Sharp Proximity Bounds for Integer Programs with Few Rows

We study the proximity between an optimal vertex of a standard form linear program with constraint matrix $A\in\mathbb{Z}^{m\times n}$ and an optimal integer solution. When $m = 2$, we prove the sharp infinity-norm bound $Δ-1$, where $Δ$ is the largest absolute maximal minor of $A$. When $m = 1$, we express the difference between an optimal LP vertex and an optimal integer solution as a nonnegative combination of certain generators in the kernel of $A$ with a bounded coefficient sum. This implies sharp proximity bounds for all norms when $m=1$. Combining this representation with its known counterpart for Hilbert bases due to Sissokho, we develop techniques that transfer results from $m=1$ to $m=2$. Along the way, for $m=2$, we prove that such a representation exists for an extended Gomory group relaxation and obtain the sharp infinity-norm bound $Δ$ for Graver basis elements. We also present a new short proof of Sissokho's result and lower bounds on the proximity for all $m$ and $Δ$.