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). 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 (FPTAS) 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 and forcing graphs of bounded tree-width of Pferschy and Schauer.
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Notes
- 1.
The proofs of the results marked with a \(\bigstar \) are omitted due to space restrictions.
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Gurski, F., Rehs, C. (2019). The Knapsack Problem with Conflict Graphs and Forcing Graphs of Bounded Clique-Width. In: Fortz, B., Labbé, M. (eds) Operations Research Proceedings 2018. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-030-18500-8_33
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DOI: https://doi.org/10.1007/978-3-030-18500-8_33
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