Machine Minimization for Scheduling Jobs with Interval Constraints

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Chuzhoy, Julia
Naor, Joseph
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The problem of scheduling jobs with interval constraints is a well-studied classical scheduling problem. The input to the problem is a collection of n jobs where each job has a set of intervals on which it can be scheduled. The goal is to minimize the total number of machines needed to schedule all jobs subject to these interval constraints. In the continuous version, the allowed intervals associated with a job form a continuous time segment, described by a release date and a deadline. In the discrete version of the problem, the set of allowed intervals for a job is given explicitly. So far, only an 0[(log n)/(log log n)]-approximation is known for either version of the problem, obtained by a randomized rounding of a natural linear programming relaxation of the problem. In fact, we show here that this analysis is tight for both versions of the problem by providing a matching lower bound on the integrality gap of the linear program. Moreover, even when all jobs can be scheduled on a single machine, the discrete case has recently been shown to be Ω(log log n)-hard to approximate. In this paper we provide improved approximation factors for the number of machines needed to schedule all jobs in the continuous version of the problem. Our main result is an O(1)-approximation algorithm when the optimal number of machines needed is bounded by a fixed constant. Thus, our results separate the approximability of the continuous and the discrete cases of the problem. For general instances, we strengthen the natural linear programming relaxation in a recursive manner by forbidding certain configurations which cannot arise in an integral feasible solution. This yields an O(OPT)-approximation, where OPT denotes the number of machines needed by an optimal solution. Combined with earlier results, our work implies an 0{√[(log n)/(log log n)]}-approximation for any value of OPT.

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2004-10-17
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Departmental Papers (CIS)
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2023-05-16T21:47:29.000
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Copyright 2004 IEEE. Reprinted from Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science 2004 (FOCS 2004), pages 81-90. Publisher URL: http://ieeexplore.ieee.org/xpl/tocresult.jsp?isNumber=29918&page=1 This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.
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