The Complexity of Finding Temporal Separators under Waiting Time Constraints

TL;DR

This paper studies the computational complexity of a temporal graph separation problem with waiting time constraints, proving it is complete for the second level of the polynomial hierarchy and exploring its parameterized complexity.

Contribution

It establishes the complexity class of Restless Temporal (s,z)-Separation as $ ext{Sigma}_2^P$-complete and provides insights into its parameterized complexity.

Findings

Restless Temporal (s,z)-Separation is $ ext{Sigma}_2^P$-complete.

The problem generalizes the NP-hard Temporal (s,z)-Separation.

Parameterization by separator size $k$ offers new complexity insights.

Abstract

In this work, we investigate the computational complexity of Restless Temporal $(s,z)$-Separation, where we are asked whether it is possible to destroy all restless temporal paths between two distinct vertices $s$ and $z$ by deleting at most $k$ vertices from a temporal graph. A temporal graph has a fixed vertex but the edges have (discrete) time stamps. A restless temporal path uses edges with non-decreasing time stamps and the time spent at each vertex must not exceed a given duration $\Delta$. Restless Temporal $(s,z)$-Separation naturally generalizes the NP-hard Temporal $(s,z)$-Separation problem. We show that Restless Temporal $(s,z)$-Separation is complete for $\Sigma_2^\text{P}$, a complexity class located in the second level of the polynomial time hierarchy. We further provide some insights in the parameterized complexity of Restless Temporal $(s,z)$-Separation parameterized by the separator size $k$.