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. 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.

中文翻译：

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Adwords 的随机在线算法

一般的 Adwords 问题在很大程度上仍未得到解决。我们定义了一个名为{\em $k$-TYPICAL}的子案例，$k \in \Zplus$，如下：所有投标人的总预算足以为每个投标人购买$k$的投标。对于“典型”实例，这似乎是一个合理的假设，至少对于 $k$ 的中等值。对于这个问题，我们给出了一个随机的在线算法，实现了 $\left(1 - {1 \over e} - {1 \over k} \right) $ 的竞争比率。我们还为其他特殊的 Adword 案例提供了随机在线算法。这些结果的关键是对 RANKING 证明的简化，RANKING 是 \cite{KVV} 中给出的在线二分匹配的最佳算法。我们的 Adwords 算法可以看作是 RANKING 的自然扩展。