Online removable knapsack with limited cuts

Authors:
Xin Han;Kazuhisa Makino
Affiliations:
School of Software, Dalian University of Technology, China;Department of Mathematical Informatics, Graduate School of Information and Technology, University of Tokyo, Tokyo, 113-8656, Japan
Venue:
Theoretical Computer Science
Year:
2010

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Abstract

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(=1) times, and (3) items are allowed to be removed from the knapsack. We obtain the following three results. (i)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. (ii)For the minimization knapsack problem, we show that no online algorithm can have a constant competitive ratio. (iii)We extend the result in (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.