Obtain a quantitative lower bound in the low-dimensional affine-hull case

Derive a uniform quantitative lower bound on the attainable uniform margin for general integer-linear-program forward problems when the affine hull of the difference-vector set has dimension less than d and contains the origin, with the bound expressed in terms of the relevant lattice or coordinate-range parameters.

Background

The paper derives explicit lower bounds on the uniform margin for general integer-linear programs with the unit-ball weight space when the convex hull of the action-difference vectors is full-dimensional, and it handles the lower-dimensional case when the affine hull does not contain the origin.

For the remaining case in which the affine hull has dimension k<d and contains the origin, the authors note that an argument on the induced lattice would require controlling norms of a dual basis. Their method does not immediately yield a uniform bound of the form (2||M||_2)-k, so the quantitative margin estimate remains unresolved.

References

In the case 0 \in \aff Z* with k < d, an isomorphic argument within the lattice induced on \aff Z* is required, but since the construction of an integral normal vector and the estimate of its norm depend on the norms of the (dual) basis of the induced lattice, a uniform constant of the type $(2| M |_2)k$ does not follow immediately from our method. We leave the quantitative lower bound in this case as unresolved (status: unknown).

— Online Inverse Integer Linear Optimization via Small-Gradient Skipping: Constant Regret and Finite Mistakes  (2609.09809 - Kitaoka, 9 Sep 2026) in Remark 2.14, Appendix 'Lower bounds on the margin by structure', subsection 'The largest attainable margin' / summary of the low-dimensional cases