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

TL;DR

This paper improves the known proximity bound for integral solutions in linear programming with integral constraints, reducing the bound from $n riangle$ to $rac{n}{2} riangle$ and exploring conditions for further improvements.

Contribution

It refines the proximity bound for integral solutions in linear programs with integral matrices and constraints, providing tighter bounds than previously established.

Findings

Bound improved from $n riangle$ to $rac{n}{2} riangle$ for $n \\geq 2$

Conditions identified where the factor $n$ can be eliminated

Enhances understanding of proximity bounds in integer linear programming

Abstract

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 Tardos proved that, given an optimal solution $x^{*}$, if an optimal integral solution $z^{*}$ exists, then it may be chosen such that $\left\Vert x^{*}-z^{*}\right\Vert _{\infty}<n\Delta$, where $\Delta$ is the largest magnitude of any subdeterminant of $A$. Since then an open question has been to improve this bound, assuming that $b$ is integral valued too. In this manuscript 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.