Online Non-clairvoyant Scheduling to Simultaneously Minimize All Convex Functions

摘要

We consider scheduling jobs online to minimize the objective ∑_(i∈[n]) w_ig_((C_i - T_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(l)-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: 1 A (2 + ε)-speed O(l)-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. 2 A (3 + ε)-speed O(l)-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(l)-competitive with speed less than 2 on a single machine. Further, any non-clairvoyent algorithm that knows the function g cannot be O(l)-competitive with speed less than 2~(1/2) on a single machine or speed less than 2 -1/mon m identical machines.