Optimal Proximity Bound and Product Function Estimates in Integer Linear Programming

We obtain an optimal proximity bound for integer linear programs in standard form max{cx: Ax=b, x nonnegative integer}, where A is an integer mxn matrix of rank m<n and b is an integer vector. Specifically, we show that the Euclidean distance from any optimal vertex solution of the LP relaxation to a nearest optimal integer solution is bounded by $\sqrt{\det (A{A}^t)}-1$ and that this estimate is asymptotically tight. We also derive bounds for the optimal integer solutions involving the product function ${\prod }_{i=1}^n(x_i+1)$ and discuss their applications in the knapsack setting.