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The knapsack problem with special neighbor constraints

Author

Listed:
  • Steffen Goebbels

    (Niederrhein University of Applied Sciences)

  • Frank Gurski

    (University of Düsseldorf)

  • Dominique Komander

    (University of Düsseldorf)

Abstract

The knapsack problem is one of the simplest and most fundamental NP-hard problems in combinatorial optimization. We consider two knapsack problems which contain additional constraints in the form of directed graphs whose vertex set corresponds to the item set. In the one-neighbor knapsack problem, an item can be chosen only if at least one of its neighbors is chosen. In the all-neighbors knapsack problem, an item can be chosen only if all its neighbors are chosen. For both problems, we consider uniform and general profits and weights. We prove upper bounds for the time complexity of these problems when restricting the graph constraints to special sets of digraphs. We discuss directed co-graphs, minimal series-parallel digraphs, and directed trees.

Suggested Citation

  • Steffen Goebbels & Frank Gurski & Dominique Komander, 2022. "The knapsack problem with special neighbor constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 1-34, February.
    • Handle: RePEc:spr:mathme:v:95:y:2022:i:1:d:10.1007_s00186-021-00767-5
    DOI: 10.1007/s00186-021-00767-5
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    References listed on IDEAS

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    1. D. S. Johnson & K. A. Niemi, 1983. "On Knapsacks, Partitions, and a New Dynamic Programming Technique for Trees," Mathematics of Operations Research, INFORMS, vol. 8(1), pages 1-14, February.
    2. Frank Gurski & Dominique Komander & Carolin Rehs, 2020. "Solutions for subset sum problems with special digraph constraints," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(2), pages 401-433, October.
    3. Frank Gurski & Dominique Komander & Carolin Rehs, 2020. "Subset Sum Problems with Special Digraph Constraints," Operations Research Proceedings, in: Janis S. Neufeld & Udo Buscher & Rainer Lasch & Dominik Möst & Jörn Schönberger (ed.), Operations Research Proceedings 2019, pages 339-346, Springer.
    4. Ulrich Pferschy & Joachim Schauer, 2017. "Approximation of knapsack problems with conflict and forcing graphs," Journal of Combinatorial Optimization, Springer, vol. 33(4), pages 1300-1323, May.
    5. Laurent Gourvès & Jérôme Monnot & Lydia Tlilane, 2018. "Subset sum problems with digraph constraints," Journal of Combinatorial Optimization, Springer, vol. 36(3), pages 937-964, October.
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