On the rectangular knapsack problem: approximation of a specific quadratic knapsack problem

• Published in: Mathematical methods of operations research (Heidelberg, Germany) Vol. 92; no. 1; pp. 107 - 132
• Main Authors: Schulze, Britta, Stiglmayr, Michael, Paquete, Luís, Fonseca, Carlos M, Willems, David, Ruzika, Stefan
• Format: Journal Article
• Language: English
• Published: Berlin, Heidelberg Springer 01.08.2020; Springer Berlin Heidelberg; Springer Nature B.V
• Subjects: Algorithms; Approximation; Approximation algorithm; Business and Management; Calculus of Variations and Optimal Control; Optimization; Hypervolume; Knapsack problem; Mathematical analysis; Mathematics; Mathematics and Statistics; Multiobjective combinatorial optimization; Operations research; Operations Research/Decision Theory; Original Article; Polynomials; Quadratic knapsack problem; Multiobjective combinatorial optimization; Approximation algorithm; Quadratic knapsack problem; Hypervolume
• ISSN: 1432-5217; 1432-2994; 1432-5217
• DOI: 10.1007/s00186-020-00702-0

In this article, we introduce the rectangular knapsack problem as a special case of the quadratic knapsack problem consisting in the maximization of the product of two separate knapsack profits subject to a cardinality constraint. We propose a polynomial time algorithm for this problem that provides a constant approximation ratio of 4.5. Our experimental results on a large number of artificially generated problem instances show that the average ratio is far from theoretical guarantee. In addition, we suggest refined versions of this approximation algorithm with the same time complexity and approximation ratio that lead to even better experimental results.