Higher-dimensional thin-ray characterization of cubic integer unboundedness

Establish that thin rays characterize cubic integer unboundedness over rational polyhedra in every dimension, extending the known characterization through dimension three.

Background

For cubic objectives over unbounded rational polyhedra, integer unboundedness need not be witnessed by descent along a single integer recession ray. The cited three-dimensional result introduces thin rays and proves that they characterize cubic integer unboundedness through dimension three. Extending this structural characterization to arbitrary fixed dimensions would provide a foundation for deciding unboundedness beyond the bounded and localized cases treated in the paper.

References

Del Pia showed that in dimension three a cubic can be unbounded below on $P\capZ3$ without being unbounded along the integer points of any single recession ray. He introduced thin rays, proved that they characterize cubic integer unboundedness through dimension three, and conjectured the same in all dimensions.

— Bounded Cubic Integer Programming in Fixed Dimension  (2609.18266 - Ari et al., 16 Sep 2026) in Section 1, Introduction; see also Section 6.4, “Recession directions and their limitations”

For cubics, the results above provide localization when the real improving sublevel is bounded, cover all coercive instances, and decide the negative leading-form case. They do not settle arbitrary unbounded polyhedra. The outstanding tasks are a general polynomial-time test for integer unboundedness and a polynomial-bit optimizer bound for finite-optimum instances with unbounded real improving sublevels.

— Bounded Cubic Integer Programming in Fixed Dimension  (2609.18266 - Ari et al., 16 Sep 2026) in Section 6.5, “The remaining general problem”