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

Abstract

Consider a linear program of the form $\max \{\boldsymbol{c}^{\top }\boldsymbol{x}:\boldsymbol{A}\boldsymbol{x}\le \boldsymbol{b}\}$, where $\boldsymbol{A}$ is an $m\times n$ integral matrix. In 1986 Cook, Gerards, Schrijver, and Tardos proved that, given an optimal solution $\boldsymbol{x}^{*}$, if an optimal integral solution $\boldsymbol{z}^{*}$ exists, then it may be chosen such that $\left\| \boldsymbol{x}^{*}-\boldsymbol{z}^{*}\right\| _{\infty }<n\varDelta$, where $\varDelta$ is the largest magnitude of any subdeterminant of $\boldsymbol{A}$. Since then an open question has been to improve this bound, assuming that $\boldsymbol{b}$ is integral valued too. In this manuscript we show that $n\varDelta$ can be replaced with whenever $n\ge 2$ and $\boldsymbol{x}^{*}$ is a vertex. We also show that, in certain circumstances, the factor n can be removed entirely.