Integrality Property in Preemptive Parallel Machine Scheduling

We consider parallel machine scheduling problems with identical machines and preemption allowed. It is shown that every such problem with chain precedence constraints and release dates and an integer-concave objective function satisfies the following integrality property : for any problem instance with integral data there exists an optimal schedule where all interruptions occur at integral dates. As a straightforward consequence of this result, for a wide class of scheduling problems with unit processing times a so-called preemption redundancy property is valid. This means that every such preemptive scheduling problem is equivalent to its non-preemptive counterpart from the viewpoint of both its optimum value and the problem complexity. The equivalence provides new and simpler proofs for some known complexity results and closes a few open questions. In the current paper we present some new structural results for preemptive scheduling problems. This work proceeds our previous research of structural properties of optimal solutions for preemptive scheduling problems initiated in [2], where some general results on the existence of optimal schedules and the existence of optimal schedules with a finite number of interruptions were established for a wide range of scheduling problems. Furthermore, two Rational Structure Theorems were proved in [2] for wide classes of scheduling problems, according to which for any problem instance heaving a nonempty set of feasible solutions there exists an optimal schedule with the following properties: (1) the total number of interruptions grows polynomially with the number of operations and with the number of fixed dates specified in that instance; (2) all operation start times and completion times and all interruptions occur at integer multiples of a rational number δ > 0 with size polynomially bounded in the input size; Research of the Russian authors is supported by RFBR grant no. 08-01-00370 and Russian-Taiwan grant no. 08-06-92000. Research of the third author is partially supported by ADTP grant 2.1.1/3235. A. Frid et al. (Eds.): CSR 2009, LNCS 5675, pp. 38–46, 2009. c © Springer-Verlag Berlin Heidelberg 2009 Integrality Property in Preemptive Parallel Machine Scheduling 39 (3) the optimal value of the objective function is an integer multiple of δ, where the size of the integer multiplier is also polynomially bounded in the input size. These results were established for a wide class of preemptive scheduling models including both classical and non-traditional machine scheduling and project scheduling models with constrained resources and a large variety of objective functions including all classical ones. An important consequence of these Rational Structure Theorems is the fact that the decision versions of preemptive scheduling problems under consideration belong to class NP . A significantly stronger structural property (compared to that formulated in the above mentioned Rational Structure Theorems) is assumed, when we speak about the integrality property. The latter means that for any problem instance there exists an optimal preemptive schedule where all interruptions occur at integral dates. This property is investigated in the current paper for parallel machine problems with identical machines. New Results. In the current paper we establish the integrality property for the preemptive version of the parallel machine scheduling problem with chain precedence constraints, release dates and an arbitrary regular integer-concave objective function. As a straightforward consequence of this result, a so-called preemption redundancy property holds for a wide class of scheduling problems with unit processing times. In particular, this closes two open questions on the preemption redundancy property of problems P |pj = 1, rj , pmtn| ∑ Tj and P |pj = 1, pmtn| ∑ wjTj (see Brucker [4]). This property also implies that every such preemptive scheduling problem is equivalent to its non-preemptive counterparts from the viewpoint of both its optimum value and the problem complexity. The equivalence provides new and sometimes simpler proofs for some known complexity results. Specifically, our Theorem 3.2 implies the NP-hardness of Pm|pj = 1, chains, pmtn| ∑ wjCj and Pm|pj = 1, chains, pmtn| ∑ Uj (previously proved in [1, 6, 19]), and the polynomial time solvability of problem P |pj = 1, rj , pmtn| ∑ wjTj whose complexity status remained open. Furthermore, due to the strong NP-hardness of problem 1|pj = 1, chains| ∑ Tj established by Leung and Young [13], our result implies the strong NP-hardness of the 1|pj = 1, chains, pmtn| ∑ Tj problem. To our knowledge (see also the Brucker’s home page [21]), the complexity status of this problem was open before. Related Results. There are few systematic studies of such structural questions in the literature on preemptive scheduling, and most known results follow from either (i) the fact that there is no advantage to preemption [3], [4], or (ii) the existence or properties of polynomial time algorithms. Results following from (i) are clearly the strongest type of structural results one could hope for for scheduling problems. We refer to the standard scheduling literature (e.g., [12]) for many such classical results; another extensive reference is the book by Tanaev, Gordon and Shafransky [18]. Structural results following from (ii) have been obtained mostly for parallel machine and open shop problems. We use the standard three-field notation [12] to describe such scheduling problems. McNaughton [14] constructs an optimal 40 P. Baptiste et al. schedule with at most m− 1 interruptions for problem P |pmtn|Cmax on m identical parallel machines and makespan objective. Sauer and Stone [16] (see also [15]) prove that for the parallel machine scheduling problem with n jobs, precedence constraints, unit processing times and the minimum makespan objective there is an optimal preemptive schedule with at most n − 1 preemption dates. Gonzalez and Sahni [8] construct an optimum schedule with at most 2(m − 1) interruptions for the uniform parallel machine versionQ|pmtn|Cmax of this problem. The bounds of McNaughton and of Gonzalez and Sahni on the number of interruptions (and preemption dates) are tight. Labetoulle et al. [10] prove that the natural greedy algorithm for the problemQ|rj , pmtn| ∑ Cj with m machines and n jobs finds an optimal solution with at most 2n−m interruptions. For the unrelated parallel machine problem R|pmtn|Cmax, Lawler et al. [12] state that a procedure of Lawler and Labetoulle [11] can be modified to yield an optimal schedule with no more than O(m) interruptions. Turning now to open shop problems, Gonzalez and Sahni [7] construct an optimal schedule for the problem O|pmtn|Cmax with m machines, n jobs and ξ operations, which has at most ξ + n+m preemption dates. Du and Leung [5] proved the corresponding result for O2|pmtn| ∑ j Cj . Little attention seems to have been given in the literature to investigating of structural properties of optimal solutions for other preemptive scheduling problems. Paper Outline. In the next section we give definitions of basic notions. In Section 2 we prove the integrality property for a class of parallel machine scheduling problems. Next we give a short conclusion in the last section.