. While Nonogram provides a rectangular grid of squares that must be filled in to create a picture, Picross 3D presents a rectangular parallelepiped (i.e., rectangular box) made of unit cubes, some of which must be removed to construct an image 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 image is complete. It is shown by Kusano et al. that Picross 3D is NP-complete. We in this paper show that the fewest clues problem of Picross 3D is Sigma_2^P-complete and that the counting version and the another solution problem of Picross 3D are #P-complete and NP-complete, respectively

On the best possible competitive ratio for the multislope ski-rental problem

The multislope ski-rental problem is an extension of the classical ski-rental problem, where the player has several lease options besides the pure rent and buy options. In this problem the hardness of an instance, which is the setting of options, significantly affects the player's performance. There is an algorithm that for a given instance, computes the best possible strategy. However, the output is given as numerical values and therefore the relational nature between an instance and the best possible performance for it has not been known. In this paper we prove that even for the easiest instance, a competitive ratio smaller than cannot be achieved. More precisely, a tight lower bound on the best possible performance is obtained in a closed form parametrized by the number of options. Furthermore, we establish a matching upper and lower bound on the competitive ratio each for the 3-option and 4-option problems.ArticleJOURNAL OF COMBINATORIAL OPTIMIZATION. 31(2): 463-490 (2016)journal articl

Online Weight Balancing on the Unit Circle

We consider a problem as follows: Given unit weights arriving in an online manner with the total cardinality unknown, upon each arrival we decide where to place it on the unit circle in R-2. The objective is to set the center of mass of the placed weights as close to the origin as possible. We apply competitive analysis defining the competitive difference as a performance measure. We first present an optimal strategy for placing unit weights which achieves a competitive difference of 1/5. We next consider a variant in which the destination of each weight must be chosen from a set of positions that equally divide the unit circle. We give a simple strategy whose competitive difference is 0.35. Moreover, in the offline setting, several conditions for the center of mass to lie at the origin are derived.ArticleIEICE TRANSACTIONS ON INFORMATION AND SYSTEMS. E99D(3): 567-574 (2016)journal articl

Competitive Analysis for the Flat-Rate Problem

We consider a problem of the choice of price plans offered by a telecommunications company: a "pay-as-you-go" plan and a "flat-rate" plan. This problem is formulated as an online optimization problem extending the ski-rental problem, and analyzed using the competitive ratio. We give a lemma for easily calculating the competitive ratio. Based on the lemma, we derive a family of optimal strategies for a realistic class of instances.ArticleIEICE TRANSACTIONS ON INFORMATION AND SYSTEMS. E99D(3): 559-566 (2016)journal articl

The Huffman Tree Problem with Unit Step Functions

A binary tree is regarded as a prefix-free binary code, in which the weighted sum of the lengths of root-leaf paths is equal to the expected codeword length. Huffman's algorithm computes an optimal tree in O(n log n) time, where n is the number of leaves. The problem was later generalized by allowing each leaf to have its own function of its depth and setting the sum of the function values as the objective function. The generalized problem was proved to be NP-hard. In this paper we study the case where every function is a unit step function, that is, a function that takes a lower constant value if the depth does not exceed a threshold, and a higher constant value otherwise. We show that for this case, the problem can be solved in O(n log n) time, by reducing it to the Coin Collector's problem.ArticleIEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES. E98A(6):1189-1196 (2015)journal articl

Analysis of Lower Bounds for the Multislope Ski-Rental Problem

The multislope ski-rental problem is an extension of the classical ski-rental problem, where the player has several options of paying both of a per-time fee and an initial fee, in addition to pure renting and buying options. Damaschke gave a lower bound of 3.62 on the competitive ratio for the case where arbitrary number of options can be offered. In this paper we propose a scheme that for the number of options given as an input, provides a lower bound on the competitive ratio, by extending the method of Damaschke. This is the first to establish a lower bound for each of the 5-or-more-option cases, for example, a lower bound of 2.95 for the 5-option case, 3.08 for the 6-option case, and 3.18 for the 7-option case. Moreover, it turns out that our lower bounds for the 3- and 4-option cases respectively coincide with the known upper bounds. We therefore conjecture that our scheme in general derives a matching lower and upper bound.ArticleIEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES. E97A(6): 1200-1205 (2014)journal articl

On Approximation Properties of the Independent Set Problem for Degree 3 Graphs

. The main problem we consider in this paper is the Independent Set problem for bounded degree graphs. It is shown that the problem remains MAX SNP--complete when the maximum degree is bounded by 3. Some related problems are also shown to be MAX SNP--complete at the lowest possible degree bounds. Next we study better poly--time approximation of the problem for degree 3 graphs, and improve the previously best ratio, 5 4 , to arbitrarily close to 6 5 . This result also provides improved poly--time approximation ratios, B+3 5 + ffl, for odd degree B. 1 Introduction The area of efficient approximation algorithms for NP--hard optimization problems has recently seen dramatic progress with a sequence of breakthrough achievements. Even when restricted only to the area of constant bound approximation the following remarkable results have been obtained in the last few years. The subclass of NP optimization problems, called MAX SNP, consisting solely of constant ratio approximable problems ..