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 1ε. For over thirty years, the best FPTAS was due to Lawler with running time O(n+1ε3) and space complexity O(n+1ε2), where n is the number of knapsack items. We present an improved FPTAS with running time O(n+1ε2log31ε) and space bound O(n+1εlog21ε). 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.