Packing a Knapsack of Unknown Capacity.
SIAM JOURNAL ON DISCRETE MATHEMATICS(2017)
摘要
We study the problem of packing a knapsack without knowing its capacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We show that there is always a policy that packs a value within factor 2 of the optimum packing, irrespective of the actual capacity. If all items have unit density, we achieve a factor equal to the golden ratio phi approximate to 618. Both factors are shown to be best possible. In fact, we obtain the above factors using packing policies that are universal in the sense that they fix a particular order of the items in the beginning and try to pack the items in this order, without changing the order later on. We give efficient algorithms computing these policies. On the other hand, we show that, for any alpha > 1, the problem of deciding whether a given universal policy achieves a factor of alpha is coNP-complete. If alpha is part of the input, the same problem is shown to be coNP-complete for items with unit densities. Finally, we show that it is coNP-hard to decide, for given alpha, whether a set of items admits a universal policy with factor alpha, even if all items have unit densities.
更多查看译文
关键词
knapsack,unknown capacity,robustness,approximation guarantees
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络