On the Complexity of Recoverable Robust Optimization in the Polynomial Hierarchy

TL;DR

This paper proves that recoverable robust optimization problems with discrete budgeted uncertainty are $ ext{Sigma}_3^p$-complete for a wide range of classical combinatorial problems, indicating high computational complexity.

Contribution

It establishes the $ ext{Sigma}_3^p$-completeness of recoverable robust problems for many classical NP-hard problems under discrete uncertainty, extending existing theoretical frameworks.

Findings

All considered problems are $ ext{Sigma}_3^p$-complete under the studied setting.

A shared abstract property implies $ ext{Sigma}_3^p$-completeness for these problems.

The results extend the framework by Grüne and Wulf.

Abstract

Recoverable robust optimization is a popular multi-stage approach, in which it is possible to adjust a first-stage solution after the uncertain cost scenario is revealed. We consider recoverable robust optimization in combination with discrete budgeted uncertainty. In this setting, it seems plausible that many problems become $\Sigma^p_3$-complete and therefore it is impossible to find compact IP formulations of them (unless the unlikely conjecture NP $= \Sigma^p_3$ holds). Even though this seems plausible, few concrete results of this kind are known. In this paper, we fill that gap of knowledge. We consider recoverable robust optimization for the nominal problems of Sat, 3Sat, vertex cover, dominating set, set cover, hitting set, feedback vertex set, feedback arc set, uncapacitated facility location, $p$-center, $p$-median, independent set, clique, subset sum, knapsack, partition, scheduling, Hamiltonian path/cycle (directed/undirected), TSP, $k$-disjoint path ($k \geq 2$), and Steiner tree. We show that for each of these problems, and for each of three widely used distance measures, the recoverable robust problem becomes $\Sigma^p_3$-complete. Concretely, we show that all these problems share a certain abstract property and prove that this property implies that their robust recoverable counterpart is $\Sigma^p_3$-complete. This reveals the insight that all the above problems are $\Sigma^p_3$-complete 'for the same reason'. Our result extends a recent framework by Gr\"une and Wulf.