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

Marcel Celaya, Stefan Kuhlmann, Joseph Paat, Robert Weismantel

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.