Online Knapsack Problems with Limited Cuts

  • Authors:

  • Xin Han;Kazuhisa Makino

  • Affiliations:

  • Department of Mathematical Informatics, Graduate School of Information and Technology, University of Tokyo, Tokyo, Japan 113-8656;Department of Mathematical Informatics, Graduate School of Information and Technology, University of Tokyo, Tokyo, Japan 113-8656

  • Venue:
  • ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
  • Year:
  • 2009

  • 2D knapsack: packing squares

    FAW-AAIM'11 Proceedings of the 5th joint international frontiers in algorithmics, and 7th international conference on Algorithmic aspects in information and management

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Abstract

The (offline) maximization (resp., minimization) knapsack problem is given a set of items with weights and sizes, and the capacity of a knapsack, to maximize (resp., minimize) the total weight of selected items under the constraint that the total size of the selected items is at most (resp., at least) the capacity of the knapsack. In this paper, we study online maximization and minimization knapsack problems with limited cuts, in which 1) items are given one by one over time, i.e., after a decision is made on the current item, the next one is given, 2) items are allowed to be cut at most k ( 驴 1) times, and 3) items are allowed to be removed from the knapsack.We obtain the following three results. For the maximization knapsack problem, we propose a (k + 1)/k-competitive online algorithm, and show that it is the best possible, i.e., no online algorithm can have a competitive ratio less than (k + 1)/k. For the minimization knapsack problem, we show that no online algorithm can have a constant competitive ratio. We extend the result (i) to the resource augmentation model, where an online algorithm is allowed to use a knapsack of capacity m ( 1), while the optimal algorithm uses a unit capacity knapsack.