A Faster FPTAS for the Unbounded Knapsack Problem

The Unbounded Knapsack Problem (UKP) is a well-known variant of the famous 0-1 Knapsack Problem (0-1 KP). In contrast to 0-1 KP, an arbitrary number of copies of every item can be taken in UKP. Since UKP is NP-hard, fully polynomial time approximation schemes (FPTAS) are of great interest. Such algorithms find a solution arbitrarily close to the optimum OPT(I), i.e. of value at least (1− ε)OPT(I) for ε > 0, and have a running time polynomial in the input length and ε . For over thirty years, the best FPTAS was due to Lawler with a running time in O(n+ 1 ε3 ) and a space complexity in O(n+ 1 ε2 ), where n is the number of knapsack items. We present an improved FPTAS with a running time in O(n+ 1 ε2 log 3 1 ε ) and a space bound in O(n+ ε log 2 1 ε ). This directly improves the running time of the fastest known approximation schemes for Bin Packing and Strip Packing, which have to approximately solve UKP instances as subproblems.