In the cup game, an adversary distributes 1 unit of water among $n$ cups every time step. The player then selects a single cup from which to remove 1 unit of water. In the bamboo trimming problem, the adversary must choose fixed rates for the cups, and the player is additionally allowed to empty the chosen cup entirely. Past work has shown that the optimal backlog in these two settings is $Θ(\log n)$ and 2 respectively. The greedy algorithm has been shown in previous work to be exactly optimal in the general cup game and asymptotically optimal in the bamboo setting. The greedy algorithm has been conjectured [16] to achieve the exactly optimal backlog of 2 in the bamboo setting as well. In this paper, we prove a lower bound of $2.076$ for the backlog of the greedy algorithm, disproving the conjecture of [16]. We also introduce a new algorithm, a hybrid greedy/Deadline-Driven, which achieves backlog $O(\log n)$ in the general cup game, and remains exactly optimal for the bamboo trimming problem and the fixed-rate cup game -- this constitutes the first algorithm that achieves asymptotically optimal performance across all three settings. Additionally, we introduce a new model, the semi-oblivious cup game, in which the player is uncertain of the exact heights of each cup. We analyze the performance of the greedy algorithm in this setting, which can be viewed as selecting an arbitrary cup within a constant multiplicative factor of the fullest cup. We prove matching upper and lower bounds showing that the greedy algorithm achieves a backlog of $Θ(n^{\frac{c-1}{c}})$ in the semi-oblivious cup game. We also establish matching upper and lower bounds of $2^{Θ(\sqrt{\log n})}$ in the semi-oblivious cup flushing game. Finally, we show that in an additive error setting, greedy is actually able to achieve backlog $Θ(\log n)$, via matching upper and lower bounds.

翻译：在杯游戏中，对手每个时间步在$n$个杯子中分配1单位水。玩家随后选择一个杯子从中移除1单位水。在竹子修剪问题中，对手必须为杯子选择固定速率，且玩家被允许完全清空所选杯子。先前研究表明，这两种设置下的最优积压量分别为$Θ(\log n)$和2。已有工作证明贪婪算法在通用杯游戏中完全最优，在竹子设置中渐近最优。文献[16]曾推测贪婪算法在竹子设置中也能达到精确最优积压量2。本文中，我们证明了贪婪算法的积压量下界为$2.076$，从而证伪了[16]的猜想。我们还提出一种新算法——贪婪/截止时间驱动混合算法，该算法在通用杯游戏中实现$O(\log n)$积压量，并在竹子修剪问题与固定速率杯游戏中保持完全最优性，这是首个在所有三种设置中均实现渐近最优性能的算法。此外，我们提出半遗忘杯游戏新模型，其中玩家无法确知每个杯子的精确高度。我们分析了贪婪算法在此设置下的性能，该设置可视为在最满杯子的常数倍乘因子范围内任选杯子。我们证明了匹配的上下界，表明贪婪算法在半遗忘杯游戏中实现$Θ(n^{\frac{c-1}{c}})$积压量。同时，我们在半遗忘杯冲刷游戏中建立了$2^{Θ(\sqrt{\log n})}$的匹配上下界。最后，我们通过匹配的上下界证明，在加性误差设置中，贪婪算法实际上能够实现$Θ(\log n)$积压量。