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).

Even after more than thirty years, these bounds are essentially the best known bounds for these measures. While some lower bounds are known for these measures, they either only work for very small values of $\varDelta$, require negative entries in the constraint matrix, or have fractional right-hand sides. Hence, these lower bounds often do not correspond to instances from algorithmic problems. This work presents for each $\varDelta > 0$ and each $d > 0$ ILPs of the above type with non-negative constraint matrices such that their proximity and sensitivity is at least $\varDelta ^{\varTheta (d)}$. Furthermore, these instances are closely related to instances of the Bin Packing problem as they form a subset of columns of the configuration ILP. We thereby show that the results of Cook et al. are indeed tight, even for instances arising naturally from problems in combinatorial optimization.

This work was supported by DFG project JA 612/20-1.