AI summaryⓘ
The authors study a situation where two decision makers, a leader and a follower, choose sets independently but their problems are linked in a bilevel optimization framework. They focus on two problems: one involving independent sets and a simpler version involving intervals. By looking at different goals for the leader and follower and how the follower reacts, they analyze how hard these problems are to solve, showing some are easy, some are NP-complete, and some fall into a higher complexity class called Σ₂^p-complete. For the simpler interval problem, the authors also provide an algorithm that solves certain cases efficiently. Their work clarifies the difficulty of these problems under various conditions.
bilevel optimizationindependent setinterval selectionpolynomial hierarchyNP-completenessΣ₂^p-completenessdynamic programmingleader-follower problemoptimistic and pessimistic reactionsum and bottleneck objectives
Abstract
We consider a bilevel optimization problem in which the ground set is partitioned between two decision makers, a leader and a follower, whose optimization problems are interleaved. We study the Bilevel Independent Set problem, and its special case, the Bilevel Interval Selection problem, on different variants emerging from a combination of the type of leader's objective function, the type of follower's objective function, and the setting in which the follower reacts, i.e., either optimistically or pessimistically. Here we consider sum and bottleneck type objective functions. We investigate the computational complexity of all these variants for the Bilevel Independent Set problem, and sort them into their respective level of the polynomial hierarchy. Our results range from $\mathsf{P}$, $\mathsf{NP}$-completeness to $Σ_2^\mathsf{p}$-completeness. For the Bilevel Interval Selection problem, we give a dynamic programming algorithm running in time $\mathcal{O}(n^4\log n)$ for the variants in which the leader and the follower have objective functions of the sum type.