Packing Groups of Items into Multiple Knapsacks

Authors Lin Chen, Guochuan Zhang



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Lin Chen
Guochuan Zhang

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Lin Chen and Guochuan Zhang. Packing Groups of Items into Multiple Knapsacks. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.STACS.2016.28

Abstract

We consider a natural generalization of the classical multiple knapsack problem in which instead of packing single items we are packing groups of items. In this problem, we have multiple knapsacks and a set of items which are partitioned into groups. Each item has an individual weight, while the profit is associated with groups rather than items. The profit of a group can be attained if and only if every item of this group is packed. Such a general model finds applications in various practical problems, e.g., delivering bundles of goods. The tractability of this problem relies heavily on how large a group could be. Deciding if a group of items of total weight 2 could be packed into two knapsacks of unit capacity is already NP-hard and it thus rules out a constant-approximation algorithm for this problem in general. We then focus on the parameterized version where the total weight of items in each group is bounded by a factor delta of the total capacity of all knapsacks. Both approximation and inapproximability results with respect to delta are derived. We also show that, depending on whether the number of knapsacks is a constant or part of the input, the approximation ratio for the problem, as a function on delta, changes substantially, which has a clear difference from the classical multiple knapsack problem.
Keywords
  • approximation algorithms
  • lower bound
  • multiple knapsack
  • bin packing

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References

  1. R. Adany, M. Feldman, E. Haramaty, R. Khandekar, B. Schieber, R. Schwartz, H. Shachnai, and T. Tamir. All-or-nothing generalized assignment with application to scheduling advertising campaigns. In Proc. of IPCO 2013, pages 13-24, 2013. Google Scholar
  2. C. Chekuri and S. Khanna. A polynomial time approximaion scheme for the multiple knapsack problem. SIAM J. Comput., 35(3):713-728, 2006. Google Scholar
  3. F. Eisenbrand, D. Pálvölgyi, and T. Rothvoss. Bin packing via discrepancy of permutations. ACM Trans. Algorithms, 9(3):39-49, 2013. Google Scholar
  4. P. C. Gilmore and R. E. Gomory. A linear programming approach to the cutting-stock problem. Operations Research, 9:39-49, 1961. Google Scholar
  5. R. L. Graham. Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math., 17(2):416-429, 1969. Google Scholar
  6. K. Jansen. Parameterized approximation scheme for the multiple knapsack problem. SIAM J. Comput., 39(4):1392-1412, 2009. Google Scholar
  7. K. Jansen. A fast approximation scheme for the multiple knapsack problem. In Proc. of SOFSEM'12, pages 313-324, 2012. Google Scholar
  8. K. Jansen, S. Kratsch, D. Marx, and l. Schlotter. Bin packing with fixed number of bins revisited. J. Comput. Syst. Sci., 79(1):39-49, 2013. Google Scholar
  9. K. Jansen, F. Land, and K. Land. Bounding the running time of algorithms for scheduling and packing problems. In Proc. of WADS'13, pages 313-324, 2013. Google Scholar
  10. D. S. Johnson. Near-optimal bin-packing algorithms. Doctoral Thesis. MIT Press, 1973. Google Scholar
  11. H. Kellerer. A polynomial time approximation scheme for the multiple knapsack problem. In Proc. of APPROX'99, pages 51-62, 1999. Google Scholar
  12. J. Matousek. Geometric discrepancy. Springer-Verlag, 1999. Google Scholar
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