Introduction A natural and well-studied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal

How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1 − 1/e for this problem. 1

In this paper we study the problem of scheduling real-time requests in distributed data servers. We assume the time to be divided into time steps of equal length called rounds. During every round a set of requests arrives at the system, and every resource is able to fulfill one request per round. Every request specifies two (distinct) resources and requires to get access to one of them. Furthermore, every request has a deadline of d, i.e. a request that arrives in round t has to be fulfilled during round t +d 1 at the latest. The number of requests which arrive during some round and the two alternative resources of every request are selected by an adversary. The goal is to maximize the number of requests that are fulfilled before their deadlines expire. We examine the scheduling problem in an online setting, i.e. new requests continuously arrive at the system, and we have to determine online an assignment of the requests to the resources in such a way that every resource has to fulfil...

. In this work an alternative online variant of the matching problem in bipartite graphs is presented. It is motivated by a scheduling problem in an online environment. In such an environment, a task is unknown up to its disclosure. However, in that moment it is not necessary to take a decision on the service of that particular task. In reality, an online scheduler has to decide on how to use the current resources. Therefore, our problem is called online request server matching (ORSM). It differs substantially from the online bipartite matching problem of Karp et al. [KVV90]. Hence, the analysis of an optimal, deterministic online algorithm for the ORSM problem results in a smaller competitive ratio of 1:5. We also introduce an extension to a weighted bipartite matching problem. A lower bound of p 5+1 2 1:618 and an upper bound of 2 is given for the competitive ratio. 1 Introduction Motivation. The problem, which is investigated here, is motivated by online schedulin...

Abstract. We consider the online metric matching problem. In this problem, we are given a graph with edge weights satisfying the triangle inequality, and k vertices that are designated as the right side of the matching. Over time up to k requests arrive at an arbitrary subset of vertices in the graph and each vertex must be matched to a right side vertex immediately upon arrival. A vertex cannot be rematched to another vertex once it is matched. The goal is to minimize the total weight of the matching. We give a O(log 2 k) competitive randomized algorithm for the problem. This improves upon the best known guarantee of O(log 3 k) due to Meyerson, Nanavati and Poplawski [19]. It is well known that no deterministic algorithm can have a competitive less than 2k − 1, and that no randomized algorithm can have a competitive ratio of less than ln k. 1

We present a poly-log-competitive deterministic online algorithm for the online transportation problem on hierarchically separated trees when the online algorithm has one extra server per site. Using metric embedding results in the literature, one can then obtain a poly-log-competitive randomized online algorithm for the online transportation on an arbitrary metric space when the online algorithm has one extra server per site. 1

Abstract. We study the online matching problem when the metric space is a single straight line. For this case, the offline matching problem is trivial but the online problem has been open and the best known competitive ratio was the trivial Θ(n) where n is the number of requests. It was conjectured that the generalized Work Function Algorithm has constant competitive ratio for this problem. We show that it is in fact Ω(log n) and O(n), and make some progress towards proving a better upper bound by establishing some structural properties of the solutions. Our technique for the upper bound doesn’t use a potential function but it reallocates the online cost in a way that the comparison with the offline cost becomes more direct. 1