On the rectangular knapsack problem

Fritz Bökler (), Markus Chimani () and Mirko H. Wagner ()
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Fritz Bökler: Osnabrück University
Markus Chimani: Osnabrück University
Mirko H. Wagner: Osnabrück University

Mathematical Methods of Operations Research, 2022, vol. 96, issue 1, No 6, 149-160

Abstract: Abstract A recent paper by Schulze et al. (Math Methods Oper Res 92(1):107–132, 2020) presented the Rectangular Knapsack Problem (Rkp) as a crucial subproblem in the study on the Cardinality-constrained Bi-objective Knapsack Problem (Cbkp). To this end, they started an investigation into its complexity and approximability. The key results are an NP -hardness proof for a more general scenario than Rkp, and a 4.5-approximation for Rkp, raising the question of improvements for either result. In this note we settle both questions conclusively: we show that (a) Rkp is indeed NP -hard in the considered setting (and even in more restricted settings), and (b) there exists both a pseudopolynomial algorithm and a fully-polynomial time approximation scheme (i.e., efficient approximability within any desired ratio $$\alpha >1$$ α > 1 ) for Rkp.

Keywords: Quadratic optimization; Knapsack problems; Multiobjective optimization; Approximation (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s00186-022-00788-8

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