We study online multidimensional variants of the generalized assignment problem which are used to model prominent real-world applications, such as the assignment of virtual machines with multiple resource requirements to physical infrastructure in cloud computing. These problems can be seen as an extension of the well known secretary problem and thus the standard online worst-case model cannot provide any performance guarantee. The prevailing model in this case is the random-order model, which provides a useful realistic and robust alternative. Using this model, we study the @math -dimensional generalized assignment problem, where we introduce a novel technique that achieves an @math -competitive algorithms and prove a matching lower bound of @math . Furthermore, our algorithm improves upon the best-known competitive-ratio for the online (one-dimensional) generalized assignment problem and the online knapsack problem.
We study online variants of weighted bipartite matching on graphs and hypergraphs. In our model for online matching, the vertices on the right-hand side of a bipartite graph are given in advance and the vertices on the left-hand side arrive online in random order. Whenever a vertex arrives, its adjacent edges with the corresponding weights are revealed and the online algorithm has to decide which of these edges should be included in the matching. The studied matching problems have applications, e.g., in online ad auctions and combinatorial auctions where the right-hand side vertices correspond to items and the left-hand side vertices to bidders. Improved thermosetting sealing materials, for example polyurethane or polysulphide compositions which can be cured in a high frequency alternating electric field, are described which contain electrically non-conducting pigments and or filler whose dielectric constant exceeds 200. In the classical secretary problem, a set S of numbers is presented to an online algorithm in random order. At any time the algorithm may stop and choose the current element, and the goal is to maximize the probability of choosing the largest element in the set. We study a variation in which the algorithm is allowed to choose k elements, and the goal is to maximize their sum. We present an algorithm whose competitive ratio is 1-O(√1 k). To our knowledge, this is the first algorithm whose competitive ratio approaches 1 as k ← ∞. As an application we solve an open problem in the theory of online auction mechanisms. 