How Many Lines to Paint the City: Exact Edge-Cover in Temporal Graphs

Abstract

Logistics and transportation networks require a large amount of resources to realize necessary connections between locations and minimizing these resources is a vital aspect of planning research. Since such networks have dynamic connections that are only available at specific times, intricate models are needed to portray them accurately. In this paper, we study the problem of minimizing the number of resources needed to realize a dynamic network, using the temporal graphs model. In a temporal graph, edges appear at specific points in time. Given a temporal graph and a natural number k, we ask whether we can cover every temporal edge exactly once using at most k temporal journeys; in a temporal journey consecutive edges have to adhere to the order of time. We conduct a thorough investigation of the complexity of the problem with respect to four dimensions: (a) whether the type of the temporal journey is a walk, a trail, or a path; (b) whether the chronological order of edges in the journey is strict or non-strict; (c) whether the temporal graph is directed or undirected; (d) whether the start and end points of each journey are given or not. We almost completely resolve the complexity of all these problems and provide dichotomies for each one of them with respect to k.

Figures (13)

• Figure 1: Illustration of the cases for possible adjacent edges for two journeys. The cases are the same for all three types of journeys, only the behaviour of the journeys may change.
• Figure 2: Illustration of the assignment gadget for the $\mathtt{NP}$-hardness reduction in \ref{['thm:UD_NS_2TTD']}.
• Figure 3: On the left is an illustration of a clause gadget for a clause $C_i = X_p\vee X_q \vee \overline{X_r}$ for the $\mathtt{NP}$-hardness reduction in \ref{['thm:UD_NS_2TTD']}. The red dotted edges have labels $2$ and $3$ and are traversed in time step $2$ by the second trail. The blue dashed edges have labels $1$ and $3$ and are traversed in time step $2$ by the second trail. The black edges have only label $3$. On the right is an example how one can traverse the clause-gadget for $C_i$ in the time step three if one of $T_i^p$ or $F_i^r$ is on the first (red dotted) trail and $T_i^q$ is on the second (blue dashed) trail in the assignment gadget.
• Figure 4: Illustration of the first trail-restriction-gadget for the $\mathtt{NP}$-hardness reduction in \ref{['thm:UD_NS_2TTD']}. The second trail-restriction-gadget is analogous, but edges have time step two, it starts at $s_2$, ends at $v_2^{start}$ and it goes through $a_i^j$ whenever the first trail-restriction-gadget goes through $b_i^j$.
• Figure 5: Illustration of the construction for the $\mathtt{NP}$-hardness reduction in \ref{['thm:SD_TPD']} for $k=3$. The assignment-gadget is on the top and the satisfaction-gadget is on the bottom. The two edges $(v^{h+1},c_1,2h(\ell+1)+1),(v^{h+1},c_1,2h(\ell+1)+2)$ connecting the assignment- and satisfaction-gadget are drawn as two separate edges. With this each edge has exactly one time label increasing with the direction of the edges.

Theorems & Definitions (49)

• Definition 3.1: Static Exact Edge-Cover by Journeys
• Definition 3.2: Temporal Exact Edge-Cover by Temporal Journeys
• Lemma 3.4
• Lemma 3.5
• Corollary 3.6
• Lemma 3.7: Strict Directed Paths
• proof
• Lemma 3.8: Non-Strict Directed Paths
• proof
• Lemma 3.9: Strict Undirected Paths