The Complexity of Temporal Vertex Cover in Small-Degree Graphs

TL;DR

This paper studies the computational complexity of the Temporal Vertex Cover problem on sparse graphs, showing NP-hardness for fixed time-windows on paths and cycles, and providing algorithms for special cases.

Contribution

It proves NP-hardness of $ ext{Δ}$-TVC on paths and cycles, resolving an open problem, and offers algorithms for graphs with bounded degree or small temporal vertex covers.

Findings

$ ext{Δ}$-TVC is NP-hard on paths and cycles for $ ext{Δ} ext{≥} 2$

Polynomial-time algorithm exists for TVC in the same setting

Algorithms are proposed for graphs with bounded degree or small temporal vertex covers

Abstract

Temporal graphs naturally model graphs whose underlying topology changes over time. Recently, the problems TEMPORAL VERTEX COVER (or TVC) and SLIDING-WINDOW TEMPORAL VERTEX COVER(or $\Delta$-TVC for time-windows of a fixed-length $\Delta$) have been established as natural extensions of the classic problem VERTEX COVER on static graphs with connections to areas such as surveillance in sensor networks. In this paper we initiate a systematic study of the complexity of TVC and $\Delta$-TVC on sparse graphs. Our main result shows that for every $\Delta\geq 2$, $\Delta$-TVC is NP-hard even when the underlying topology is described by a path or a cycle. This resolves an open problem from literature and shows a surprising contrast between $\Delta$-TVC and TVC for which we provide a polynomial-time algorithm in the same setting. To circumvent this hardness, we present a number of exact and approximation algorithms for temporal graphs whose underlying topologies are given by a path, that have bounded vertex degree in every time step, or that admit a small-sized temporal vertex cover.