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 $\sqrt{\det(AA^t)}-1$ 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.

Source: arXiv:2606.13579v1 - http://arxiv.org/abs/2606.13579v1 PDF: https://arxiv.org/pdf/2606.13579v1 Original Link: http://arxiv.org/abs/2606.13579v1