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arXiv · 2607.15369

On the Computational Complexity of Bilevel Integer Linear Programming

Nagisa Sugishita·Margarida Carvalho
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Abstract

We investigate the computational complexity of bilevel integer linear programming. While Jeroslow~(1985) established that the decision version of this problem is $\Sigma^p_2$-complete when restricted to binary variables, we prove that this $\Sigma^p_2$-completeness persists even for general integer variables, settling a question that remained open for over 40 years. Furthermore, we analyze the impact of various structural assumptions on computational complexity. Notably, we strengthen the result of K\"oppe et al.~(2010) by proving polynomial-time solvability whenever the total number of upper- and lower-level variables is fixed, without any additional assumptions.

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Nagisa Sugishita, Margarida Carvalho. 2026-07-16. On the Computational Complexity of Bilevel Integer Linear Programming. https://arxiv.org/abs/2607.15369

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