Foremost, Fastest, Shortest: Temporal Graph Realization under Various Path Metrics

TL;DR

This work investigates temporal graph realization with respect to three fundamental path metrics—Foremost, Fastest, and Shortest—under a range of settings (strict/non-strict paths, periodic/non-periodic graphs, and bounded edge-labels). It delivers polynomial-time algorithms for Foremost-path TGR with unbounded labeling and various constraints, while delineating hardness when labeling is restricted (e.g., one label per edge) or when the distance matrix admits multiple values, supplemented by an FPT algorithm parameterized by the number of multi-valued entries. For Fastest-path TGR, the authors establish W[1]-hardness with respect to vertex cover number plus the largest entry, improving earlier results, and they extend hardness to non-periodic and periodic variants; they also show that NS-fastest-path TGR remains hard. Shortest-path TGR is NP-hard in non-periodic settings but becomes polynomial-time solvable in the periodic case. The results map the complexity landscape of temporal network design and verification, offering constructive algorithms, reductions, and parameterized insights with implications for temporal network design and analysis.

Abstract

In this work, we follow the current trend on temporal graph realization, where one is given a property P and the goal is to determine whether there is a temporal graph, that is, a graph where the edge set changes over time, with property P . We consider the problems where as property P , we are given a prescribed matrix for the duration, length, or earliest arrival time of pairwise temporal paths. That is, we are given a matrix D and ask whether there is a temporal graph such that for any ordered pair of vertices (s, t), Ds,t equals the duration (length, or earliest arrival time, respectively) of any temporal path from s to t minimizing that specific temporal path metric. For shortest and earliest arrival temporal paths, we are the first to consider these problems as far as we know. We analyze these problems for many settings like: strict and non-strict paths, periodic and non-periodic temporal graphs, and limited number of labels per edge (that is, limited occurrence number per edge over time). In contrast to all other path metrics, we show that for the earliest arrival times, we can achieve polynomial-time algorithms in periodic and non-periodic temporal graphs and for strict and and non-strict paths. However, the problem becomes NP-hard when the matrix does not contain a single integer but a set or range of possible allowed values. As we show, the problem can still be solved efficiently in this scenario, when the number of entries with more than one value is small, that is, we develop an FPT-algorithm for the number of such entries. For the setting of fastest paths, we achieve new hardness results that answers an open question by Klobas, Mertzios, Molter, and Spirakis [Theor. Comput. Sci. '25] about the parameterized complexity of the problem with respect to the vertex cover number and significantly improves over a previous hardness result for the feedback vertex set number. When considering shortest paths, we show that the periodic versions are polynomial-time solvable whereas the non-periodic versions become NP-hard.

Figures (5)

• Figure 1: The variable gadget from the reduction of \ref{['foremost simple']} with a true assignment on the left side and a false assignment on the right side. Here, the clause $c_i$ contains the literal $x$ and the clause $c_j$ contains the literal $\overline{x}$. The matrix only shows the entries of value smaller than 4. Each clause vertex $c_i$ aims to reach $\top$ by time $3$, but for each variable $x$, one of the edges $\{x,\top\}$ or $\{\overline{x},\top\}$ has to receive label $2$.
• Figure 2: The variable gadget from the reduction of \ref{['range size 2 hard']} with a true assignment on the left side and a false assignment on the right side. The matrix only shows the entries of value smaller than 5. Each clause vertex $c_i$ aims to reach $\top$ by time $2$ and $\bot$ by time $3$, but $\top$ and $\bot$ should pairwise not reach each other prior to time 5.
• Figure 3: An illustration of the underlying graph from the reduction behind \ref{['hardness fast strict']}. The edges of the biclique $(L\cup \{s',s",t',t"\}, V \cup X)$ are not depicted. Note that $L$ is a clique of size $2k+2$.
• Figure 4: An illustration of the labeling of the first described time block for a realization of the instance of Fastest-path TGR from the reduction behind \ref{['hardness fast strict']}. For a better readability, $\gamma$ substitutes $2k+3$ and the addition by the lower label $\alpha$ of the time block is omitted on all labels. Only the edges that receive labels are depicted.
• Figure 5: An example of the reduction behind \ref{['hardness shortest']} for the formula $(x \lor y \lor z) \land (\overline{x} \lor y) \land (\overline{y} \lor \overline{z})$. A labeling that realizes the matrix $D$ is depicted, where the dashed arcs receive the label set $\{1,2,7,8\}$. Moreover this labeling corresponds to a satisfying truth assignment ($x$ = False, $y$ = True, $z$ = False).

Theorems & Definitions (29)

• Theorem 1
• Proposition 1
• Proposition 2
• Definition 1: Edge compatibility
• Definition 2
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 2
• Lemma 4