We study a new online assignment problem, called the \textit{Online Task Assignment with Controllable Processing Time}. In a bipartite graph, a set of online vertices (tasks) should be assigned to a set of offline vertices (machines) under the known adversarial distribution (KAD) assumption. We are the first to study controllable processing time in this scenario: There are multiple processing levels for each task and higher level brings larger utility but also larger processing delay. A machine can reject an assignment at the cost of a rejection penalty, taken from a pre-determined rejection budget. Different processing levels cause different penalties. We propose the Online Machine and Level Assignment (OMLA) Algorithm to simultaneously assign an offline machine and a processing level to each online task. We prove that OMLA achieves $1/2$-competitive ratio if each machine has unlimited rejection budget and $\Delta/(3\Delta-1)$-competitive ratio if each machine has an initial rejection budget up to $\Delta$. Interestingly, the competitive ratios do not change under different settings on the controllable processing time and we can conclude that OMLA is "insensitive" to the controllable processing time.

翻译：我们研究了一种新的在线分配问题，称为“在线任务分配与可控处理时间”。在二部图中，一组在线顶点（任务）应在已知对抗分布假设下分配给一组离线顶点（机器）。我们首次在此场景中研究可控处理时间：每个任务具有多个处理层级，更高层级带来更大效用但也会导致更长的处理延迟。机器可以拒绝分配，但需承担来自预定义拒绝预算的拒绝惩罚。不同处理层级导致不同惩罚。我们提出在线机器与层级分配（OMLA）算法，以同时为每个在线任务分配离线机器和处理层级。我们证明，若每台机器具有无限拒绝预算，OMLA可实现$1/2$竞争比；若每台机器初始拒绝预算不超过$\Delta$，则可实现$\Delta/(3\Delta-1)$竞争比。有趣的是，竞争比在可控处理时间的不同设置下保持不变，由此可推断OMLA对可控处理时间具有“不敏感性”。