Realizing temporal graphs from fastest travel times

TL;DR

This work initiates the study of realizing temporal graphs from fastest travel times in $\Delta$-periodic temporal graphs. It proves NP-hardness in general and shows tractability on trees, followed by a tight parameterized complexity landscape: the problem is W[1]-hard w.r.t. the feedback vertex number but fixed-parameter tractable (via Lenstra ILP) when parameterized by the feedback edge number. The authors present a polynomial-time tree algorithm and an ILP-based FPT algorithm, underpinned by preprocessing, structured guessing of temporal-path configurations, and a careful encoding of constraints. Collectively, the results map the complexity frontier for temporal graph realization and point to promising directions for future parameterizations and problem variants with practical relevance to network design and verification in periodically changing networks.

Abstract

In this paper we initiate the study of the temporal graph realization problem with respect to the fastest path durations among its vertices, while we focus on periodic temporal graphs. Given an $n \times n$ matrix $D$ and a $Δ\in \mathbb{N}$, the goal is to construct a $Δ$-periodic temporal graph with $n$ vertices such that the duration of a fastest path from $v_i$ to $v_j$ is equal to $D_{i,j}$, or to decide that such a temporal graph does not exist. The variations of the problem on static graphs have been well studied and understood since the 1960's (e.g. [Erdős and Gallai, 1960], [Hakimi and Yau, 1965]). As it turns out, the periodic temporal graph realization problem has a very different computational complexity behavior than its static (i.e. non-temporal) counterpart. First, we show that the problem is NP-hard in general, but polynomial-time solvable if the so-called underlying graph is a tree. Building upon those results, we investigate its parameterized computational complexity with respect to structural parameters of the underlying static graph which measure the ``tree-likeness''. We prove a tight classification between such parameters that allow fixed-parameter tractability (FPT) and those which imply W[1]-hardness. We show that our problem is W[1]-hard when parameterized by the feedback vertex number (and therefore also any smaller parameter such as treewidth, degeneracy, and cliquewidth) of the underlying graph, while we show that it is in FPT when parameterized by the feedback edge number (and therefore also any larger parameter such as maximum leaf number) of the underlying graph.

Figures (6)

• Figure 1: An example of a $\Delta$-periodic temporal graph $(G,\lambda,\Delta)$, where $\Delta = 10$ and the 10-periodic labeling $\lambda: E \rightarrow \{1,2,\ldots,10\}$ is as follows: $\lambda(v_1 v_2)=7$, $\lambda(v_2 v_3)=3$, $\lambda(v_3 v_4)=5$, and $\lambda(v_4 v_5)=1$. Here, the fastest temporal path from $v_1$ to $v_2$ traverses the first edge $v_1v_2$ at time $7$, second edge $v_2v_3$ a time $13$, third edge $v_3v_4$ at time $15$ and the last edge $v_4v_5$ at time $21$. This results in the total duration of $21 - 7 + 1 = 15$ for the fastest temporal path from $v_1$ to $v_5$.
• Figure 5: An example of a temporal graph (with $\Delta \geq 9$), where the fastest temporal path $P_{u,v}$ (in blue) from $u$ to $v$ is of duration $7$, while the fastest temporal path $P_{u,w}$ (in red) from $u$ to a vertex $w$, that is on a path $P_{u,v}$, is of duration $1$ and is not a subpath of $P_{u,v}$.
• Figure 6: An example of a graph with its important vertices: $U$ (in blue), $U^*$ (in green) and $Z^*$ (in orange). Corresponding feedback edges are marked with a thick red line, while dashed edges represent the edges (and vertices) "removed" from $G'$ at the initial step.
• Figure 7: In the above graph vertices $v_1, v_{11}, w$ are in $U$, while $v_2, v_{10}$ are in $U^*$. Numbers above all $v_i$ represent the values of the fastest temporal paths from $w$ to each of them (i. e., the entries in the $w$-th row of matrix $D$). From the basic guesses we know the fastest temporal path $P$ from $w$ to $v_2$ (depicted in blue) and the fastest temporal path $Q$ from $w$ to $v_{10}$. From the values of durations from $w$ to each $v_i$ we cannot determine the fastest paths from $w$ to all $v_i$. More precisely, we know that $w$ reaches $v_2, v_3, v_4, v_5$ (resp. $v_{10}, v_{9}, v_{9}, v_{7}$) by first using the path $P$ (resp. $Q$) and then proceeding through the vertices, but we do not know how $w$ reaches $v_6$ the fastest. Therefore we have to introduce some more guesses.
• Figure 8: Illustration of the guesses G-\ref{['FPT-guessFTPamongv2z2']}, G-\ref{['FPT:guess-uToSegmentz2']}, G-\ref{['FPT:guess-splitFromAnotherSegmentAndPaths']}, and G-\ref{['FPT:guess-splitFromUtoAnotherSegment']}.

Theorems & Definitions (20)

• Definition 1: temporal graph KKK00
• Definition 2: fastest temporal path
• Theorem 3
• Theorem 4
• Theorem 22
• Theorem 23
• Corollary 24
• Definition 25
• Corollary 26
• Definition 27: Travel delays