Preemptive models of scheduling with controllable processing times and of scheduling with imprecise computation: A review of solution approaches

This paper provides a review of recent results on scheduling with controllable processing times. The stress is on the methodological aspects that include parametric flow techniques and methods for solving mathematical programming problems with submodular constraints. We show that the use of these methodologies yields fast algorithms for solving problems on single machine or parallel machines, with either one or several objective functions. For a wide range of problems with controllable processing times we report algorithms with the running times which match those known for the corresponding problems with fixed processing times. As a by-product, we present the best possible algorithms for a number of problems on parallel machines that are traditionally studied within the body of research on scheduling with imprecise computation

Decomposition algorithms for submodular optimization with applications to parallel machine scheduling with controllable processing times

In this paper we present a decomposition algorithm for maximizing a linear function over a submodular polyhedron intersected with a box. Apart from this contribution to submodular optimization, our results extend the toolkit available in deterministic machine scheduling with controllable processing times. We demonstrate how this method can be applied to developing fast algorithms for minimizing total compression cost for preemptive schedules on parallel machines with respect to given release dates and a common deadline. Obtained scheduling algorithms are faster and easier to justify than those previously known in the scheduling literature

Scheduling divisible loads with time and cost constraints

In distributed computing, divisible load theory provides an important system model for allocation of data-intensive computations to processing units working in parallel. The main task is to define how a computation job should be split into parts, to which processors those parts should be allocated and in which sequence. The model is characterized by multiple parameters describing processor availability in time, transfer times of job parts to processors, their computation times and processor usage costs. The main criteria are usually the schedule length and cost minimization. In this paper, we provide the generalized formulation of the problem, combining key features of divisible load models studied in the literature, and prove its NP-hardness even for unrestricted processor availability windows. We formulate a linear program for the version of the problem with a fixed number of processors. For the case with an arbitrary number of processors, we close the gaps in the study of special cases, developing efficient algorithms for single criterion and bicriteria versions of the problem, when transfer times are negligible

Proportionate flow shop games

htmlabstractIn a proportionate flow shop problem several jobs have to be processed through a fixed sequence of machines and the processing time of each job is equal on all machines. By identifying jobs with agents, whose costs linearly depend on the completion time of their jobs, and assuming an initial processing order on the jobs, we face two problems: the first one is how to obtain an optimal order that minimizes the total processing cost, the second one is how to allocate the cost savings obtained by ordering the jobs optimally. In this paper we focus on the allocation problem. PFS games are defined as cooperative games associated to proportionate flow shop problems. It is seen that PFS games have a nonempty core. Moreover, it is shown that PFS games are convex if the jobs are initially ordered in decreasing urgency. For this case an explicit game independent expression for the Shapley value is provid

A note on reverse scheduling with maximum lateness objective

The inverse and reverse counterparts of the single-machine scheduling problem 1||Lmax are studied in [2], in which the complexity classification is provided for various combinations of adjustable parameters (due dates and processing times) and for five different types of norm: ℓ1,ℓ2,ℓ∞,ℓΣH , and ℓmaxH . It appears that the O(n2) -time algorithm for the reverse problem with adjustable due dates contains a flaw. In this note, we present the structural properties of the reverse model, establishing a link with the forward scheduling problem with due dates and deadlines. For the four norms ℓ1,ℓ∞,ℓΣH , and ℓmaxH , the complexity results are derived based on the properties of the corresponding forward problems, while the case of the norm ℓ2 is treated separately. As a by-product, we resolve an open question on the complexity of problem 1||∑αjT2j

Models and algorithms for energy-efficient scheduling with immediate start of jobs

We study a scheduling model with speed scaling for machines and the immediate start requirement for jobs. Speed scaling improves the system performance, but incurs the energy cost. The immediate start condition implies that each job should be started exactly at its release time. Such a condition is typical for modern Cloud computing systems with abundant resources. We consider two cost functions, one that represents the quality of service and the other that corresponds to the cost of running. We demonstrate that the basic scheduling model to minimize the aggregated cost function with n jobs is solvable in O(nlogn) time in the single-machine case and in O(n²m) time in the case of m parallel machines. We also address additional features, e.g., the cost of job rejection or the cost of initiating a machine. In the case of a single machine, we present algorithms for minimizing one of the cost functions subject to an upper bound on the value of the other, as well as for finding a Pareto-optimal solution

NP-hardness of shop-scheduling problems with three jobs

AbstractThis paper deals with the problem of scheduling n jobs on m machines in order to minimize the maximum completion time or mean flow time of jobs. We extend the results obtained in Sotskov (1989, 1990, 1991) on the complexity of shop-scheduling problems with n = 3. The main result of this paper is an NP-hardness proof for scheduling 3 jobs on 3 machines, whether preemptions of operations are allowed or forbidden

Complexity of mixed shop scheduling problems: A survey

We survey recent results on the computational complexity of mixed shop scheduling problems. In a mixed shop, some jobs have fixed machine orders (as in the job shop), while the operations of the other jobs may be processed in arbitrary order (as in the open shop). The main attention is devoted to establishing the boundary between polynomially solvable and NP-hard problems. When the number of operations per job is unlimited, we focus on problems with a fixed number of jobs