The Modified Integer Round-Up Conjecture for bin packing (Scheithauer-Terno)

Theorem 1.1: for every integer $c\ge0$ there are a finite rational instance $I$, every item larger than $1/6$ (so at most five items per bin), and an integer $B$ with $LP(I)=B$ and $OPT(I)>B+c$; the same LP value holds for the stronger formulation whose patterns are subsets of individual item copies. So the additive integrality gap of the configuration LP is unbounded and the Modified Integer Round-Up Conjecture is false. Not shown: a gap growing like $\log OPT$ in the instance size, matching the Hoberg-Rothvoss upper bound; the paper says it obtains no such lower bound.

Theorem 1.1: for every integer $c\ge0$ there are a finite rational instance $I$, every item larger than $1/6$ (so at most five items per bin), and an integer $B$ with $LP(I)=B$ and $OPT(I)>B+c$; the same LP value holds for the stronger formulation whose patterns are subsets of individual item copies. So the additive integrality gap of the configuration LP is unbounded and the Modified Integer Round-Up Conjecture is false. Not shown: a gap growing like $\log OPT$ in the instance size, matching the Hoberg-Rothvoss upper bound; the paper says it obtains no such lower bound.