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.

The performance of online algorithms for the bin packing problem is usually measured by the asymptotic approximation ratio. However, even if an online algorithm is explicitly described, it is in general difficult to obtain the exact value of the asymptotic approximation ratio. In this paper we show a theorem that gives the exact value of the asymptotic approximation ratio in a closed form when the item sizes and the online algorithm satisfy some conditions. Moreover, we demonstrate that our theorem serves as a powerful tool for the design of online algorithms combined with mathematical optimization.

This paper presents a new method to translate a regular expression into a nondeterministic finite automaton (an NFA for short). Let r be a regular expression and let M be a Thompson automaton for r. We first introduce a labeled Thompson automaton defined by assigning two types of expressions which denote prefixes and suffixes of words in L(r) to each state of M. Then we give new ϵ-free NFAs constructed from a labeled Thompson automaton. These NFAs are called a prefix equation automaton and a suffix equation automaton. We show that a suffix equation automaton is isomorphic to an equation automaton defined by Antimirov. Furthermore we give an NFA called a unified equation automaton by joining two NFAs. Thus the number of states of a unified equation automaton can be smaller than that of an equation automaton.

The multislope ski-rental problem is an online optimization problem that generalizes the classical ski-rental problem. The player is offered not only a buy and a rent options but also other options that charge both initial and per-time fees. The competitive ratio of the classical ski-rental problem is known to be 2. In contrast, the best known so far on the competitive ratio of the multislope ski-rental problem is an upper bound of 4 and a lower bound of 3.62. In this paper we consider a parametric version of the multislope ski-rental problem, regarding the number of options as a parameter. We prove an upper bound for the parametric problem which is strictly less than 4. Moreover, we give a simple recurrence relation that yields an equation having a lower bound value as its root.

A searchable symmetric encryption (SSE) scheme is a method that searches encrypted data without decrypting it. In this paper, we address the substring search problem such that for a set D of documents and a pattern p, we find all occurrences of p in D. Here, a document and a pattern are defined as a string. A directed acyclic word graph (DAWG), which is a deterministic finite automaton, is known for solving a substring search problem on a plaintext. We improve a DAWG so that all transitions of a DAWG have distinct symbols. Besides, we present a space-efficient and secure substring SSE scheme using an improved DAWG. The proposed substring SSE scheme consists of an index with a simple structure, and the size is O(n) for the total size n of documents.

In the bin packing problem, we are asked to place given items, each being of size between zero and one, into bins of capacity one. The goal is to minimize the number of bins that contain at least one item. An online algorithm for the bin packing problem decides where to place each item one by one when it arrives. The asymptotic approximation ratio of the bin packing problem is defined as the performance of an optimal online algorithm for the problem. That value indicates the intrinsic hardness of the bin packing problem. In this paper we study the bin packing problem in which every item is of either size α or size β (≤ α). While the asymptotic approximation ratio for $alpha > rac{1}{2}$ was already identified, that for $alpha leq rac{1}{2}$ is only partially known. This paper is the first to give a lower bound on the asymptotic approximation ratio for any $alpha leq rac{1}{2}$, by formulating linear optimization problems. Furthermore, we derive another lower bound in a closed form by constructing dual feasible solutions.

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.

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.

Let us consider a regular expression r of length m and a text string T of length n over an alphabet Σ. Then, the RE minimal substring search problem is to find all minimal substrings of T matching r. Yamamoto proposed O(mn) time and O(m) space algorithm using a Thompson automaton. In this paper, we improve Yamamoto's algorithm by introducing parallelism. The proposed algorithm runs in O(mn) time in the worst case and in O(mn/p) time in the best case, where p denotes the number of processors. Besides, we show a parameter related to the parallel time of the proposed algorithm. We evaluate the algorithm experimentally.

The bin packing problem is a problem of finding an assignment of a sequence of items to a minimum number of bins, each of capacity one. An online algorithm for the bin packing problem is an algorithm that irrevocably assigns each item one by one from the head of the sequence. Gutin, Jensen, and Yeo (2006) considered a version in which all items are only of two different sizes and the online algorithm knows the two possible sizes in advance, and gave an optimal online algorithm for the case when the larger size exceeds 1/2. In this paper we provide an optimal online algorithm for some of the cases when the larger size is at most 1/2, on the basis of a framework that facilitates the design and analysis of algorithms.

