Proximity Bounds for Random Integer Programs

Abstract

We study proximity bounds within a natural model of random integer programs of the type $\max \varvec{c}^{\top }\varvec{x}:\varvec{A}\varvec{x}=\varvec{b},\,\varvec{x}\in \mathbb {Z}_{\ge 0}$, where $\varvec{A}\in \mathbb {Z}^{m\times n}$ is of rank m, $\varvec{b}\in \mathbb {Z}^{m}$ and $\varvec{c}\in \mathbb {Z}^{n}$. In particular, we seek bounds for proximity in terms of the parameter $\varDelta (\varvec{A})$, which is the square root of the determinant of the Gram matrix $\varvec{A}\varvec{A}^{\top }$ of $\varvec{A}$. We prove that, up to constants depending on n and m, the proximity is “generally” bounded by $\varDelta (\varvec{A})^{1/(n-m)}$, which is significantly better than the best deterministic bounds which are, again up to dimension constants, linear in $\varDelta (\varvec{A})$.

The first author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—The Berlin Mathematics Research Center MATH+(EXC-2046/1, project ID: 390685689).