Minimizing Reachability Times on Temporal Graphs via Shifting Labels

TL;DR

The paper studies ReachFast, a problem that optimizes information spread in temporal graphs by shifting edge labels to minimize the maximum reaching-time from a set of sources. It establishes hardness for constrained shift variants, provides a polynomial-time algorithm for the unconstrained single-source case, proves NP-hardness for two sources, and identifies tractable regimes including trees, bounded-treewidth graphs via MSO/Courcelle, and parallel-paths with two sources. It thus delineates the boundary between intractable and tractable instances and offers practical, class-specific algorithms. The results leverage parameterized complexity and logic-based methods to deliver both hardness proofs and constructive FPT algorithms with clear implications for dynamic networks and scheduling disciplines.

Abstract

We study how we can accelerate the spreading of information in temporal graphs via shifting operations; a problem that captures real-world applications varying from information flows to distribution schedules. In a temporal graph there is a set of fixed vertices and the available connections between them change over time in a predefined manner. We observe that, in some cases, shifting some connections, i.e., advancing or delaying them, can decrease the travel time from some vertex (source) to another vertex. We study how we can minimize the maximum time a set of sources needs to reach every vertex, when we are allowed to shift some of the connections. If we restrict the allowed number of changes, we prove that, already for a single source, the problem is NP-hard, and W[2]-hard when parameterized by the number of changes. Then we focus on unconstrained number of changes. We derive a polynomial-time algorithm when there is a single source. When there are two sources, we show that the problem becomes NP-hard; on the other hand, we design an FPT algorithm parameterized by the treewidth of the graph plus the lifetime of the optimal solution, that works for any number of sources. Finally, we provide polynomial-time algorithms for several graph classes.

Figures (7)

• Figure 1: Left: The original timetable. Right: The modified timetable with the delayed meeting time.
• Figure 2: Top: The original temporal graph of a network where we assume that every label has traversal time equal to $1$. Note that vertex $C$ reaches vertex $A$ at time step $12$ and vertex $E$ is unreachable. Bottom: The modified temporal graph by delaying edge $AB$ and advancing edge $CD$. Note that now, vertex $C$ reaches vertex $A$ at time step $4$ and vertex $E$ can now be reached at time step $3$.
• Figure 3: Illustration of the construction used in Theorem \ref{['thm:one-source-hard']}. Here, the subsets of the HittingSet instance are $S_1=\{1,3,4\},S_2,=\{3,4\},\ldots,S_m=\{4,n\}$.
• Figure 4: The construction used in Theorem \ref{['thm:two-sources-hard']}. The first clause of the MonLinNAE3SAT instance is $c_1 =(x_1 \vee x_2 \vee x_3)$.
• Figure 5: The labelling of $\tilde{\mathcal{G}\xspace}(X)$ for a given satisfying assignment of MonLinNAE3SAT, where in clause $c_j = (x_{i_1} \vee x_{i_2} \vee x_{i_3})$ we have $x_{i_1} = \mathtt{True}$ and $x_{i_3} = \mathtt{False}$; the assignment of $x_{i_2}$ does not affect the reachability. The coloured directed paths, are the temporal paths from the corresponding source. The edges with two colors and two arrows are used at the same time step by both colours.

Theorems & Definitions (19)

• Theorem 1
• proof
• Corollary 2
• Theorem 3
• proof
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof