# Additive approximation to bin packing

The bin packing problem is an NP-hard optimization problem that has many constant-factor approximation algorithms. I am looking for an additive approximation. I.e., given a set $$I$$ of items and bin capacity $$B$$, find a packing in which the number of bins is at most $$OPT(I,B)+d$$, where $$OPT(I,B)$$ is the optimal number of bins and $$d$$ is some small constant (ideally $$d=1$$).

Here I found an algorithm that finds a packing in at most $$OPT(I,B)+O(log{OPT(I,B)})$$ bins, but not $$OPT(I,B)+$$constant. This leads me to suspect that it may be impossible unless P=NP. But I did not find a proof of impossibility either.

Is such an additive approximation possible in polynomial time?