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Approximation of knapsack problems with conflict and forcing graphs

Author

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  • Ulrich Pferschy

    (University of Graz)

  • Joachim Schauer

    (University of Graz)

Abstract

We study the classical 0–1 knapsack problem with additional restrictions on pairs of items. A conflict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be included in the knapsack. A natural way for representing these constraints is the use of conflict (resp. forcing) graphs. By modifying a recent result of Lokstanov et al. (Proceedings of the 25th annual ACM-SIAM symposium on discrete algorithms, SODA, pp 570–581, 2014) we derive a fairly complicated FPTAS for the knapsack problem on weakly chordal conflict graphs. Next, we show that the techniques of modular decompositions and clique separators, widely used in the literature for solving the independent set problem on special graph classes, can be applied to the knapsack problem with conflict graphs. In particular, we can show that every positive approximation result for the atoms of prime graphs arising from such a decomposition carries over to the original graph. We point out a number of structural results from the literature which can be used to show the existence of an FPTAS for several graph classes characterized by the exclusion of certain induced subgraphs. Finally, a PTAS for the knapsack problem with H-minor free conflict graph is derived. This includes planar graphs and, more general, graphs of bounded genus. The PTAS is obtained by expanding a general result of Demaine et al. (Proceedings of 46th annual IEEE symposium on foundations of computer science, FOCS 2005, pp 637–646, 2005). The knapsack problem with forcing graphs can be transformed into a minimization knapsack problem with conflict graphs. It follows immediately that all our FPTAS results of the current and a previous paper carry over from conflict graphs to forcing graphs. In contrast, the forcing graph variant is already inapproximable on planar graphs.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:33:y:2017:i:4:d:10.1007_s10878-016-0035-7
    DOI: 10.1007/s10878-016-0035-7
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    References listed on IDEAS

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    1. Ulrich Pferschy & Joachim Schauer, 2013. "The maximum flow problem with disjunctive constraints," Journal of Combinatorial Optimization, Springer, vol. 26(1), pages 109-119, July.
    2. Ruslan Sadykov & François Vanderbeck, 2013. "Bin Packing with Conflicts: A Generic Branch-and-Price Algorithm," INFORMS Journal on Computing, INFORMS, vol. 25(2), pages 244-255, May.
    3. Albert E. Fernandes Muritiba & Manuel Iori & Enrico Malaguti & Paolo Toth, 2010. "Algorithms for the Bin Packing Problem with Conflicts," INFORMS Journal on Computing, INFORMS, vol. 22(3), pages 401-415, August.
    4. Mhand Hifi & Nabil Otmani, 2012. "An algorithm for the disjunctively constrained knapsack problem," International Journal of Operational Research, Inderscience Enterprises Ltd, vol. 13(1), pages 22-43.
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    Cited by:

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    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, 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.
    4. 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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