Online Non-clairvoyant Scheduling to Simultaneously Minimize All Convex Functions

Abstract

AbstractWe consider scheduling jobs online to minimize the objective ∑ i ∈ [n] w i g(C i − r i ), where w i is the weight of job i, r i is its release time, C i is its completion time and g is any non-decreasing convex function. Previously, it was known that the clairvoyant algorithm Highest-Density-First (HDF) is (2 + ε)-speed O(1)-competitive for this objective on a single machine for any fixed 0 < ε < 1 [1]. We show the first non-trivial results for this problem when g is not concave and the algorithm must be non-clairvoyant. More specifically, our results include: A (2 + ε)-speed O(1)-competitive non-clairovyant algorithm on a single machine for all non-decreasing convex g, matching the performance of HDF for any fixed 0 < ε < 1. A (3 + ε)-speed O(1)-competitive non-clairovyant algorithm on multiple identical machines for all non-decreasing convex g for any fixed 0 < ε < 1. Our positive result on multiple machines is the first non-trivial one even when the algorithm is clairvoyant. Interestingly, all performance guarantees above hold for all non-decreasing convex functions g simultaneously. We supplement our positive results by showing any algorithm that is oblivious to g is not O(1)-competitive with speed less than 2 on a single machine. Further, any non-clairvoyent algorithm that knows the function g cannot be O(1)-competitive with speed less than \(\sqrt{2}\) on a single machine or speed less than \(2-\frac{1}{m}\) on m identical machines.KeywordsSingle MachineCompetitive RatioOnline AlgorithmIdentical MachineOnline ScheduleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.