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Subset sum problems with digraph constraints

Author

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  • Laurent Gourvès

    (Université Paris-Dauphine, PSL Research University)

  • Jérôme Monnot

    (Université Paris-Dauphine, PSL Research University)

  • Lydia Tlilane

    (Université Paris-Dauphine, PSL Research University)

Abstract

We introduce and study optimization problems which are related to the well-known Subset Sum problem. In each new problem, a node-weighted digraph is given and one has to select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints called digraph constraints and maximality need to be satisfied. The digraph constraint imposes that a node must belong to the solution if at least one of its predecessors is in the solution. An alternative of this constraint says that a node must belong to the solution if all its predecessors are in the solution. The maximality constraint ensures that no superset of a feasible solution is also feasible. The combination of these constraints provides four problems. We study their complexity and present some approximation results according to the type of input digraph, such as directed acyclic graphs and oriented trees.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:36:y:2018:i:3:d:10.1007_s10878-018-0262-1
    DOI: 10.1007/s10878-018-0262-1
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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. Geon Cho & Dong X. Shaw, 1997. "A Depth-First Dynamic Programming Algorithm for the Tree Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 9(4), pages 431-438, November.
    3. Sebastian Bervoets & Vincent Merlin & Gerhard J. Woeginger, 2015. "Vote trading and subset sums," Post-Print halshs-01102568, HAL.
    4. Ling Gai & Guochuan Zhang, 2008. "On lazy bureaucrat scheduling with common deadlines," Journal of Combinatorial Optimization, Springer, vol. 15(2), pages 191-199, February.
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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.
    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.

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