The Geometry of Scheduling

Abstract

We consider the following general scheduling problem. The input consists of n jobs, each with an arbitrary release time, size, and 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 total weighted flow time, total weighted tardiness, and sum of flow time squared. We give an O(log log P ) approximation for this problem, where P is the ratio of the maximum to minimum job size. We also give an O(1) approximation in the special case of identical release times. These results are obtained by reducing the scheduling problem to a geometric capacitated set cover problem in two dimensions. Introduction.We consider the following general offline scheduling problem. General scheduling problem (GSP). The input consists of a collection of n jobs, and for each job j there is a positive integer release time r j , a positive integer size p j and a nondecreasing cost (or weight) function w j (t) ≥ 0 specifying a nonnegative cost for each time t > r j . (We will specify later how these weight functions are represented.) A feasible solution is a preemptive schedule, which assigns to each job j time slots [t, t + 1] (not necessarily consecutive and satisfying t ≥ r j ), during which j is run. A job is completed once it has been run for p j units of time. If job j completes at time t, then a cost of w j (t) is incurred for that job. The objective is to minimize the total cost,

On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

“…Since individual cost functions allow one to model general tardiness cost g j : t → max{0, t − d j } with job-specific deadlines d j -for which the problem is known to be strongly NP-hard [Lawler 1977]-the problem 1 || w j g j (C j ) is obviously also strongly NP-hard. Bansal and Pruhs [2010a] have given a geometric interpretation of the flow time problem variant that yields a O(log log P)-approximation in the presence of release dates and preemption where P is again the ratio of the maximum and minimum processing time. In the special case of uniform release dates, their method achieves the constant factor of 16.…”

Online Scheduling with General Cost Functions

“…Recently, Bansal and Pruhs considered the offline version of this problem, where each job J j has an individual cost function g j (x) [2]. The main result in [2] is a polynomial-time O(log log nP )-approximation algorithm, where P is the ratio of the size of the largest job to the size of the smallest job.…”

“…The main result in [2] is a polynomial-time O(log log nP )-approximation algorithm, where P is the ratio of the size of the largest job to the size of the smallest job. This result is without speed augmentation.…”

A Polynomial Time Constant Approximation For Minimizing Total Weighted Flow-time

“…Chekuri and Khanna [8] . Substantial progress towards getting a polynomial time constant approximation algorithm was made by Bansal and Pruhs [4], who gave a very elegant O(log logP ) approximation to the problem. Their main insight was to reduce min-WPFT to a geometric set-cover problem, and argue that the geometry of the resulting objects leads to O(log logP ) approximation to this set-cover problem.…”