Parallel machine scheduling with nested job assignment restrictions
Introduction
Scheduling with job assignment restrictions. Consider independent jobs that are to be scheduled on parallel (but not necessarily identical) machines . Every job can only be processed on a certain subset of the machines, and on those machines it takes time units of uninterrupted processing to complete. All jobs are available for processing at time 0. The goal is to schedule the jobs on the machines such that the maximum job completion time is minimized. This problem is called scheduling with job assignment restrictions, and we will denote it by ; see Lawler, Lenstra, Rinnooy Kan and Shmoys [1] for more information on the standard three-field scheduling notation.
The approximability behavior of problem is not fully understood: The results of Lenstra, Shmoys and Tardos [2] yield a polynomial time approximation algorithm with worst case ratio 2 for the more general scheduling problem ; this is also the strongest known positive approximability result for problem . On the negative side, [2] prove that (unless P=NP) one cannot reach any worst case ratio better than 3/2 for in polynomial time. Ebenlendr, Krcal and Sgall [3] extend this negative result even to the highly restricted special case where holds for all jobs. The exact location of the approximability threshold between and 2 is totally unclear, and forms an outstanding open problem in the area of machine scheduling (see for instance Schuurman and Woeginger [4]).
Special cases with ordered and nested machine sets. Hwang, Chang and Lee [5] and Glass and Kellerer [6] discuss special cases of where the family of sets carries certain nice combinatorial structures. In one of these special cases, the machine sets are totally ordered by inclusion:
This special case arises for instance in the assignment of computer programs to processors, where each program has a memory requirement and where each processor has a memory capacity. A program can only be assigned to processors with sufficiently large memories. Clearly, if the processing requirement of is larger than the processing requirement of , then . Hwang, Chang and Lee [5] derived a polynomial time approximation algorithm with worst case ratio for this totally ordered special case. Glass and Kellerer [6] improved the worst case ratio to , and finally Ou, Leung and Li [7] came up with a polynomial time approximation scheme.For any two jobs and , either or .
Glass and Kellerer [6] also discuss another special case of where the machine sets are nested. (Note that the totally ordered special case as discussed above forms a subproblem of the nested special case.)
This special case arises for instance in the drying stage of flour mills in the United Kingdom; see Glass and Mills [8]. Glass and Kellerer [6] derive a polynomial time approximation algorithm with worst case ratio for this nested special case. The existence of a polynomial time approximation scheme for the nested special case is posed as an open research problem in the conclusions section of [6].For any two jobs and , either , or , or .
Our result. We derive a polynomial time approximation scheme for the special case of where the machine sets are nested. Our result generalizes the approximation scheme of [7] and answers the open problem of [6]. Our approach is heavily built on the ideas of Hochbaum and Shmoys [9] and Alon, Azar, Woeginger and Yadid [10].
The note is structured as follows. Section 2 summarizes several easy observations on an auxiliary packing problem in trees. Section 3 shows how to simplify a scheduling instance into a so-called rounded instance, and Section 4 shows how to solve this rounded instance in polynomial time. Section 5 wraps things up and presents the approximation scheme.
Section snippets
Trees and packings
Tree structures will play a central role in our investigations. As usual, a rooted tree is a connected acyclic graph, where the edges in correspond to father–son relations between the vertices in . With the exception of one distinguished vertex (that is called the root), every vertex in has a unique father; the root has no father. The vertices without sons are called leaves. The descendants of a vertex are vertex itself, the sons of , the sons of its sons, and so on.
Every
Definition of the rounded instance
We now return to the scheduling problem, and we consider an arbitrary instance of the nested special case of . We enumerate all machine sets occurring in this instance, and we add every missing one-element machine set and also the set to this enumeration. We organize these machine sets in a tree :
- •
Every vertex corresponds to a set of machines in the enumeration. The leaves correspond to the one-element machine sets, and the root corresponds to
Solution of the rounded instance
Throughout this section we will assume that the integer is a fixed constant that does not depend on the input. Our goal is to design a fast algorithm for the rounded instance . Recall from the preceding section that in such a rounded instance , all job processing times are integer multiples of lying in the range from to .
We represent subsets of the jobs in instance as vectors where the non-negative integer denotes the number of jobs in the subset
The approximation scheme
Finally, let us put things together and derive the desired PTAS. Consider some fixed real number with . For a given instance of with nested machine sets, we perform the following three steps (S1 through S3) to get an approximation algorithm with worst case ratio .
- (S1)
Define a positive integer . Compute the instance from .
- (S2)
Use Theorem 4.2 to solve instance to optimality, and call the resulting optimal schedule .
- (S3)
Transform schedule into a
Acknowledgements
This research was supported by the Netherlands Organisation for Scientific Research (NWO), grant 639.033.403, by the EU Research Project 015964 (AEOLUS: Algorithmic Principles for Building Efficient Overlay Computers) and by BSIK grant 03018 (BRICKS: Basic Research in Informatics for Creating the Knowledge Society).
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