# Decision Problems in Multilevel Linear Programming

arXiv: https://arxiv.org/abs/2605.04929 · HTML: https://mlsanity.com/paper/2605.04929/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 May 2026.

## Built on by (most cited first, 1 of 1)

- [First-order Constrained Trilevel Optimization Over Distributed Networks for Robust Coreset Selection](https://mlsanity.com/paper/2607.27632/) (2026)

## Abstract (arXiv)

We study the computational complexity of decision problems in k-level linear programming (LP). Seminal work by Jeroslow establishes that determining whether the optimal objective value of a k-level LP is at least as good as a given threshold is Sigmapk-1-hard. In this paper, we demonstrate the matching upper bound and thereby prove that this problem is Sigmapk-1-complete. To this end, we show that the feasible region of a k-level LP can be expressed as a union of sets defined by weak and strict linear inequalities. Moreover, we show that the decision of the unboundedness is Sigmapk-1-complete. Finally, we discuss the extension of our results to the mixed-binary cases. In short, this work closes lasting open questions in multilevel programming.