Completeness in the Polynomial Hierarchy for many natural Problems in Bilevel and Robust Optimization
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912722406342656 |
|---|---|
| author | Grüne, Christoph Wulf, Lasse |
| author_facet | Grüne, Christoph Wulf, Lasse |
| contents | In bilevel and robust optimization we are concerned with combinatorial min-max problems, for example from the areas of min-max regret robust optimization, network interdiction, most vital vertex problems, blocker problems, and two-stage adjustable robust optimization. Even though these areas are well-researched for over two decades and one would naturally expect many (if not most) of the problems occurring in these areas to be complete for the classes $Σ^p_2$ or $Σ^p_3$ from the polynomial hierarchy, almost no hardness results in this regime are currently known. However, such complexity insights are important, since they imply that no polynomial-sized integer program for these min-max problems exist, and hence conventional IP-based approaches fail. We address this lack of knowledge by introducing over 70 new $Σ^p_2$-complete and $Σ^p_3$-complete problems. The majority of all earlier publications on $Σ^p_2$- and $Σ^p_3$-completeness in said areas are special cases of our meta-theorem. Precisely, we introduce a large list of problems for which the meta-theorem is applicable (including clique, vertex cover, knapsack, TSP, facility location and many more). We show that for every single of these problems, the corresponding min-max (i.e. interdiction/regret) variant is $Σ^p_2$- and the min-max-min (i.e. two-stage) variant is $Σ^p_3$-complete. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_10540 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Completeness in the Polynomial Hierarchy for many natural Problems in Bilevel and Robust Optimization Grüne, Christoph Wulf, Lasse Computational Complexity Discrete Mathematics Optimization and Control F.2.2 In bilevel and robust optimization we are concerned with combinatorial min-max problems, for example from the areas of min-max regret robust optimization, network interdiction, most vital vertex problems, blocker problems, and two-stage adjustable robust optimization. Even though these areas are well-researched for over two decades and one would naturally expect many (if not most) of the problems occurring in these areas to be complete for the classes $Σ^p_2$ or $Σ^p_3$ from the polynomial hierarchy, almost no hardness results in this regime are currently known. However, such complexity insights are important, since they imply that no polynomial-sized integer program for these min-max problems exist, and hence conventional IP-based approaches fail. We address this lack of knowledge by introducing over 70 new $Σ^p_2$-complete and $Σ^p_3$-complete problems. The majority of all earlier publications on $Σ^p_2$- and $Σ^p_3$-completeness in said areas are special cases of our meta-theorem. Precisely, we introduce a large list of problems for which the meta-theorem is applicable (including clique, vertex cover, knapsack, TSP, facility location and many more). We show that for every single of these problems, the corresponding min-max (i.e. interdiction/regret) variant is $Σ^p_2$- and the min-max-min (i.e. two-stage) variant is $Σ^p_3$-complete. |
| title | Completeness in the Polynomial Hierarchy for many natural Problems in Bilevel and Robust Optimization |
| topic | Computational Complexity Discrete Mathematics Optimization and Control F.2.2 |
| url | https://arxiv.org/abs/2311.10540 |