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

2025-10-09

AI Summary

This paper investigates the Temporal Vertex Cover and Temporal Dominating Set problems on temporal graphs: the former requires covering all edges in every snapshot, while the latter demands dominating all vertices in each snapshot. It presents the first systematic parameterized modeling and complexity analysis of the Dominating Set problem on temporal graphs, introducing a novel parameterization framework based on vertex-interval membership width (vimw) and interval membership width (imw). Leveraging dynamic programming and width-based techniques, the authors design a unified fixed-parameter tractable (FPT) algorithm for Temporal Vertex Cover parameterized by vimw + k + ℓ, improving upon prior results. Crucially, they establish FPT solvability for Temporal Dominating Set parameterized by imw + k + ℓ—while Temporal Vertex Cover remains NP-hard under the same parameter—thereby revealing, for the first time, a fundamental separation in parameterized complexity between these two canonical problems.

Technology Category

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

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: Topic discovery and tracking
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

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 extsc{Timeline Vertex Cover} (also known as extsc{MinTimeline$_infty$}), a temporal analogue to the classical extsc{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 extsc{Timeline Vertex Cover} and introduce the extsc{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 extsc{Timeline Vertex Cover}. Surprisingly, for $imw+ k+ell$, extsc{Timeline Dominating Set} turns out to be easier than extsc{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 $t$.

Problem

• Studying Timeline Vertex Cover and Timeline Dominating Set problems in temporal graphs
• Analyzing parameterized complexity using vimw, imw, k, and l parameters
• Developing FPT-algorithms for temporal graph problems with specific parameter combinations

Innovation

• FPT-algorithms using vertex-interval-membership-width parameterization
• Timeline Dominating Set solved via interval-membership-width parameterization
• Partial problem versions covering t edges or vertices