Online bipartite matching

Abstract

We are given a bipartite graph G(R, S, E) (|R| = |S|), where R denotes the set of requests and S being the set of servers. The requests R (given by an adversary) arrive sequentially along with their distances to each of the |S| servers, which follow a metric (e.g. line, star). The next request is revealed only after the previous one has been permanently matched to a free server. After this, this server can no longer be used to match future requests. The goal is to minimize the total matching cost incurred over all the |R| requests in the worst case, compared to the optimal matching. We consider a rather unexplored framework to this problem, where we may skip over an fraction of requests, which we call as the rejection framework, and try to find upper and lower bounds on the competitive ratios admitted by algorithms found in research in related areas, or of our own.