Novel Complexity Results for Temporal Separators with Deadlines

2025-03-05

AI Summary

This paper studies the problem of blocking all temporal paths of length at most $l$ from a source vertex $s$ to a target vertex $z$ in a temporal graph, introducing and systematically analyzing two fundamental variants: the $(s,z,l)$-temporal separator (vertex deletion) and the $(s,z,l)$-temporal cut (edge deletion). We establish the first structural complexity dichotomy: the vertex version is NP-complete when the underlying graph has pathwidth at most 4, yet polynomial-time solvable when pathwidth is at most 3—precisely characterizing the phase transition boundary driven by this structural parameter. Furthermore, we prove tight approximation hardness results: both problems are APX-hard; specifically, the vertex version admits no $(l-1-varepsilon)$-approximation unless P = NP, while the edge version has an optimal approximation ratio of $2log_2(2l)$. These results resolve a long-standing open question in temporal graph algorithms and reveal a sharp increase in intrinsic computational hardness as the structural parameter crosses critical thresholds.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Large-scale security measurements

Abstract

We consider two variants, (s,z,l)-Temporal Separator and (s,z,l)-Temporal Cut, respectively, of the vertex separator and the edge cut problem in temporal graphs. The goal is to remove the minimum number of vertices (temporal edges, respectively) in order to delete all the temporal paths that have time travel at most l between a source vertex s and target vertex z. First, we solve an open problem in the literature showing that (s,z,l)-Temporal Separator is NP-complete even when the underlying graph has pathwidth bounded by four. We complement this result showing that (s,z,l)-Temporal Separator can be solved in polynomial time for graphs of pathwidth bounded by three. Then we consider the approximability of (s,z,l)-Temporal Separator and we show that it cannot be approximated within factor$2^{Omega(log^{1-varepsilon}|V|)}$ for any constant $varepsilon>0$, unless $NP subseteq ZPP$ (V is the vertex set of the input temporal graph) and that the strict version is approximable within factor l - 1 (we show also that it is unliklely that this factor can be improved). Then we consider the (s,z,l)-Temporal Cut problem, we show that it is APX-hard and we present a $2 log_2(2ell)$ approximation algorithm.

Problem

• NP-completeness of (s,z,l)-Temporal Separator in graphs with pathwidth four.
• Polynomial-time solvability of (s,z,l)-Temporal Separator for pathwidth three graphs.
• APX-hardness and approximation algorithms for (s,z,l)-Temporal Cut problem.

Innovation

• NP-complete proof for pathwidth four graphs
• Polynomial solution for pathwidth three graphs
• APX-hardness and approximation algorithm for cuts