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

Consider a linear program of the form max{c(+) x : Ax <= b}, where A is an m x 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 parallel to x* - z* parallel to(infinity) < 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 n/2 center dot Delta whenever n >= 2 and x* is a vertex. We also show that, in certain circumstances, the factor n can be removed entirely.