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Consider a linear program of the form $\max\;c^{\top}x:Ax\leq b$, where $A$is an $m\times n$ integral matrix. In 1986 Cook, Gerards, Schrijver, and Tardosproved that, given an optimal solution $x^{*}$, if an optimal integral solution$z^{*}$ exists, then it may be chosen such that $\left\Vertx^{*}-z^{*}\right\Vert _{\infty}<n\Delta$, where $\Delta$ is the largestmagnitude of any subdeterminant of $A$. Since then an open question has been toimprove this bound, assuming that $b$ is integral valued too. In thismanuscript we show that $n\Delta$ can be replaced with $\frac{n}{2}\cdot\Delta$whenever $n\geq2$. We also show that, in certain circumstances, the factor $n$can be removed entirely.

Improving the Cook et al. Proximity Bound Given Integral Valued Constraints