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Counting and enumerating independent sets with applications to combinatorial optimization problems

Author

Listed:
  • Frank Gurski

    (University of Düsseldorf)

  • Carolin Rehs

    (University of Düsseldorf)

Abstract

Counting and enumerating maximal and maximum independent sets are well-studied problems in graph theory. In this paper we introduce methods to count and enumerate maximal/maximum independent sets in threshold graphs and k-threshold graphs and improve former results for these problems. The results can be applied to combinatorial optimization problems, and in particular to different variations of the knapsack problem. As feasible solutions for instances of those problems correspond to independent sets in threshold graphs and k-threshold graphs, we obtain polynomial time results for special knapsack and multidimensional knapsack instances. Also, we show lower and upper bounds for the number of necessary bins in several bin packing problems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:91:y:2020:i:3:d:10.1007_s00186-019-00696-4
    DOI: 10.1007/s00186-019-00696-4
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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. Carolin Rehs & Frank Gurski, 2018. "A Graph Theoretic Approach to Solve Special Knapsack Problems in Polynomial Time," Operations Research Proceedings, in: Natalia Kliewer & Jan Fabian Ehmke & Ralf Borndörfer (ed.), Operations Research Proceedings 2017, pages 295-301, Springer.
    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. 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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