Novel Complexity Results for Temporal Separators with Deadlines

TL;DR

The paper investigates two robustness problems on temporal graphs with deadlines: the (s,z,$\ell$)-Temporal Separator (vertex-based) and the (s,z,$\ell$)-Temporal Cut (edge-based). It establishes sharp complexity boundaries, proving NP-hardness for bounded-pathwidth $4$ and $\ell=1$ while yielding a polynomial-time algorithm for pathwidth $3$, and showing strong inapproximability for the separator alongside APX-hardness for the temporal cut. It then provides concrete algorithms: a greedy $\ell-1$-approximation for the strict separator and a $2\log_2(2\ell)$-approximation for the temporal cut, with reductions from classical problems (Directed Multicut, Multiway Cut) and a structural pathwidth analysis. Together, these results map the trade-offs between temporal deadlines, graph structure, and algorithmic tractability, informing robustness analysis in time-evolving networks.

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$2^{Ω(\log^{1-\varepsilon}|V|)}$ 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 $2 \log_2(2\ell)$ approximation algorithm.

Figures (2)

• Figure 1: Top left: main structure of the reduction: there is a vertex-chossing phase leading to paths to $r$, followed by an edge vertification phase (note, not all edges are shown). Middle: a $V_i$ gadget, with $t_{green}$ edges in green, $t_{red}$ edges in red, and $t_u$ edges in blue. Right: an edge gadget for $e = uv$, with $t_{green}$ edges in green and $t_u, t_v$ edges in blue.
• Figure 2: An illustration of the structure described by Lemma \ref{['lem:pwstructure']}. The top part shows the temporal graph, the bottom part shows the bags that contain both $s$ and $z$ (the left and right areas represent $A$ and $C$, and the middle box represents $B$).

Theorems & Definitions (36)

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