With the rapid developments in the Internet of Things (IoT), low power wide area networks (LPWAN) framework, which is a low-power, long-distance communication method, is attracting attention. However, in LPWAN, the access time is limited by Duty Cycle (DC) to avoid mutual interference. Packet-level index modulation (PLIM) is a modulation scheme that uses a combination of the transmission time and frequency channel of a packet as an index, enabling throughput expansion even under DC constraints. The indexes used in PLIM are transmitted according to the mapping. However, when many sensors access the same index, packet collisions occur owing to selecting the same index. Therefore, we propose a mapping design for PLIM using mathematical optimization. The mapping was designed and modeled as a quadratic integer programming problem. The results of the computer simulation evaluations were used to realize the design of PLIM, which achieved excellent sensor information aggregation in terms of environmental monitoring accuracy.

In Variant 4 of the one-way trading game [El-Yaniv, Fiat, Karp, and Turpin, 2001], a player has one dollar at the beginning and wants to convert it to yen only by one-way conversion. The exchange rate is guaranteed to fluctuate between m and M, and only the maximum fluctuation ratio φ = M/m is informed to the player in advance. The performance of an algorithm for this game is measured by the competitive ratio. El-Yaniv et al. derived the best possible competitive ratio over all algorithms for this game. However, it seems that the behavior of the best possible algorithm itself has not been explicitly described. In this paper we reveal the behavior of the best possible algorithm by solving a linear optimization problem. The behavior turns out to be quite different from that of the best possible algorithm for Variant 2 in which the player knows m and M in advance.

Flat Panel Displays (FPDs) are manufactured using many pieces of different processing equipment arranged sequentially in a line. Although the constant inter-arrival time (i.e., the tact time) of glass substrates in the line should be kept as short as possible, the collision probability between glass substrates increases as tact time decreases. Since the glass substrate is expensive and fragile, collisions should be avoided. In this paper, we derive a closed form formula of the approximate collision probability for a model, in which the processing time on each piece of equipment is assumed to follow Erlang distribution. We also compare some numerical results of the closed form and computer simulation results of the collision probability.

In the 3-slope ski-rental problem, the player is asked to determine a strategy, that is, (i) whether to buy a ski wear and then a ski set separately, or to buy them at once for a discount price, and (ii) when to buy these goods. If the player has not got any thing, he/she can rent it for some price. The objective is to minimize the total cost, under the assumption that the player does not know how many times he/she goes skiing in the future. We reveal that even with a large discount for buying at once available, there is some price setting for which to buy the goods separately is a more reasonable choice. We also show that the performance of the optimal strategy may become arbitrarily worse, when a large discount is offered.

Information exchange through a relay node is attracting attention for applying machine-to-machine communications. If the node demodulates the received signal in relay processing confidentially, the information leakage through the relay station is a problem. In wireless MIMO switching, the frequency spectrum usage efficiency can be improved owing to the completion of information exchange within a short time. This study proposes a novel wireless MIMO switching method for secure information exchange. An overloaded situation, in which the access nodes are one larger than the number of antennas in the relay node, makes the demodulation of the relay node difficult. The access schedule of nodes is required for maintaining the overload situation and the high information exchange efficiency. This study derives the equation model of the access schedule and constructs an access schedule with fewer time periods in the integer programming problem. From the computer simulation, we confirm that the secure capacity of the proposed MIMO switching is larger than that of the original one, and the constructed access schedule is as large as the ideal and minimum time period for information exchange completion.

In the online removable knapsack problem, a sequence of items, each labeled with its value and its size, is given one by one. At each arrival of an item, a player has to decide whether to put it into a knapsack or to discard it. The player is also allowed to discard some of the items that are already in the knapsack. The objective is to maximize the total value of the knapsack. Iwama and Taketomi gave an optimal algorithm for the case where the value of each item is equal to its size. In this paper we consider a case with an additional constraint that the capacity of the knapsack is a positive integer N and that the sizes of items are all integral. For each positive integer N, we design an algorithm and prove its optimality. It is revealed that the competitive ratio is not monotonic with respect to N.

The construction of a Huffman code can be understood as the problem of finding a full binary tree such that each leaf is associated with a linear function of the depth of the leaf and the sum of the function values is minimized. Fujiwara and Jacobs extended this to a general function and proved the extended problem to be NP-hard. The authors also showed the case where the functions associated with leaves are each non-decreasing and convex is solvable in polynomial time. However, the complexity of the case of non-decreasing non-convex functions remains unknown. In this paper we try to reveal the complexity by considering non-decreasing non-convex functions each of which takes the smaller value of either a linear function or a constant. As a result, we provide a polynomial-time algorithm for two subclasses of such functions.

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 $mathbb{R}^{2}$. 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 $rac{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.