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Solutions for the knapsack problem with conflict and forcing graphs of bounded clique-width

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  • Frank Gurski

    (University of Düsseldorf)

  • Carolin Rehs

    (University of Düsseldorf)

Abstract

The 0–1-knapsack problem is a well-known NP-hard problem in combinatorial optimization. We consider the extensions to the knapsack problem with conflict graph (KCG) and the knapsack problem with forcing graph (KFG). KCG has first been introduced by Yamada et al. and represents incompatibilities between items of the knapsack instance. KFG has been introduced by Pferschy and Schauer and represents the necessity of items for other items. Within this paper we provide pseudo-polynomial solutions for KCG and KFG with co-graphs as conflict and forcing graphs and extend these solutions to conflict and forcing graphs of bounded clique-width. Our solutions are based on dynamic programming using the tree-structure representing the conflict graph and the forcing graph. Further we conclude fully polynomial time approximation schemes for KCG on conflict graphs of bounded clique-width and KFG on forcing graphs of bounded clique-width. This generalizes the known results for conflict graphs and forcing graphs of bounded tree-width of Pferschy and Schauer.

Suggested Citation

  • Frank Gurski & Carolin Rehs, 2019. "Solutions for the knapsack problem with conflict and forcing graphs of bounded clique-width," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(3), pages 411-432, June.
    • Handle: RePEc:spr:mathme:v:89:y:2019:i:3:d:10.1007_s00186-019-00664-y
    DOI: 10.1007/s00186-019-00664-y
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    References listed on IDEAS

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    1. Frank Gurski & Carolin Rehs, 2019. "The Knapsack Problem with Conflict Graphs and Forcing Graphs of Bounded Clique-Width," Operations Research Proceedings, in: Bernard Fortz & Martine Labbé (ed.), Operations Research Proceedings 2018, pages 259-265, Springer.
    2. 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.
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    Cited by:

    1. Luiz Viana & Manoel Campêlo & Ignasi Sau & Ana Silva, 2021. "A unifying model for locally constrained spanning tree problems," Journal of Combinatorial Optimization, Springer, vol. 42(1), pages 125-150, July.
    2. Wei, Zequn & Hao, Jin-Kao & Ren, Jintong & Glover, Fred, 2023. "Responsive strategic oscillation for solving the disjunctively constrained knapsack problem," European Journal of Operational Research, Elsevier, vol. 309(3), pages 993-1009.
    3. Frank Gurski & Carolin Rehs, 2020. "Counting and enumerating independent sets with applications to combinatorial optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(3), pages 439-463, June.

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