Non-clairvoyantly Scheduling to Minimize Convex Functions

The paper considers 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. It is 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 (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+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 . 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 g simultaneously . 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 √(2) on a single machine or (2-1/m) on m identical machines.