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Solutions for subset sum problems with special digraph constraints

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  • Frank Gurski

    (University of Düsseldorf)

  • Dominique Komander

    (University of Düsseldorf)

  • Carolin Rehs

    (University of Düsseldorf)

Abstract

The subset sum problem is one of the simplest and most fundamental NP-hard problems in combinatorial optimization. We consider two extensions of this problem: The subset sum problem with digraph constraint (SSG) and subset sum problem with weak digraph constraint (SSGW). In both problems there is given a digraph with sizes assigned to the vertices. Within SSG we want to find a subset of vertices whose total size does not exceed a given capacity and which contains a vertex if at least one of its predecessors is part of the solution. Within SSGW we want to find a subset of vertices whose total size does not exceed a given capacity and which contains a vertex if all its predecessors are part of the solution. SSG and SSGW have been introduced recently by Gourvès et al. who studied their complexity for directed acyclic graphs and oriented trees. We show that both problems are NP-hard even on oriented co-graphs and minimal series-parallel digraphs. Further, we provide pseudo-polynomial solutions for SSG and SSGW with digraph constraints given by directed co-graphs and series-parallel digraphs.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:92:y:2020:i:2:d:10.1007_s00186-020-00718-6
    DOI: 10.1007/s00186-020-00718-6
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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. 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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    Cited by:

    1. 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.

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