A context in sponsored search is additional information about a query, such as the user's age, gender or location, that can change an advertisement's relevance or an advertiser's value for that query. Given a set of contexts, advertiser welfare is maximized if the search engine runs a separate auction for each context; however, due to lack of competition within contexts, this can lead to a significant loss in revenue. In general, neither separate auctions nor pure bundling need maximize revenue. With this motivation, we study the algorithmic question of computing the revenue-maximizing partition of a set of items under a second-price mechanism and additive valuations for bundles. We show that the problem is strongly NP-hard, and present an algorithm that yields a 1/2-approximation of the revenue from the optimal partition. The algorithm simultaneously yields a 1/2-approximation of the optimal welfare, thus ensuring that the gain in revenue is not at the cost of welfare. Finally we show that our algorithm can be applied to the sponsored search setting with multiple slots, to obtain a constant factor approximation of the revenue from the optimal partition.
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