A lifted-space dynamic programming algorithm for the Quadratic Knapsack Problem - Djeumou Fomeni, Franklin (Discrete Applied Mathematics, 2023)
The Quadratic Knapsack Problem (QKP) is a well-known combinatorial optimization problem which amounts to maximizing a quadratic function of binary variables, subject to a single linear constraint. It has many applications in finance, logistics, telecommunications, facility location, etc. The QKP is NP-hard in the strong sense and the state-of-the-art algorithm for solving the QKP can only handle problems of small and moderate sizes. In this paper, we present a novel deterministic heuristic algorithm for finding good QKP feasible solutions. This algorithm consists of combining the dynamic programming approach with a local search procedure, with the novelty that both are adapted and implemented in the space of lifted variables of the QKP. The algorithm runs in O(n3c) time and is able to find optimal solutions to more than 97% of standard instances, about 80% of some well-known hard QKP instances, as well as optimality gaps of 0.1% or less for other instances.