Timeline Problems in Temporal Graphs: Vertex Cover vs. Dominating Set

TL;DR

This work analyzes Timeline Vertex Cover and Timeline Dominating Set on temporal graphs, introducing structural parameters $\mathrm{vimw}$ and $\mathrm{imw}$ to study classical and parameterized complexity, including partial variants. It proves NP-hardness under tight restrictions, and provides a comprehensive FPT landscape: all problems are fixed-parameter tractable when parameterized by $\mathrm{vimw}+k+\ell$, with refined bounds using $\mathrm{vimw}[x]$; Timeline DS is FPT for $\mathrm{imw}+k+\ell$ while Timeline VC remains harder under the same parameter. The paper also delivers kernel results and $2^{O(t)}$-time algorithms for the vertex-cover/dominating-set variants, and contrasts the power of $\mathrm{vimw}$ vs $\mathrm{imw}$ in temporal graphs. These findings offer sharper, structure-aware algorithms for timeline problems in temporal graphs and outline directions for further optimization and MinTimeline variants.

Abstract

A temporal graph is a finite sequence of graphs, called snapshots, over the same vertex set. Many temporal graph problems turn out to be much more difficult than their static counterparts. One such problem is \textsc{Timeline Vertex Cover} (also known as \textsc{MinTimeline$_\infty$}), a temporal analogue to the classical \textsc{Vertex Cover} problem. In this problem, one is given a temporal graph $\mathcal{G}$ and two integers $k$ and $\ell$, and the goal is to cover each edge of each snapshot by selecting for each vertex at most $k$ activity intervals of length at most $\ell$ each. Here, an edge $uv$ in the $i$th snapshot is covered, if an activity interval of $u$ or $v$ is active at time $i$. In this work, we continue the algorithmic study of \textsc{Timeline Vertex Cover} and introduce the \textsc{Timeline Dominating Set} problem where we want to dominate all vertices in each snapshot by the selected activity intervals. We analyze both problems from a classical and parameterized point of view and also consider partial problem versions, where the goal is to cover (dominate) at least $t$ edges (vertices) of the snapshots. With respect to the parameterized complexity, we consider the temporal graph parameters vertex-interval-membership-width $(vimw)$ and interval-membership-width $(imw)$. We show that all considered problems admit FPT-algorithms when parameterized by $vimw + k+\ell$. This provides a smaller parameter combination than the ones used for previously known FPT-algorithms for \textsc{Timeline Vertex Cover}. Surprisingly, for $imw+ k+\ell$, \textsc{Timeline Dominating Set} turns out to be easier than \textsc{Timeline Vertex Cover}, by also admitting an FPT-algorithm, whereas the vertex cover version is NP-hard even if $imw+\, k+\ell$ is constant. We also consider parameterization by combinations of $n$, the vertex set size, with $k$ or $\ell$ and parameterization by $t$.

Figures (1)

• Figure 1: The short red boxes represent a 2-activity 0-timeline dominating all temporal vertices and the long blue boxes represent a 1-activity 2-timeline covering all temporal edges. Observe that the interval length is defined in such a way that an interval starting and ending in the same snapshot has length 0.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Theorem 4
• Proposition 5
• Theorem 6
• Proposition 7
• Theorem 8
• Theorem 9
• Theorem 10
• Theorem 11