PartiallyOrdered Knapsack and Applications to Scheduling
 Stavros G. Kolliopoulos,
 George Steiner
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Abstract
In the partiallyordered knapsack problem (POK) we are given a set N of items and a partial order ≺p on N. Each item has a size and an associated weight. The objective is to pack a set N’⊆ N of maximum weight in a knapsack of bounded size. N’ should be precedenceclosed, i.e., be a valid prefix of ≺p . POK is a natural generalization, for which very little is known, of the classical Knapsack problem. In this paper we advance the stateoftheart for the problem through both positive and negative results. We give an FPTAS for the important case of a 2dimensional partial order, a class of partial orders which is a substantial generalization of the seriesparallel class, and we identify the first nontrivial special case for which a polynomialtime algorithm exists. We also characterize cases where the natural linear relaxation for POK is useful for approximation and we demonstrate its limitations. Our results have implications for approximation algorithms for scheduling precedenceconstrained jobs on a single machine to minimize the sum of weighted completion times, a problem closely related to POK.
 Title
 PartiallyOrdered Knapsack and Applications to Scheduling
 Book Title
 Algorithms — ESA 2002
 Book Subtitle
 10th Annual European Symposium Rome, Italy, September 17–21, 2002 Proceedings
 Pages
 pp 612624
 Copyright
 2002
 DOI
 10.1007/3540457496_54
 Print ISBN
 9783540441809
 Online ISBN
 9783540457497
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 2461
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Rolf Möhring ^{(4)}
 Rajeev Raman ^{(5)}
 Editor Affiliations

 4. Fakultät II: Mathematik und Naturwissenschaften, Technische Universität Berlin
 5. Department of Mathematics and Computer Science, University of Leicester
 Authors

 Stavros G. Kolliopoulos ^{(6)}
 George Steiner ^{(7)}
 Author Affiliations

 6. Department of Computing and Software, McMaster University, USA
 7. Management Science and Information Systems, McMaster University, USA
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