Temporal Graph Realization With Bounded Stretch

TL;DR

This work studies 60 Temporal Graph Realization under a bounded stretch, formalizing STGR: given a static graph G, a period Δ, and a target factor α, assign time labels in {1,...,Δ} to edges so that the fastest s→t temporal path duration is at most α times the static distance D_G(s,t) for all vertex pairs. It provides a comprehensive treatment spanning hardness, approximation, and parameterized results: (i) NP-hardness and strong inapproximability (δ within Δ^{1−ε} or 2^{n^c}); (ii) a simple radius-based polynomial-time algorithm achieving stretch strictly below Δ, often optimal, and tighter bounds on trees; (iii) MSO-based fixed-parameter algorithms yielding FPT results for parameters like neighborhood diversity and treewidth with diameter; (iv) a local-search variant LS STGR with XP algorithms in the search radius k but W[2]-hardness, underscoring a nuanced landscape of tractability; (v) novel gadget constructions (Δ-sunglasses) underpin the hardness proofs. Collectively, the results delineate the boundary between tractable and intractable instances of scheduling periodic temporal networks under a natural stretch objective, with implications for network design and temporal routing where latency scales with physical distance. The combination of combinatorial reductions, logic-based meta-theorems, and parameterized algorithms provides a rich toolkit for understanding temporal realization problems under bounded stretch.

Abstract

A periodic temporal graph, in its simplest form, is a graph in which every edge appears exactly once in the first $Δ$ time steps, and then it reappears recurrently every $Δ$ time steps, where $Δ$ is a given period length. This model offers a natural abstraction of transportation networks where each transportation link connects two destinations periodically. From a network design perspective, a crucial task is to assign the time-labels on the edges in a way that optimizes some criterion. In this paper we introduce a very natural optimality criterion that captures how the temporal distances of all vertex pairs are `stretched', compared to their physical distances, i.e. their distances in the underlying static (non-temporal) graph. Given a static graph $G$, the task is to assign to each edge one time-label between 1 and $Δ$ such that, in the resulting periodic temporal graph with period~$Δ$, the duration of the fastest temporal path from any vertex $u$ to any other vertex $v$ is at most $α$ times the distance between $u$ and $v$ in $G$. Here, the value of $α$ measures how much the shortest paths are allowed to be \emph{stretched} once we assign the periodic time-labels. Our results span three different directions: First, we provide a series of approximation and NP-hardness results. Second, we provide approximation and fixed-parameter algorithms. Among them, we provide a simple polynomial-time algorithm (the \textit{radius-algorithm}) which always guarantees an approximation strictly smaller than $Δ$, and which also computes the optimum stretch in some cases. Third, we consider a parameterized local search extension of the problem where we are given the temporal labeling of the graph, but we are allowed to change the time-labels of at most $k$ edges; for this problem we prove that it is W[2]-hard but admits an XP algorithm with respect to $k$.

Figures (5)

• Figure 1: Example of a graph $G$ with radius $3$, where $v_x\in V(G)$ is a vertex of eccentricity equal to the radius. The gray areas depict the distance 1-3 neighborhoods of $v_x$. The labels given by the radius algorithm are illustrated. Edges between vertices in the same neighborhood are not depicted and are given arbitrary labels by the algorithm.
• Figure 2: The sunglasses gadgets for $\Delta=3$ with the sunglasses labeling. The black vertices are the docking points and the white vertices are the central vertices.
• Figure 3: The sunglasses gadgets for odd $\Delta > 3$. The black vertices indicate the docking points, the black edges indicate the edges of the chronological paths and cycles, the teal edges indicate the four zigzag paths and the red edge are the parallel edges. For $\Delta = 5$, the parallel edges coincide with edges of the chronological cycles. The shown labeling is the sunglasses labeling. Formally, this labeling labels (a) the chronological paths increasingly from $1$ to $\Delta$, (b) the zigzag paths increasingly from $1$ to $\frac{\Delta -2}{2}$, and (c) all other edges (namely, all vertically drawn edges) with label $\frac{\Delta +1}{2}$. Note that in this way, both chronological cycles are assigned increasing labels from $1$ to $\Delta$.
• Figure 4: The sunglasses gadgets for even $\Delta\geq 4$. Again, the black vertices indicate the docking points, the black edges indicate the edges of the chronological paths and cycles, the teal edges indicate the four zigzag paths and the red edge are the parallel edges. For $\Delta \in \{4,6\}$, the parallel edges coincide with edges of the chronological cycles, or the edge (indicated in blue) between the central vertices. The latter described edge is formally not part of the sunglasses gadget for $\Delta > 4$ but is added in the NP-hardness reduction anyway. The shown labeling is the sunglasses labeling. Formally, this labeling labels /a) the chronological paths increasingly from $1$ to $\Delta$, (b) the zigzag paths increasingly from $1$ to $\frac{\Delta }{2}-1$, and (c) all other edges (namely, all vertically drawn edges) with label $\frac{\Delta }{2}$. Note that in this way, both chronological cycles are assigned strictly increasing labels.
• Figure 5: An illustration of the additional vertices $\{c1,c2,c_3\}$ and edges (as well as their labels) for the reduction in the special case of $\Delta = 4$.

Theorems & Definitions (28)

• Lemma 2
• Theorem 3
• Theorem 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Definition 10: Sunglasses gadgets for $\Delta=3$
• Definition 11