Online Bipartite Matching and Adwords
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A simple and optimal online algorithm for online bipartite matching, called RANKING, was given in <cit.>; however, its analysis was difficult to comprehend. Over the years, several researchers contributed valuable ideas to simplifying its proof. We start by observing that the past proofs were incomplete; we have identified the missing piece and named it the no-surpassing property. The simplicity of the final proof naturally raised the possibility of extending RANKING to the adwords problem. We first managed to extend it to a subcase of the latter, called SINGLE-VALUED. However, a further extension, to the subcase called SMALL, in which bids are small compared to budgets, faced a major hurdle, namely failure of the no-surpassing property. Since SMALL has been of considerable practical significance in ad auctions <cit.> and our approach has distinct advantages over <cit.>, we have stated our result as a Conditional Theorem after assuming the no-surpassing property. We leave the open problem of removing this assumption and salvaging the (substantial) ideas that were needed to analyze the algorithm, modulo the assumption.
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