d-Dimensional Knapsack in the Streaming Model

Abstract

We study the d-dimensional knapsack problem in the data streaming model. The knapsack is modelled as a d-dimensional integer vector of capacities. For simplicity, we assume that the input is scaled such that all capacities are 1. There is an input stream of n items, each item is modelled as a d-dimensional integer column of non-negative integer weights and a scalar profit. The input instance has to be processed in an online fashion using sub-linear space. After the items have arrived, an approximation for the cost of an optimal solution as well as a template for an approximate solution is output.

Our algorithm achieves an approximation ratio $(2(\frac{1}{2}+\sqrt{2 d+\frac{1}{4}}))^{-1}$ using space O(2^O(d) ·log^d + 1 d ·log^d + 1 Δ·logn) bits, where $\{\frac{1}{\Delta}, \frac{2}{\Delta}, \dots, 1\}$, Δ ≥ 2 is the set of possible profits and weights in any dimension. We also show that any data streaming algorithm for the t(t − 1)-dimensional knapsack problem that uses space $o(\sqrt{\Delta}/t^2)$ cannot achieve an approximation ratio that is better than 1/t. Thus, even using space Δ^γ, for γ< 1/2, i.e. space polynomial in Δ, will not help to break the $1/t \approx 1/\sqrt{d}$ barrier in the approximation ratio.