On the Complexity of Bilevel Independent Set Problem
This work provides a complexity landscape for bilevel optimization on independent set problems, which is foundational for understanding the limits of such hierarchical decision-making.
The paper studies the computational complexity of the Bilevel Independent Set problem and its special case, Bilevel Interval Selection, under various leader/follower objective functions and optimistic/pessimistic settings, classifying variants from P to Σ₂ᵖ-complete. For Bilevel Interval Selection with sum-type objectives, they provide an O(n⁴ log n) dynamic programming algorithm.
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.