The Geometry of Scheduling

We consider the following general scheduling problem: The input consists of $n$ jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as weighted flow, weighted tardiness, and sum of flow squared. The main contribution of this paper is a randomized polynomial-time algorithm with an approximation ratio $O(\log \log nP )$, where $P$ is the maximum job size. We also give an $O(1)$ approximation in the special case when all jobs have identical release times. Initially, we show how to reduce this scheduling problem to a particular geometric set-cover problem. We then consider a natural linear programming formulation of this geometric set-cover problem, strengthened by adding knapsack cover inequalities, and show that rounding the solution of this linear program can be reduced to other particular geometric set-cover problems. We then develop algorithms for these sub-problems using the local ratio technique, and Varadarajan's quasi-uniform sampling technique. This general algorithmic approach improves the best known approximation ratios by at least an exponential factor (and much more in some cases) for essentially all of the nontrivial common special cases of this problem. We believe that this geometric interpretation of scheduling is of independent interest.