Parallel machine scheduling with job assignment restrictions

Abstract

Abstract:In the classical multiprocessor scheduling problem independent jobs must be assigned to parallel, identical machines with the objective of minimizing the makespan. This article explores the effect of assignment restrictions on the jobs for multiprocessor scheduling problems. This means that each job can only be processed on a specific subset of the machines. Particular attention is given to the case of processing times restricted to one of two values, 1 and λ, differing by at most 2. A matching based polynomial time ε-approximation algorithm is developed that has a performance ratio tending to 2 − 1 1+λ. This algorithm is shown to have the best possible performance, tending to 3/2, for processing times 1 and 2. For the special case of nested processing sets, i.e., when the sets of machines upon which individual jobs may be assigned are non-overlapping, the behavior of list scheduling algorithms is explored. Finally, for assignment restrictions determined by just one characteristic of the machines, such as disc storage or memory constraint in the case of high performance computing, we contribute an algorithm that provides a 3/2 worst case bound and runs in time linear in the number of jobs.

Matching with sizes (or scheduling with processing set restrictions)

“…We note that Glass and Kellerer [6] gave an algorithm resulting in a 1.5-approximation for P |M j , p j = {1, 2}|C max , however, our algorithm is simpler than theirs, and has a faster running time.…”

“…The corresponding scheduling problem is denoted by P |M j , p j = {1, 2}|C max . Both decision problems are NP-complete, even if c(v j ) = 2 for all v j ∈ V [3] or when we have to decide whether C max ≤ 2 or C max ≥ 3 for the related instance of P |M j , p j = {1, 2}|C max [6]. Note that the latter result implies that we cannot give a polynomial-time approximation for P |M j , p j = {1, 2}|C max with a factor better than 1.5 (unless P=NP).…”

“…We will define the problem more precisely in Section 2. Meanwhile we refer to a survey [13] on scheduling with processing set restrictions and two papers [6,7] that contain results closely related to those presented in this paper (we will specify these in Section 2).…”

Stable Assignment with Couples: Parameterized Complexity and Local Search

“…If couples are involved, the problem becomes hard. More precisely, the decision version of this problem is NP-complete [13,3], even in the special case where each hospital has a capacity of 2, and the acceptable hospital pairs for a couple are always of the form (h, h) for some h ∈ H. However, if the number of couples is small, which is a reasonable assumption in many practical applications, Maximum Matching with Couples becomes tractable, as shown by Theorem 1. We will use the following lemma to solve a special case of the Maximum Matching with Couples problem.…”

“…This problem was proved to be NPcomplete even if p = 2 (see [13,3]), so investigating the parameterized complexity of this problem might be of practical importance.…”