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 using space O(2 O(d) •log d+1 d •log d+1 Δ•logn) bits, where , 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 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 barrier in the approximation ratio. © 2009 Springer Berlin Heidelberg.