Randomized Online Algorithms for Adwords

The general adwords problem has remained largely unresolved. We define a subcase called {\em $k$-TYPICAL}, $k \in \Zplus$, as follows: the total budget of all the bidders is sufficient to buy $k$ bids for each bidder. This seems a reasonable assumption for a "typical" instance, at least for moderate values of $k$. We give a randomized online algorithm achieving a competitive ratio of $\left(1 - {1 \over e} - {1 \over k} \right) $ for this problem. We also give randomized online algorithms for other special cases of adwords. We give randomized online algorithms for other special cases of adwords as well, including an optimal algorithm for the case when the bids are small compared to budgets. This case captures a central computational issue that arises in the context of ad auctions, for instance in Google's AdWords marketplace. Previous algorithms for this case were deterministic \cite{MSVV, Buchbinder2007online} and were based on LP-duality. The key to these results is a simplification of the proof for RANKING, the optimal algorithm for online bipartite matching, given in \cite{KVV}. Our algorithms for adwords can be seen as natural extensions of RANKING.