Scheduling with a minimum number of machines
Introduction
A typical scheduling problem consists of three factors, namely the machine environment, the job description and the criteria. One aims at a feasible schedule to optimize the criterion subject to the resource constraints on machines and the information on jobs. Usually the machine environment is known and fixed. In other words we know what machines are available before we make use of them to schedule jobs efficiently. In this paper we are going to deal with a scheduling problem in a dual fashion. Given a set of jobs, each with a time window and a length, we must schedule each job within its time window without any overlap with other jobs on its machine. Clearly we are interested in the minimum number of machines needed to complete all jobs. This problem can be formally defined as follows.
Problem statement. We are given an infinite number of identical machines, and a set of jobs. Associated with each job there are a release date , a deadline , and a processing time , where are all non-negative integers. Job can be denoted as a tuple . The goal is to schedule all jobs with a minimum number of machines so that every job starts no earlier than its release date and completes by its deadline. This problem is denoted as SRDM (scheduling with release dates and deadlines on a minimum number of machines) [1]. The model is motivated by a variety of practical applications including train scheduling [2], scheduling of maintenance work for trains in a service station and runway scheduling. In addition, let us consider service scheduling, in which there are a number of clients to be served. Each client has his own time window during which he is available and a required service time. The question is to hire as few servers as possible to satisfy all clients under the condition that a client can be served by only one server and each server can serve at most one client at a time. If the system can only accept clients at very beginning it becomes a special case of SRDM where all jobs have a common release time (equivalently, if all servers share a common working deadline by which the assigned jobs have been completed, a special case that all jobs have the same deadlines occur). If the system only provides some identical service, we can assume that all job processing times are the same–it is another special case of SRDM. In this paper we will concentrate on approximation algorithms for the above two special cases.
For each job we may consider the interval as the time window of in which it has to be processed, and is the slack of . The slack gives the maximum amount the job (or an interval of length ) can be moved within its window. If all jobs have identical time windows, the problem SRDM is reduced to the well-known bin packing problem which is -hard [3], [4]. If all jobs are of slack zero, then occupies the whole window on some machine. It is exactly the Interval Partition Problem (also referred to as the Interval Coloring Problem) which can be solved in polynomial time [5], [6].
Related results. A decision version of the SRDM problem asking if one machine suffices to schedule all the jobs is strongly -complete [4]. Another variant of the problem is the Job Interval Selection Problem (JISP) [7], [8], in which there are a set of jobs and an integral number . Each job consists of a number of intervals on the real line. The goal is to find a subset of intervals with maximum cardinality such that at most one interval is selected from each job, and for any point on the real line at most intervals containing . Even the single machine case () was shown to be -hard and a greedy algorithm of bound two was presented in [9]. Chuzhoy et al. [7] improved the bound by giving a -approximation algorithm for this problem.
The SRDM problem was considered by Cieliebak et al. [1]. They distinguished the easy case and the hard case by proving that it is -hard if the maximum slack while it is polynomially solvable if . It was further shown that the SRDM problem cannot be approximated within a factor less than 2 unless . The greedy algorithm by Spieksma [9] can be recursively applied to the SRDM problem which gives an upper bound of . They developed an approximation algorithm, called Greedy Best-Fit (GBF), and showed that its asymptotic approximation ratio is at most 9 for the case of equal processing times. However GBF is within a factor of at least for the general problem. Another approximation algorithm called Divide Best-Fit was derived for the case of equal release dates which has an asymptotic approximation ration of at most .
Chuzhoy et al. [10] extensively studied the general SRDM problem (in their paper they referred the problem as the continuous version of machine minimization for scheduling jobs with interval constraints). They showed an -approximation algorithm when the optimal number of machines needed is bounded by a constant. It further implies an -approximation for the general case.
Our contribution. In this paper we deal with the two special cases as in [1]. When all jobs are of the same release dates we derive an approximation algorithm with a ratio of 2. When all jobs are of equal processing times, we reduce the upper bound of algorithm GBF from 9 down to 6 and show a lower bound of 4. For both cases we significantly improve the previous results. Moreover, our bounds are not asymptotic but absolute (without an additive constant).
A machine is open if it is assigned a job. For an instance we denote, respectively, by and the number of machines used in an optimal schedule and the number of machines opened by an approximation algorithm. The approximation ratio of the algorithm is thus defined as the supremum of the ratio over all instances .
Section snippets
Common release dates
The Greedy Best-Fit (GBF) algorithm by Cieliebak et al. [1]. works in a greedy way. It processes the shiftable intervals in the nondecreasing order of window sizes. For job GBF checks if there is a time window available for on an open machine. If yes is assigned to some open machine; otherwise a new machine is needed for . In both cases is scheduled with a leftmost placement. Here by leftmost placement we mean that the job is started at the earliest possible time over all open
Equal processing times
In this section we consider the case of equal processing times. Assume that all jobs are of processing time . Cieliebak et al. [1]proved that algorithm GBF is an asymptotic 9-approximation algorithm. We will show that the approximation ratio can be reduced to 6. Note that the two algorithms GBF and GIS are the same for equal processing times.
Theorem 4 GBF is a 6-approximation algorithm for the case of equal processing times.
Proof Let be any instance and let . Assume that job opens the
Acknowledgements
We would like to thank the anonymous referee for the insightful comments. The research of the second author has been partially supported by NSFC (60573020).
References (10)
- et al.
Interval selection: Applications, algorithms and lower bounds
Journal of Algorithms
(2003) - M. Cieliebak, T. Erlebach, F. Hennecke, B. Weber, P. Widmayer, Scheduling with release times and deadlines on a minimum...
- S. Eidenbenz, A. Pagourtzis, P. Widmayer, Flexible train rostering, in: Proceedings of the 14th International Symposium...
- et al.
Approximation algorithms for bin packing: A survey
- et al.
Computers and Intractability
(1979)
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