Online Non-clairvoyant Scheduling to Simultaneously Minimize All Convex Functions

We consider scheduling jobs online to minimize the objective ∑ i∈[n] wig(Ci− ri), where wi is the weight of job i, ri is its release time, Ci is its completion time and g is any non-decreasing convex function. Previously, it was known that the clairvoyant algorithm Highest-DensityFirst (HDF) is (2 + )-speed O(1)-competitive for this objective on a single machine for any fixed 0 < < 1 [21]. 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 + )-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 < < 1. • A (3 + )-speed O(1)-competitive non-clairovyant algorithm on multiple identical machines for all non-decreasing convex g for any fixed 0 < < 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 √ 2 on a single machine or speed less than 2− 1 m on m identical machines.