Online Non-clairvoyant Scheduling to Simultaneously Minimize All Convex Functions

We consider scheduling jobs online to minimize the objective ∑  _{i ∈ [n]} w _ig(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:

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">2–√2 on a single machine or speed less than 2−1m">2−1m2−1m on m identical machines.