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.

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.

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.

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.

We investigate an online problem of a robot exploring the outer boundary of an unknown simple polygon P. The robot starts from a specified vertex s and walks an exploration tour outside P. It has to see all points of the polygon's outer boundary and to return to the start. We provide lower and upper bounds on the ratio of the distance traveled by the robot in comparison to the length of the shortest path. We consider P in two scenarios: convex polygon and concave polygon. For the first scenario, we prove a lower bound of 5 and propose a 23.78-competitive strategy. For the second scenario, we prove a lower bound of 5.03 and propose a 26.5-competitive strategy.

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.

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.

The online unit clustering problem is one of the most basic clustering problems proposed by Chan and Zarrabi-Zadeh (WAOA2007 and Theory of Computing Systems 45(3), 2009). Several variants of this problem have been extensively studied. In this letter, we propose a new variant of the online unit clustering problem, called the online unit clustering problem with capacity constraints. For this problem, we use competitive analysis to evaluate the performance of an online algorithm. Then, we develop an online algorithm whose competitive ratio is at most 3.178, and show that a lower bound on the competitive ratio of any online algorithm is 2.

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.

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.

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 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.

This paper considers online vertex exploration problems in a simple polygon where starting from a point in the inside of a simple polygon, a searcher is required to explore a simple polygon to visit all its vertices and finally return to the initial position as quickly as possible. The information of the polygon is given online. As the exploration proceeds, the searcher gains more information of the polygon. We give a 1.219-competitive algorithm for this problem. We also study the case of a rectilinear simple polygon, and give a 1.167-competitive algorithm.

The purpose of the online graph exploration problem is to visit all the nodes of a given graph and come back to the starting node with the minimum total traverse cost. However, unlike the classical Traveling Salesperson Problem, information of the graph is given online. When an online algorithm (called a searcher) visits a node v, then it learns information on nodes and edges adjacent to v. The searcher must decide which node to visit next depending on partial and incomplete information of the graph that it has gained in its searching process. The goodness of the algorithm is evaluated by the competitive analysis. If input graphs to be explored are restricted to trees, the depth-first search always returns an optimal tour. However, if graphs have cycles, the problem is non-trivial. In this paper we consider two simple cases. First, we treat the problem on simple cycles. Recently, Asahiro et al. proved that there is a 1.5-competitive online algorithm, while no online algorithm can be (1.25-ε)-competitive for any positive constant ε. In this paper, we give an optimal online algorithm for this problem; namely, we give a (1.366)-competitive algorithm, and prove that there is no (-ε)-competitive algorithm for any positive constant ε. Furthermore, we consider the problem on unweighted graphs. We also give an optimal result; namely we give a 2-competitive algorithm and prove that there is no (2-ε)-competitive online algorithm for any positive constant ε.

The online buffer management problem formulates the problem of queuing policies of network switches supporting QoS (Quality of Service) guarantee. In this paper, we consider one of the most standard models, called multi-queue switches model. In this model, Albers et al. gave a lower bound , and Azar et al. gave an upper bound on the competitive ratio when m, the number of input ports, is large. They are tight, but there still remains a gap for small m. In this paper, we consider the case where m=2, namely, a switch is equipped with two ports, which is called a bicordal buffer model. We propose an online algorithm called Segmental Greedy Algorithm (SG) and show that its competitive ratio is at most ( 1.231), improving the previous upper bound by ( 1.286). This matches the lower bound given by Schmidt.

The online buffer management problem formulates the problem of queueing policies of network switches supporting QoS (Quality of Service) guarantee. For this problem, several models are considered.In this paper, we focus on shared memory switches with preemption. We prove that the competitive ratio of the Longest Queue Drop (LQD) policy is (4M-4)/(3M-2) in the case of N=2, where N is the number of output ports in a switch and M is the size of the buffer.This matches the lower bound given by Hahne, Kesselman and Mansour.Also, in the case of arbitrary N, we improve the competitive ratio of LQD from 2 to 2 - (1/M) minK = 1, 2, ..., N{M/K + K - 1}.

We study the performance of oblivious routing algorithms that follow minimal (shortest) paths, referred to as minimal oblivious routing algorithms in this paper, using competitive analysis on a d-dimensional, N = 2d-node hypercube. We assume that packets are injected into the hypercube arbitrarily and continuously, without any (e.g., probabilistic) assumption on the arrival pattern of the packets. Minimal algorithms reduce the total load in the network in the first place and they preserve locality. First we show that the well known deterministic oblivious routing algorithm, namely, the greedy routing algorithm, has competitive ratio Ω(N1/2). Then we show a problem lower bound of Ω(Nlog 2 (5/4)/log5 N). We also give a natural randomized minimal oblivious routing algorithm whose competitive ratio is close to the problem lower bound we provide.