Online Resource Allocation with Time-Flexible Customers

In classic online resource allocation problems, a decision-maker tries to maximize her reward through making immediate and irrevocable choices regarding arriving demand points (agents). However, in many settings, some arriving agents may be patient and willing to wait a short amount of time for the resource. Motivated by this, we study the online resource allocation problem in the presence of time-flexible agents under an adversarial online arrival model. We present a setting with flexible and inflexible agents who seek a resource or service that replenishes periodically. Inflexible agents demand the resource immediately upon arrival while flexible agents are willing to wait a short period of time. Our work presents a class of POLYtope-based Resource Allocation (POLYRA) algorithms that achieve optimal or near-optimal competitive ratios under an adversarial arrival process. Such POLYRA algorithms work by consulting a particular polytope and only making decisions that guarantee the algorithm's state remains feasible in this polytope. When the number of agent types is either two or three, POLYRA algorithms can obtain the optimal competitive ratio. We design these polytopes by constructing an upper bound on the competitive ratio of any algorithm, which is characterized via a linear program (LP) that considers a collection of overlapping worst-case input sequences. Our designed POLYRA algorithms then mimic the optimal solution of this upper bound LP via its polytope's definition, obtaining the optimal competitive ratio. When there are more than three types, we show that our overlapping worst-case input sequences do not result in an attainable competitive ratio, adding an additional challenge to the problem. Considering this, we present a near-optimal nested POLYRA algorithm which achieves at least 80% of the optimal competitive ratio while having a simple and interpretable structure.