Improved Approximation Schemes for Scheduling Unrelated Parallel Machines

Abstract

We consider the problem of scheduling n independent jobs on m unrelated parallel machines where each job has to be processed by exactly one machine, processing job j on machine i requires p ij time units, and the objective is to minimize the makespan, i.e., the maximum job completion time. Focusing on the case when m is fixed, we present for both preemptive and nonpreemptive variants of the problem fully polynomial approximation schemes whose running times depend only linearly on n. We also study an extension of the problem where processing job j on machine i incurs a cost of c ij , and thus there are two optimization criteria: makespan and cost. We show that, for any fixed m, there is a fully polynomial approximation scheme that, given values T and C, computes for any fixed > 0 a schedule in O n time with makespan at most 1 + T and cost at most 1 + C, if there exists a schedule of makespan T and cost C.1. Introduction. Let n and m denote the number of jobs and machines, respectively. Suppose that the execution time of job j on machine i is p ij , i = 1 m j = 1 n. The objective is to compute a schedule such that each job is processed by exactly one machine and the makespan, i.e., the largest completion time is minimized. For the nonpreemptive variant of the problem, Lenstra et al. (1990) gave a polynomial-time 2-approximation algorithm, which, for the general case, currently is the best-known approximation ratio achieved in polynomial time. They also proved that, for any positive < 1/2, no polynomial-time 1 + -approximation algorithm exists unless P = NP. Since the problem is NP-hard even for m = 2, it is natural to ask how well the optimum can be approximated when there are only a constant number of machines.In contrast to the previously mentioned inapproximability result for the general case, there exists a fully polynomial-time approximation scheme for the problem when m is fixed. Horowitz and Sahni (1976) proved that, for any > 0, an -approximate solution can be computed in O nm nm/ m−1 time, which is polynomial in both n and 1/ if m is constant. Lenstra et al. (1990) also gave an approximation scheme for the problem with running time bounded by the product of n + 1 m/ and a polynomial of the input size. Even though for fixed m their algorithm is not fully polynomial, it has a much smaller space complexity than the one in Horowitz and Sahni (1976). In this paper we present a new approximation scheme for the problem whose running time is n m/ O m . If there are only a constant number of machines, it gives a fully polynomial-time approximation scheme that computes an -approximate solution in O n time for any fixed > 0. This linear complexity bound is a substantial improvement in terms of n compared to the above mentioned results. To obtain the linear running time for the case when m is fixed, we consider "long" and "short" jobs separately, combine the previous (dynamic (Horowitz and Sahni 1976) and linear (Lenstra et al. 1990) programming) approaches, use the recent price-directive decomposition method proposed...

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