Bin packing within an additive constant: is there a polynomial-time algorithm using at most OPT + C bins?

Theorem 1.2: for every fixed integer $c\ge0$ it is NP-hard to distinguish pairs $(I,B)$ where $I$ packs into $B$ bins from pairs where $I$ needs more than $B+c$ bins, with rational sizes all greater than $1/6$. Corollary 1.3: a deterministic polynomial-time algorithm using at most $OPT(I)+C$ bins for an absolute constant $C$ exists if and only if P = NP. The reduction goes from vertex cover with Håstad's gap through a competing two-tree interval construction encoded in five-item bins. Not shown: a lower bound of order $\log OPT$ matching Hoberg-Rothvoss, or hardness for a fixed number of item sizes (polynomial by Goemans-Rothvoss).

Theorem 1.2: for every fixed integer $c\ge0$ it is NP-hard to distinguish pairs $(I,B)$ where $I$ packs into $B$ bins from pairs where $I$ needs more than $B+c$ bins, with rational sizes all greater than $1/6$. Corollary 1.3: a deterministic polynomial-time algorithm using at most $OPT(I)+C$ bins for an absolute constant $C$ exists if and only if P = NP. The reduction goes from vertex cover with Håstad's gap through a competing two-tree interval construction encoded in five-item bins. Not shown: a lower bound of order $\log OPT$ matching Hoberg-Rothvoss, or hardness for a fixed number of item sizes (polynomial by Goemans-Rothvoss).