The Fully Polynomial Approximation Algorithm for the 0-1 Knapsack Problem

A modified fast approximation algorithm for the 0-1 knapsack problem with improved complexity is presented, based on the schemes of Ibarra, Kim and Babat. By using a new partition of items, the algorithm solves the n -item 0-1 knapsack problem to any relative error tolerance ε > 0 in the scaled profit space P*/K = O ( 1/ ε 1+δ ) with time O(n log(1/ε )+1/ ε^{2+2δ}) and space O(n +1/ ɛ^{2+δ}), where P^{*} and b are the maximal profit and the weight capacity of the knapsack problem, respectively, K is a problem-dependent scaling factor, δ={α}/(1+α) and α=O( log b ). This algorithm reduces the exponent of the complexity bound in [5].