The essence of the simplest buy-at-bulk network design problem is buying network capacity "wholesale " to guarantee connectivity from all network nodes to a certain central network switch. Capacity is sold with "volume discount": the more capacity is bought, the cheaper is the price per unit of bandwidth. We provide O(log 2 n) randomized approximation algorithm for the problem. This solves the open problem in [15]. The only previously known solutions were restricted to special cases (Euclidean graphs) [15]. We solve additional natural variations of the problem, such as multi-sink network design, as well as selective network design. These problems can be viewed as generalizations of the the Generalized Steiner Connectivity and Prize-collecting salesman (K-MST) problems. In the selective network design problem, some subset of k wells must be connected to the (single) refinery, so that the total cost is minimized. 1 Introduction 1.1 The basic problem Consider an oil company that wi...

We consider the problem of selecting threshold times to transition a device to low-power sleep states during an idle period. The two-state case in which there is a single active and a single sleep state is a continuous version of the ski-rental problem. We consider a generalized version in which there is more than one sleep state, each with its own power consumption rate and transition costs. We give an algorithm that, given a system, produces a deterministic strategy whose competitive ratio is arbitrarily close to optimal. We also give an algorithm to produce the optimal online strategy given a system and a probability distribution that generates the length of the idle period. We also give a simple algorithm that achieves a competitive ratio of 3 + 2 √ 2 ≈ 5.828 for any system.

In this paper, we generalize the Ski-Rental Problem to the Bahncard Problem which is an online problem of practical relevance for all travelers. The Bahncard is a railway pass of the Deutsche Bundesbahn (the German railway company) which entitles its holder to a 50% price reduction on nearly all train tickets. It costs 240 DM, and it is valid for 12 months. Similar bus or railway passes can be found in many other countries. For the common traveler, the decision at which time to buy a Bahncard is a typical online problem, because she usually does not know when and where she will travel next. We show that the greedy algorithm applied by most travelers and clerks at ticket offices is not better in the worst case than the trivial algorithm which never buys a Bahncard. We present two optimal deterministic online algorithms, an optimistic one and a pessimistic one. We further give a lower bound for randomized online algorithms and present an algorithm which we conjecture to be optimal; a proof of the conjecture is given for a special case of the problem. It turns out that the optimal competitive ratio only depends on the price reduction factor (50% for the German Bahncard Problem), but does not depend on the price or validity period of a Bahncard. Keywords: Bahncard Problem, competitive analysis, online algorithm, SkiRental Problem ? The author was partially supported by the EU ESPRIT LTR Project No. 20244 (ALCOM-IT). He was further supported by a Habilitation Scholarship of the German Research Foundation (DFG). 1 1

Abstract not found

Most on-line analysis assumes that, at each time step, all relevant information up to that time step is available and a decision has an immediate effect. In many on-line problems, however, the time relevant information is available and the time a decision has an effect may be decoupled. For example, when making an investment, one might not have completely up-to-date information on market prices. Similarly, a buy or sell order might only be executed some time later in the future. We introduce and explore natural delayed models for several well-known on-line problems. Our analyses demonstrate the importance of considering timeliness in determining the competitive ratio of an on-line algorithm. For many problems, we demonstrate that there exist algorithms with small competitive ratios even when large delays affect the timeliness of information and the effect of decisions.