The Geometry of Scheduling

TL;DR

This paper introduces a new geometric approach to a broad class of scheduling problems, providing improved approximation algorithms with significant theoretical guarantees for minimizing aggregate costs.

Contribution

It presents a novel geometric set-cover reduction and randomized algorithms that achieve better approximation ratios for general scheduling objectives.

Findings

Achieves an O(log log n P) approximation ratio for the general scheduling problem.

Provides an O(1) approximation for jobs with identical release times.

Improves previous approximation bounds exponentially for many special cases.

Abstract

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. Our main result 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. The main idea is to reduce this scheduling problem to a particular geometric set-cover problem which is then solved 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. Our geometric interpretation of scheduling may be of independent interest.