Additive Rounding for LP-Tight Three-Item Bin Packing
We consider bin-packing instances with \(3m\) items of sizes in \((1/4,1/2)\), total size \(m\), and configuration linear programming value \(m\). Let \(d\) be the relative dimension of the polytope of zero-sum multiplicity patterns obtained by centering sizes at \(1/3\). We determine the maximum additive integrality gap for every \(d\le10\). It is zero for \(d\le4\) and one for \(5\le d\le10\). More generally, a representation of the multiplicity vector by at most five zero-sum patterns admits an integral packing, and a representation by at most eleven patterns requires at most one…
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