Parameterized Complexity of Temporal Connected Components: Treewidth and k-Path Graphs

TL;DR

This paper investigates the parameterized complexity of maximum temporal connected components (tccs) in temporal graphs, focusing on open (oTCC) and closed (cTCC) variants under structural parameter treewidth $tw$ and temporal parameter $tpn$. It shows that neither $tw$ nor $tpn$ alone yields fixed-parameter tractability: both variants are $ ext{NP-hard}$ even at $tw=9$, and $cTCC$ is $ ext{NP-hard}$ for $tpn=6$, while $oTCC$ is in $ ext{XP}$ with respect to $tpn$. The authors identify tractable regimes by combining parameters, proving $ ext{FPT}$ algorithms for $(tw+tpn)$, $(tw+ abla)$, and $(tw+ ext{Δ}^t)$, and provide an MSO-based framework; they also obtain an $ ext{XP}$ algorithm for $oTCC$ on $k$-path graphs with running time $O(n^{2\mathrm{tpn}+1})$ via a VC-dimension argument. Additionally, they establish $ ext{NP-hard}$ness via a construction that uses non-transitivity gadgets to keep treewidth bounded, and discuss extensions to monotone path graphs where $tpn$ alone yields $ ext{FPT}$. Overall, the results map a nuanced landscape for tractability depending on how structural and temporal constraints are combined.

Abstract

We study the parameterized complexity of maximum temporal connected components (tccs) in temporal graphs, i.e., graphs that deterministically change over time. In a tcc, any pair of vertices must be able to reach each other via a time-respecting path. We consider both problems of maximum open tccs (openTCC), which allow temporal paths through vertices outside the component, and closed tccs (closedTCC) which require at least one temporal path entirely within the component for every pair. We focus on the structural parameter of treewidth, tw, and the recently introduced temporal parameter of temporal path number, tpn, which is the minimum number of paths needed to fully describe a temporal graph. We prove that these parameters on their own are not sufficient for fixed parameter tractability: both openTCC and closedTCC are NP-hard even when tw=9, and closedTCC is NP-hard when tpn=6. In contrast, we prove that openTCC is in XP when parameterized by tpn. On the positive side, we show that both problem become fixed parameter tractable under various combinations of structural and temporal parameters that include, tw plus tpn, tw plus the lifetime of the graph, and tw plus the maximum temporal degree.

Figures (9)

• Figure 1: Overview of our results. "para-$\mathtt{NP}$-h", "$\mathtt{W}[1]$-h", "$\mathtt{XP}$", and "$\mathtt{FPT}$" abbreviate para-$\mathtt{NP}$-hard, $\mathtt{W}[1]$-hard, exponential-time algorithm, and fixed-parameter tractable, respectively. Complexities for cTCC are indicated by the blue square, and for oTCC by the orange circle; the orange oval in the $\mathsf{tpn}$ column indicates that we provide an $\mathtt{XP}$ algorithm while it remains open whether the problem is $\mathtt{FPT}$ or $\mathtt{W}[1]$-hard. The numbers right of the indicator reference the corresponding statement in the paper and for $\Lambda$ the literature reference. All results hold on strict and non-strict, directed and undirected temporal graphs.
• Figure 2: Illustration (left) and mathematical description (right) of the compatibilities between the $V$-vertex and $E$-vertex sets. The red "inc" edges represent the incidence compatibility between a $V$-vertex set and its $E$-vertex sets, while the orange "id" edges, represent the identity compatibility between inverse $E$-vertex sets. Each of these compatibilities is realized via a separator-vertex and there are consequently no direct edges between these sets.
• Figure 3: Identity compatibility between $E_{12}$ and $E_{21}$ via $s_1$ and $s_2$. For example, the edge $(b,E)$ is the third in $<_{12}$, so the edges $((b,E),s_1)$ and $((E,b),s_2)$ are labeled with time 5, while the edges $(s_1,(E,b))$ and $(s_2,(b,E))$ are labeled with time 6.
• Figure 4: Global arrangement at $s_1$ (left) and $s_2$ (right). At $s_1$, the blocks $\mathcal{E}\xspace[E_{ij}\xrightarrow{id}{E_{ji}}]\xspace$ (\ref{['fig:tw pNP special s1s2']}) are arranged in lexicographic order of the index pairs $(i,j)$, with connecting temporal edges inserted between consecutive blocks: e. g., $(e_{21},s_1,9)$ for each $e_{21}\in E_{21}$ and $(s_1,e_{13},10)$ for each $e_{13}\in E_{13}$. At $s_2$, the arrangement is reversed.
• Figure 5: Incidence compatibility via $s_3$ and $s_4$ between $V_1$ and $E_{12}$. Since $b \in V_1$ is second in $<_1$, the edges $(b,s_3)$, $((b,E),s_4)$ and $((b,F),s_4)$ have time label $3$, while $(s_3,(b,E))$, $(s_3,(b,F))$ and $(s_4,b)$ have label $4$. Note that the order of the time labels between $V_2$ and $E_{21}$ follows the lexicographic order on $V_2$, although the vertices of $E_{21}$ are not arranged in that order in the drawing.

Theorems & Definitions (7)

• Definition 1: $k$-Path Graphs
• Definition 2: open/closed Temporal Connected Component
• Definition 3: kernelization Fomin_Lokshtanov_Saurabh_Zehavi_2019
• Definition 4: Tree Decomposition, Treewidth
• Theorem 5
• Lemma 6: non-transitivity gadgets
• Remark 7: meta-vertices