Exact and approximate algorithms for discounted {0-1} knapsack problem

The Discounted {0-1} Knapsack Problem (D{0-1}KP) is an extension of the classical 0-1 knapsack problem (0-1 KP) that consists of selecting a set of item groups where each group includes three items and at most one of the three items can be selected. The D{0-1}KP is more challenging than the 0-1 KP because four choices of items in an item group diversify the selection of the items. In this paper, we systematically studied the exact and approximate algorithms for solving D{0-1}KP. Firstly, a new exact algorithm based on the dynamic programming and its corresponding fully polynomial time approximation scheme were designed. Secondly, a 2-approximation algorithm for D{0-1}KP was developed. Thirdly, a greedy repair algorithm for handling the infeasible solutions of D{0-1}KP was proposed and we further studied how to use binary particle swarm optimization and greedy repair algorithm to solve the D{0-1}KP. Finally, we used four different kinds of instances to compare the approximate rate and solving time of the exact and approximate algorithms. The experimental results and theoretical analysis showed that the approximate algorithms worked well for D{0-1}KP instances with large value, weight, and size coefficients, while the exact algorithm was good at solving D{0-1}KP instances with small value, weight, and size coefficients.