Abstract
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 $\varphi\approx1.618$. 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 ${\mathsf{coNP}}$-complete. If $\alpha$ is part of the input, the same problem is shown to be ${\mathsf...
Highlights
In the standard knapsack problem we are given a set of items, each associated with a size and a value, and a capacity of the knapsack
We show that the oblivious knapsack problem always admits a robustness factor of 2
We show that, for given α, it is coNP-hard to decide whether an instance of the oblivious knapsack problem admits a universal policy with robustness factor α, even when all items have unit density
Summary
In the standard knapsack problem we are given a set of items, each associated with a size and a value, and a capacity of the knapsack. We show that the oblivious knapsack problem always admits a robustness factor of 2 This robustness factor can be achieved with a policy that packs the items according to a fixed order, irrespective of the observations made while packing. We give a different efficient algorithm for the case that all items have unit density, i.e., size and value of each item coincide This algorithm produces a universal policy with a robustness factor of at most the golden ratio φ ≈ 1.618. We show that, for given α, it is coNP-hard to decide whether an instance of the oblivious knapsack problem admits a universal policy with robustness factor α, even when all items have unit density
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