Packing a Knapsack of Unknown Capacity

TL;DR

This paper investigates packing a knapsack without knowing its capacity, presenting universal policies that guarantee near-optimal packing ratios and analyzing their computational complexity.

Contribution

It introduces universal packing policies with provable approximation factors and establishes their optimality and computational hardness.

Findings

Universal policies achieve a factor 2 approximation for general items.

For unit density items, the approximation factor equals the golden ratio.

Deciding the effectiveness of a given universal policy is coNP-complete.

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. 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 and try to pack the items in this order, independent of the observations made while packing. 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.