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An Approximation Scheme for Bin Packing with Conflicts

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  • Klaus Jansen

    (Christian-Albrechts-University of Kiel)

Abstract

In this paper we consider the following bin packing problem with conflicts. Given a set of items V = {1,..., n} with sizes s1,..., s ∈ (0,1) and a conflict graph G = (V, E), we consider the problem to find a packing for the items into bins of size one such that adjacent items (j, j′) ∈ E are assigned to different bins. The goal is to find an assignment with a minimum number of bins. This problem is a natural generalization of the classical bin packing problem. We propose an asymptotic approximation scheme for the bin packing problem with conflicts restricted to d-inductive graphs with constant d. This graph class contains trees, grid graphs, planar graphs and graphs with constant treewidth. The algorithm finds an assignment for the items such that the generated number of bins is within a factor of (1 + ∈) of optimal provided that the optimum number is sufficiently large. The running time of the algorithm is polynomial both in n and in $$\tfrac{1}{\varepsilon}$$ .

Suggested Citation

  • Klaus Jansen, 1999. "An Approximation Scheme for Bin Packing with Conflicts," Journal of Combinatorial Optimization, Springer, vol. 3(4), pages 363-377, December.
    • Handle: RePEc:spr:jcomop:v:3:y:1999:i:4:d:10.1023_a:1009871302966
    DOI: 10.1023/A:1009871302966
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    1. de Werra, D., 1985. "An introduction to timetabling," European Journal of Operational Research, Elsevier, vol. 19(2), pages 151-162, February.
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    1. Fleszar, Krzysztof, 2022. "A MILP model and two heuristics for the Bin Packing Problem with Conflicts and Item Fragmentation," European Journal of Operational Research, Elsevier, vol. 303(1), pages 37-53.
    2. Gyorgy Dosa & Zsolt Tuza & Deshi Ye, 2013. "Bin packing with “Largest In Bottom” constraint: tighter bounds and generalizations," Journal of Combinatorial Optimization, Springer, vol. 26(3), pages 416-436, October.
    3. Walaa H. El-Ashmawi & Ahmad Salah & Mahmoud Bekhit & Guoqing Xiao & Khalil Al Ruqeishi & Ahmed Fathalla, 2023. "An Adaptive Jellyfish Search Algorithm for Packing Items with Conflict," Mathematics, MDPI, vol. 11(14), pages 1-28, July.
    4. Ananya Christman & Christine Chung & Nicholas Jaczko & Scott Westvold & David S. Yuen, 2022. "Robustly assigning unstable items," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1556-1577, October.
    5. 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.
    6. 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.
    7. Ulrich Pferschy & Joachim Schauer, 2013. "The maximum flow problem with disjunctive constraints," Journal of Combinatorial Optimization, Springer, vol. 26(1), pages 109-119, July.
    8. Samir Elhedhli & Lingzi Li & Mariem Gzara & Joe Naoum-Sawaya, 2011. "A Branch-and-Price Algorithm for the Bin Packing Problem with Conflicts," INFORMS Journal on Computing, INFORMS, vol. 23(3), pages 404-415, August.
    9. Renatha Capua & Yuri Frota & Luiz Satoru Ochi & Thibaut Vidal, 2018. "A study on exponential-size neighborhoods for the bin packing problem with conflicts," Journal of Heuristics, Springer, vol. 24(4), pages 667-695, August.
    10. Borja Ena & Alberto Gomez & Borja Ponte & Paolo Priore & Diego Diaz, 2022. "Homogeneous grouping of non-prime steel products for online auctions: a case study," Annals of Operations Research, Springer, vol. 315(1), pages 591-621, August.
    11. Ananya Christman & Christine Chung & Nicholas Jaczko & Scott Westvold & David S. Yuen, 0. "Robustly assigning unstable items," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-22.
    12. 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.
    13. Lijun Wei & Zhixing Luo, & Roberto Baldacci & Andrew Lim, 2020. "A New Branch-and-Price-and-Cut Algorithm for One-Dimensional Bin-Packing Problems," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 428-443, April.
    14. Ekici, Ali, 2023. "A large neighborhood search algorithm and lower bounds for the variable-Sized bin packing problem with conflicts," European Journal of Operational Research, Elsevier, vol. 308(3), pages 1007-1020.
    15. Gyorgy Dosa, 2017. "Batched bin packing revisited," Journal of Scheduling, Springer, vol. 20(2), pages 199-209, April.
    16. Jasmin Grabenschweiger & Karl F. Doerner & Richard F. Hartl & Martin W. P. Savelsbergh, 2021. "The vehicle routing problem with heterogeneous locker boxes," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 29(1), pages 113-142, March.
    17. Sergio García & Valentina Cacchiani & Lieselot Vanhaverbeke & Martin Bischoff, 2014. "The table placement problem: a research challenge at the EWI 2007," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 208-226, April.

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