{"ID":2850456,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.21126","arxiv_id":"2510.21126","title":"Complexity of Bilevel Linear Programming with a Single Upper-Level Variable","abstract":"Bilevel linear programming (LP) is one of the simplest classes of bilevel optimization problems, yet it is known to be NP-hard in general. Specifically, determining whether the optimal objective value of a bilevel LP is at least as good as a given threshold, a standard decision version of the problem, is NP-complete. However, this decision problem becomes tractable when either the number of lower-level variables or the number of lower-level constraints is fixed, which prompts the question: What if restrictions are placed on the upper-level problem? In this paper, we address this gap by showing that the decision version of bilevel LP remains NP-complete even when there is only a single upper-level variable, no upper-level constraints (apart from the constraint enforcing optimality of the lower-level decision) and all variables are bounded between 0 and 1. This result implies that fixing the number of variables or constraints in the upper-level problem alone does not lead to tractability in general. On the positive side, we show that there is a polynomial-time algorithm that finds a local optimal solution of such a rational bilevel LP instance. We also demonstrate that many combinatorial optimization problems, such as the knapsack problem and the traveling salesman problem, can be written as such a bilevel LP instance.","short_abstract":"Bilevel linear programming (LP) is one of the simplest classes of bilevel optimization problems, yet it is known to be NP-hard in general. Specifically, determining whether the optimal objective value of a bilevel LP is at least as good as a given threshold, a standard decision version of the problem, is NP-complete. H...","url_abs":"https://arxiv.org/abs/2510.21126","url_pdf":"https://arxiv.org/pdf/2510.21126v2","authors":"[\"Nagisa Sugishita\",\"Margarida Carvalho\"]","published":"2025-10-24T03:29:54Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}