A number of recent papers have addressed the algorithmic problem of allocating advertisement space for keywords in sponsored search auctions so as to maximize revenue, most of which assume that pricing is done via a first-price auction. This does not realistically model the Generalized Second Price (GSP) auction used in practice, in which bidders pay the next-highest bid for keywords that they are allocated. Towards the goal of more realistically modelling these auctions, we introduce the Second-Price Ad Auctions problem, in which bidders ’ payments are determined by the GSP mechanism. We show that the complexity of the Second-Price Ad Auctions problem is quite different than that of the more studied First-Price Ad Auctions problem. First, unlike the first-price variant, for which small constant-factor approximations are known, we show that it is NP-hard to approximate the Second-Price Ad Auctions problem to any non-trivial factor, even when the bids are small compared to the budgets. Second, we show that this discrepancy extends even to the 0-1 special case that we call the Second-Price Matching problem (2PM). In particular, offline 2PM is APX-hard, and for online 2PM there is no deterministic algorithm achieving
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