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, IBPN for 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 Lp-norm, ∀p∈{1,2,…,∞}, IBPN turns out to be NP-hard in the strong sense. IBPN under 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.