The Complexity of Temporal Vertex Cover in Small-Degree Graphs

TL;DR

The 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, and presents 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.

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 $Δ$-TVC for time-windows of a fixed-length $Δ$) 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 $Δ$-TVC on sparse graphs. Our main result shows that for every $Δ\geq 2$, $Δ$-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 $Δ$-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.

Figures (13)

• Figure 1: An example of visualizing a temporal path graph $\mathcal{G}$ as a $2$-dimensional array, in which every edge corresponds to a time-edge of $\mathcal{G}$.
• Figure 2: An example of an instance of a planar monotone rectilinear 3SAT $\phi = (x_2 \lor x_3 \lor x_4) \land (x_1 \lor x_2 \lor x_4) \land (x_1 \lor x_4 \lor x_5) \land (\overline{x_2} \lor \overline{x_3} \lor \overline{x_5}) \land (\overline{x_1} \lor \overline{x_2} \lor \overline{x_5})$. For visual purposes, the line segments for variables and for clauses are illustrated here with boxes.
• Figure 3: Example of a segment block construction.
• Figure 4: An example where dummy time-edges, and vertical and horizontal paths are depicted.
• Figure 5: An example of two optimum covers of a segment block: (i) with the "orange and green"-colored, or (ii) with the "orange and red"-colored vertex appearances.

Theorems & Definitions (38)

• Definition 1: Temporal Graph
• Lemma 2
• proof
• Lemma 3
• proof
• Corollary 4
• proof
• Lemma 5
• proof
• Lemma 6