Online Task Assignment with Controllable Processing Time

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.