The Near Exact Bin Covering Problem

We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant Δ, and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and 1+Δ. Our goal is to maximize the number of near exact covered bins. If Δ=0 or Δ>0 is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that P≠NP). However, for the case where Δ>0 is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.