The knapsack problem with a minimum filling constraint

We study a knapsack problem with an additional minimum filling constraint, such that the total weight of selected items cannot be less than a given threshold. The problem has several applications in shipping, e-commerce, and transportation service procurement. When the threshold equals the knapsack capacity, even finding a feasible solution to the problem is NP-hard. Therefore, we consider the case when the ratio α of threshold to capacity is less than 1. For this case, we develop an approximation scheme that returns a feasible solution with a total profit not less than (1 - ε) times the total profit of an optimal solution for any ε > 0, and with a running time polynomial in the number of items, 1/ε, and 1/(1-α).