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

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 $t$.