On the rectangular knapsack problem

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 α>1) for Rkp .