Scheduling meets n-fold integer programming

Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter. In this paper, we continue this study and show that several...

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Bibliographic Details
Published in:Journal of scheduling Vol. 21; no. 5; pp. 493 - 503
Main Authors: Knop, Dušan, Koutecký, Martin
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2018
Springer Nature B.V
Subjects:
Artificial Intelligence
Business and Management
Calculus of Variations and Optimal Control; Optimization
Combinatorial analysis
Integer programming
Job shops
Operations Research/Decision Theory
Optimization
Production scheduling
Scheduling
Supply Chain Management
Scheduling on parallel machines
90C10
03D15
Fixed parameterized tractability
90B35
ISSN:1094-6136, 1099-1425
Online Access:Get full text
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Summary:Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter. In this paper, we continue this study and show that several additional cases of fundamental scheduling problems are fixed-parameter tractable for some natural parameters. Our main tool is n -fold integer programming, a recent variable dimension technique which we believe to be highly relevant for the parameterized complexity community. This paper serves to showcase and highlight this technique. Specifically, we show the following four scheduling problems to be fixed-parameter tractable, where p max is the maximum processing time of a job and w max is the maximum weight of a job: Makespan minimization on uniformly related machines ( Q | | C max ) parameterized by p max , Makespan minimization on unrelated machines ( R | | C max ) parameterized by p max and the number of kinds of machines (defined later), Sum of weighted completion times minimization on unrelated machines ( R | | ∑ w j C j ) parameterized by p max + w max and the number of kinds of machines, The same problem, R | | ∑ w j C j , parameterized by the number of distinct job times and the number of machines.
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ISSN:1094-6136
1099-1425
DOI:10.1007/s10951-017-0550-0