Notes on inverse bin-packing problems

In bin-packing problems, given items need to be packed using a minimum number of bins. Inverse bin-packing number problems,IBPNfor short, assume a given set of items and number of bins. The objective is to achieve the minimum perturbation to the item-size vector so that all the items can be packed into the prescribed number of bins. In this paper, complexity status and approximation behavior for IBPN were investigated. Under the L_p-norm, p ∈{1, 2, ⋯,∞},IBPN turns out to be NP-hard in the strong sense.IBPNunder the L1-norm admits a polynomial time differential approximation scheme, and a fully polynomial time approximation scheme if a constant number of machines is provided as input. We also consider another IBPN variant where a specified feasible solution is given instead of a target bin number. The objective is to make the given solution optimal with minimum modification. We provide the hardness result for this problem.