Online request server matching

In this report we study a new variant of an online bipartite matching problem, which can be interpreted as a scheduling problem. Then, we have one resource, called server, which is available for one unit in every step of a discrete time model. At most one task with unit demand can occur per time step. It is called request, and speciies a set of time steps when the serving is accepted. These times must not be situated in the past and neither need to be consecutive nor include the time of the arrival of the request. After the arrival of a request in time step i, an online algorithm has to decide which request will be served in step i. This decision has to be made without knowledge of future requests. The objective is a maximization of the number of served requests. It is obvious, how to model this scheduling problem with a matching problem in a bipartite graph. However, this online bipartite matching problem diiers considerably from the model in KVV90]. So we call our model online request server matching (ORSM). Indeed, it is a special case of the roommates problem, that was introduced in BR93]. To investigate the ORSM problem, we use competitive analysis. A lower bound of the competitive ratio of 1.5 is shown, and an optimal 1.5-competitive online algorithm is presented. Additionally, a weighted variant of the ORSM problem (wORSM) is investigated. Here, every edge has a weight and the objective is the online construction of a matching with maximal overall weight. For the wORSM problem a general lower bound of 1.618 for the competitive ratio is shown. A presented online algorithm is proven to be 2-competitive. A few concluding remarks and a discussion of open problems complete this report.