We prove that bin packing is NP-hard to approximate within an additive factor of $o(\ls\OPT)$, even when all item sizes are arbitrarily close to $1/3$. This gives the first lower bound for bin packing with an additive factor that depends explicitly on the optimum value. In addition, we give a polynomial-time reduction from any bin packing instance into one with item sizes arbitrarily close to $1/3$, preserving its additive configuration-LP gap up to constant factors. Thus, in the sense of additive configuration LP gap and up to a constant, we confirm the conjecture that 3-partition instances are the hardest bin packing instances. In particular, we obtain an unconditional $Ω(\ls\OPT)$ gap for 3-partition instances. We build on the recent OpenAI construction that uses five items per bin and rules out any constant additive gap.

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