We introduce and study the online pause and resume problem. In this problem, a player attempts to find the $k$ lowest (alternatively, highest) prices in a sequence of fixed length $T$, which is revealed sequentially. At each time step, the player is presented with a price and decides whether to accept or reject it. The player incurs a switching cost whenever their decision changes in consecutive time steps, i.e., whenever they pause or resume purchasing. This online problem is motivated by the goal of carbon-aware load shifting, where a workload may be paused during periods of high carbon intensity and resumed during periods of low carbon intensity and incurs a cost when saving or restoring its state. It has strong connections to existing problems studied in the literature on online optimization, though it introduces unique technical challenges that prevent the direct application of existing algorithms. Extending prior work on threshold-based algorithms, we introduce double-threshold algorithms for both the minimization and maximization variants of this problem. We further show that the competitive ratios achieved by these algorithms are the best achievable by any deterministic online algorithm. Finally, we empirically validate our proposed algorithm through case studies on the application of carbon-aware load shifting using real carbon trace data and existing baseline algorithms.

翻译：我们引入并研究在线暂停与恢复问题。在该问题中，参与者试图在一个固定长度T的序列中寻找k个最低（或最高）价格，该序列逐步揭示。在每个时间步，参与者面对一个价格并决定接受或拒绝。每当参与者的决策在连续时间步中发生变化（即暂停或恢复购买时），将产生切换成本。该在线问题源于碳感知负载迁移的目标——工作负载可在高碳强度时段暂停，在低碳强度时段恢复，并在保存或恢复状态时产生成本。该问题与在线优化文献中已有研究的问题具有密切联系，但引入了独特的技术挑战，导致现有算法无法直接应用。通过扩展基于阈值的算法研究，我们针对该问题的最小化与最大化变体提出了双阈值算法。进一步证明，这些算法实现的竞争比是任意确定性在线算法所能达到的最优值。最后，我们利用真实碳迹数据与现有基线算法，通过碳感知负载迁移应用的案例研究，对所提算法进行了实证验证。