Approximation algorithms for fractional knapsack problems

We consider the combinatorial optimization problem where the objective function is the ratio of two linear functions. This type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function. We propose for the original problem an approximation scheme based on the one hand upon an approximation scheme for the auxiliary problem and on the other hand upon a constant approximation algorithm for the original problem. As an example of the method we propose an O(n²) time 1/2-approximation algorithm for the fractional 0–1 knapsack problem (BFKP) which combined with a known approximation scheme for the 0–1 linear knapsack problem running in O(n³/ϵ) leads to a fully polynomial-time approximation scheme (FPTAS) for BFKP with time complexity O(n³/ϵ). In the same way we propose an FPTAS for the unbounded fractional knapsack problem with time complexity O(n² + n log(1/ϵ) + (1/ϵ)³).