Non-clairvoyantly Scheduling to Minimize Convex Functions

Abstract

The paper considers scheduling jobs online to minimize the objective $\sum _{i \in [n]}w_ig(C_i-r_i)$, 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. It is known that the clairvoyant algorithm Highest-Density-First (HDF) is $(2+\epsilon )$-speed O(1)-competitive for this objective on a single machine for any fixed $0< \epsilon < 1$ (Im et al., in: ACM-SIAM symposium on discrete algorithms, pp 1254–1265, 2012). In this paper, we give the first non-trivial results for this problem when g is a non-decreasing convex function and the algorithm must be non-clairvoyant. More specifically, our results include:

• A $(2+\epsilon )$-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 $0< \epsilon < 1$.

• A $(3+\epsilon )$-speed O(1)-competitive non-clairovyant algorithm on multiple identical machines for all non-decreasing convex g for any fixed $0< \epsilon < 1$.

The paper gives the first non-trivial upper-bound on multiple machines even if the algorithm is allowed to be clairvoyant. All performance guarantees above hold for all non-decreasing convex functions gsimultaneously. The positive results are supplemented by almost matching lower bounds. We show that any algorithm that is oblivious to g is not O(1)-competitive with speed augmentation less than 2 on a single machine. Further, any non-clairvoyent algorithm that knows the function g cannot be O(1)-competitive with speed augmentation less than $\sqrt{2}$ on a single machine or  $(2-\frac{1}{m})$ on m identical machines.