Complexity of Bilevel Linear Programming with a Single Upper-Level Variable

TL;DR

This work proves that even a highly restricted bilevel linear program with a single upper-level variable and unit-bound variables remains NP-complete in its decision form, and it rules out the existence of a PTAS unless $\text{NP}=\text{P}$; verifying global optimality is coNP-complete. On the positive side, it shows how to compute a locally optimal solution in polynomial time and demonstrates that classical combinatorial problems like knapsack and the traveling salesman problem can be encoded as \textsc{BLP}_{\textsc{single}} instances. The authors develop a chain of reformulations—lexicographic, weighted, and penalty—to transform multi-variable upper-level problems into the single-variable form while preserving optimal values. These results indicate that restricting the upper level alone does not guarantee tractability and reveal a rich connection between bilevel LPs and classic combinatorial optimization problems.

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.

Figures (1)

• Figure 1: Plot of $u$ and $z$ in $\mathcal{S}(\ref{['eq:SATBilevelLexProgrammingLowerLevel']})$. The solid black lines represent $u_3$ and $z_3$, the dashed blue lines represent $u_2$ and $z_2$, and the dotted red lines represent $u_1$ and $z_1$.

Theorems & Definitions (33)

• theorem thmcountertheorem
• lemma thmcounterlemma
• lemma thmcounterlemma
• lemma thmcounterlemma
• lemma thmcounterlemma
• lemma thmcounterlemma
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• lemma thmcounterlemma