A 3-space dynamic programming heuristic for the cubic knapsack problem

The cubic knapsack problem (CKP) is a combinatorial optimization problem, which can be seen both as a generalization of the quadratic knapsack problem (QKP) and of the linear Knapsack problem (KP). This problem consists of maximizing a cubic function of binary decision variables subject to one linear knapsack constraint. It has many applications in biology, project selection, capital budgeting problem, and in logistics. The QKP is known to be strongly NP-hard, which implies that the CKP is also NP-hard in the strong sense. Unlike its linear and quadratic counterparts, the CKP has not received much of attention in the literature. Thus the few exact solution methods known for this problem can only handle problems with up to 60 decision variables. In this paper, we propose a deterministic dynamic programming-based heuristic algorithm for finding a good quality solution for the CKP. The novelty of this algorithm is that it operates in three different space variables and can produce up to three different solutions with different levels of computational effort. The algorithm has been tested on a set of 1570 test instances, which include both standard and challenging instances. The computational results show that our algorithm can find optimal solutions for nearly 98% of the standard test instances that could be solved to optimality and for 92% for the challenging instances. Finally, the computational experiments present comparisons between our algorithm, an existing heuristic algorithm for the CKP found in the literature, as well as adaptations to the CKP of some heuristic algorithms designed for the QKP. The results show that our algorithm outperforms all these methods.