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

TL;DR

This paper investigates the computational complexity of bilevel linear programming with a single upper-level variable, showing it remains NP-complete even under restrictive conditions, but also provides a polynomial-time local optimality algorithm.

Contribution

It proves NP-completeness persists for bilevel LP with one upper-level variable and no upper-level constraints, and introduces a polynomial-time local optimality algorithm.

Findings

NP-completeness persists with a single upper-level variable

A polynomial-time algorithm for local optimal solutions is developed

Many combinatorial problems can be formulated as such bilevel LPs

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.