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. Another subcase, when bids are small compared to budgets, has been of considerable practical significance in ad auctions \cite{MSVV}. For this case, we give an optimal randomized online algorithm achieving a competitive ratio of $\left(1 - {1 \over e} \right)$. Previous algorithms for this case were based on LP-duality; the impact of our new approach remains to be seen. 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.