Three partition refinement algorithms
SIAM Journal on Computing
Computing a perfect edge without vertex elimination ordering of a chordal bipartite graph
Information Processing Letters
Approximation algorithms for NP-hard problems
Scheduling to minimize average completion time: off-line and on-line approximation algorithms
Mathematics of Operations Research
Graph classes: a survey
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems
Journal of the ACM (JACM)
Strengthening integrality gaps for capacitated network design and covering problems
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Precedence constrained scheduling to minimize sum of weighted completion times on a single machine
Discrete Applied Mathematics
Two-Dimensional Gantt Charts and a Scheduling Algorithm of Lawler
SIAM Journal on Discrete Mathematics
Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On the Approximability of Average Completion Time Scheduling under Precedence Constraints
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
Operations Research Letters
Partially ordered knapsack and applications to scheduling
Discrete Applied Mathematics
Single-Machine Scheduling with Precedence Constraints
Mathematics of Operations Research
Single machine precedence constrained scheduling is a vertex cover problem
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Optimizing data popularity conscious bloom filters
Proceedings of the twenty-seventh ACM symposium on Principles of distributed computing
Scheduling with Precedence Constraints of Low Fractional Dimension
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
On the Approximability of Single-Machine Scheduling with Precedence Constraints
Mathematics of Operations Research
Approximating precedence-constrained single machine scheduling by coloring
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
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In the partially-ordered knapsack problem (POK) we are given a set N of items and a partial order 驴P on N. Each item has a size and an associated weight. The objective is to pack a set N驴 驴 N of maximum weight in a knapsack of bounded size. N驴 should be precedence-closed, i.e., be a valid prefix of 驴P. POK is a natural generalization, for which very little is known, of the classical Knapsack problem. In this paper we advance the state-of-the-art for the problem through both positive and negative results. We give an FPTAS for the important case of a 2-dimensional partial order, a class of partial orders which is a substantial generalization of the series-parallel class, and we identify the first non-trivial special case for which a polynomial-time 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 precedence-constrained jobs on a single machine to minimize the sum of weighted completion times, a problem closely related to POK.