Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Friday, March 22, 2013, 12:15 pm

Duration: 30 minutes

Location: CAB G11

Speaker: Jan Vondrak (IBM Almaden Research Center)

Online Algorithms for Welfare Maximization in Allocation Problems

In the online welfare maximization problem, items arriving online should be allocated to n agents, in order to maximize their total happiness (welfare). Variants of the problem arise depending on the valuations functions, i.e. how the agents value different subsets of items. The simplest form of the problem is a bipartite matching problem where each agent would like to receive exactly one 1 item, one of its neighbors in a given bipartite graph. The greedy algorithm gives a 1/2-approximation for bipartite matching, and more generally for online welfare maximization, whenever the valuation functions are monotone submodular. In 1990, Karp, Vazirani & Vazirani showed that a careful randomization of the greedy algorithm gives an improved (1-1/e)-approximation for online bipartite matching. Recently, there has been a lot of work on extensions of this result, motivated by applications in online advertising such as Google Adwords. In this work, we show that no online algorithm (even randomized) gives approximation better than 1/2 for the class of "coverage valuations", a subclass of submodular valuations, hence proving the optimality of the greedy algorithm for both of these classes. We also provide some negative evidence for the class of budget-additive valuations which is of particular interest in applications. Joint work with Michael Kapralov and Ian Post.

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