Matching and Edge Cover in Temporal Graphs

TL;DR

The paper investigates computational aspects of Temporal Edge Cover and Temporal Matching in temporal graphs, proving NP-hardness even for small lifetimes or when the underlying graph is a tree. It delivers fixed-parameter tractable algorithms parameterized by the lifetime $\tau$ and by treewidth, using dynamic programming on nice tree decompositions, with a running time of $O^*(2^{w^2}\,8^{w\tau})$. On the approximation front, Temporal Edge Cover admits an $O(\log\tau)$-approximation, while it is hard to approximate within $b\log\tau$ for any constant $0<b<1$, via reductions from Min Set Cover; Temporal Matching has a harder approximation barrier, with no $O(\tau^{1-\varepsilon})$-approximation unless $\mathrm{P}=\mathrm{NP}$, but a simple $\tau$-approximation is available by snapshot-wise maximum matchings. The authors also show that the duality between maximum matching and minimum edge cover in static graphs does not extend to the temporal setting, with NP-hardness persisting in both directions and no temporal Gallai theorem. Overall, the work characterizes complexity, fixed-parameter tractability, and approximability for these temporal problems and highlights fundamental differences from the classical static case.

Abstract

Temporal graphs are a special class of graphs for which a temporal component is added to edges, that is, each edge possesses a set of times at which it is available and can be traversed. Many classical problems on graphs can be translated to temporal graphs, and the results may differ. In this paper, we define the Temporal Edge Cover and Temporal Matching problems and show that they are NP-complete even when fixing the lifetime or when the underlying graph is a tree. We then describe two FPT algorithms, with parameters lifetime and treewidth, that solve the two problems. We also find lower bounds for the approximation of the two problems and give two approximation algorithms which match these bounds. Finally, we discuss the differences between the problems in the temporal and the static framework.

Figures (9)

• Figure 1: Two minimal temporal edge covers of a temporal graph. The one on the right has minimum cardinality.
• Figure 2: Two maximal temporal matchings of a temporal graph. The one on the right has maximum cardinality.
• Figure 3: Graph $L_i$.
• Figure 4: Graph $T_{j,k,l}$.
• Figure 5: Three ways to cover the vertices of $T_{j,k,l}$ with six edges in addition to those marked $j$, $k$, $l$.

Theorems & Definitions (20)

• Definition 1: Temporal Edge Cover
• Definition 2: Temporal Matching
• Theorem 3
• Theorem 4
• Definition 5
• Lemma 6
• Lemma 7
• Lemma 8
• Theorem 9
• Theorem 10