Strengths and Limitations of Greedy in Cup Games

Abstract

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.