Improved Bounds for Scheduling Flows under Endpoint Capacity Constraints

Abstract

We consider the problem of scheduling flows subject to endpoint capacity constraints. We are given a set of capacitated nodes and an online sequence of requests where each request has a release time and a demand that needs to be routed between two nodes. A schedule specifies, for each time step, the requests that are routed in that step under the constraint that the total demand routed on a node in any step is at most its capacity. A key performance metric in such a scheduling scenario is the response time (or flow time) of a request, which is the difference between the time the request is completed and its release time. Previous work has shown that it is impossible to achieve bounded competitive ratio for average response time without resource augmentation, and that a constant factor competitive ratio is achievable with augmentation exceeding two (Dinitz-Moseley Infocom 2020). For the maximum response time objective, the best known result is a 2-competitive algorithm with a resource augmentation at least 4 (Jahanjou et al SPAA 2020). In this paper, we present improved bounds for the above flow scheduling problem under various response time objectives. Our first result is a lower bound showing that, without resource augmentation, the best competitive ratio for the maximum response time objective is Ω(n), where n is the number of nodes. The remaining results present simple, resource-augmented algorithms that are competitive for the maximum response time objective in their respective settings. Our first algorithm, Proportional Allocation, uses (1 + ε) resource augmentation to achieve a (1/ε)-competitive ratio for maximum response time in the setting with general demands, general capacities, and splittable jobs, for any ε > 0. Our second algorithm, Batch Decomposition, is 2-competitive (resp., matches optimum) for maximum response time using resource augmentation 2 (resp., 4) in the setting with unit demands, unit capacities, and unsplittable jobs. We also derive bounds for the simultaneous approximation of average and maximum response time metrics.