d-Dimensional Knapsack in the Streaming Model

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( 1 2 + √ 2d+ 1 4 ))−1 using space O(2 ⋅log d⋅log ⋅logn) bits, where { 1 , 2 , . . . , 1}, ≥ 2 is the set of possible profits and weights in any dimension, and P is the ratio between the minimum and maximum profit. We also show that any data streaming algorithm for the t(t− 1)-dimensional knapsack problem that uses space o( √ /t) cannot achieve an approximation ratio that is better than 1/t. Thus, even using space , for < 1/2, i.e. space polynomial in , will not help to break the 1/t ≈ 1/ √ d barrier in the approximation ratio.