Online Non-clairvoyant Scheduling to Simultaneously Minimize All Convex Functions

Abstract

We 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.