Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs

TL;DR

This paper demonstrates that classical bounds on the proximity and sensitivity of integer linear programs are tight by constructing instances that match these bounds, even for naturally arising combinatorial problems.

Contribution

The authors construct ILP instances with non-negative matrices that match the known upper bounds on proximity and sensitivity, confirming the bounds are tight.

Findings

Constructed ILP instances with tight proximity bounds

Constructed ILP instances with tight sensitivity bounds

Instances relate to natural combinatorial optimization problems

Abstract

We consider ILPs, where each variable corresponds to an integral point within a polytope $\mathcal{P}$, 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 proximity) is an important measure. A classical result by Cook et al.~(Math. Program., 1986) shows that it is at most $\Delta^{\Theta(d)}$ where $\Delta$ 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 sensitivity) is similarly bounded, i. e., $\lVert b-b' \rVert_{1}\cdot \Delta^{\Theta(d)}$, also shown by Cook et al. 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 $\Delta$, 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 $\Delta > 0$ and each $d > 0$ ILPs of the above type with non-negative constraint matrices such that their proximity and sensitivity is at least $\Delta^{\Theta(d)}$. 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.