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 + ∈)-speed O(1)-competitive for this objective on a single machine for any fixed 0 < ∈ < 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: 1. 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. 2. 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 {the square root of}2 on a single machine or speed less than 2 - 1/m on m identical machines.
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