Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs

Abstract

We consider Integer Linear Programs (ILPs), where each variable corresponds to an integral point within a polytope \(\mathcal {P}\subseteq \mathbb {R}^{d}\), i. e., ILPs of the form \(\min \{c^{\top }x\mid \sum _{p\in \mathcal {P}\cap \mathbb {Z}^d} x_p p = b, x\in \mathbb {Z}^{|\mathcal {P}\cap \mathbb {Z}^d|}_{\ge 0}\}\). The distance between an optimal fractional solution and an optimal integral solution (called the proximity) is an important measure. A classical result by Cook et al. (Math. Program., 1986) shows that it is at most \(\varDelta ^{\varTheta (d)}\) where \(\varDelta =\Vert \mathcal {P}\cap \mathbb {Z}^{d} \Vert _{\infty }\) is the largest coefficient in the constraint matrix. Another important measure studies the change in an optimal solution if the right-hand side b is replaced by another right-hand side \(b'\). The distance between an optimal solution x w.r.t. b and an optimal solution \(x'\) w.r.t. \(b'\) (called the sensitivity) is similarly bounded, i. e., \(\Vert b-b' \Vert _{1}\cdot \varDelta ^{\varTheta (d)}\), also shown by Cook et al. (Math. Program., 1986).