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Families of tractable problems with respect to vertex-interval-membership width and its generalisations

Jessica Enright, Samuel D. Hand, Laura Larios-Jones, Kitty Meeks

TL;DR

This work introduces tree-interval-membership (TIM) width as a unifying generalization of existing temporal-graph width measures and develops two meta-algorithms that yield fixed-parameter tractability results for broad families of temporal problems. TIM width, together with VIM width, is used to characterise when problems admit efficient FPT algorithms, with complete characterisations tying tractability to simple algorithmic properties (local temporally uniformity and component-exchangeability). The authors apply these meta-algorithms to temporal Hamiltonian Path, Temporal Dominating Set, Delta-Temporal Matching, and Temporal Reachability Edge Deletion, delivering explicit FPT runtimes that scale with width parameters. The results highlight the practical impact of TIM width as a powerful toolkit for proving tractability across diverse temporal problems, and establish theoretical links to the treewidth of static expansions and to prior width notions. They also outline promising directions, including computing TIM width in polynomial time and extending the framework to edge-based width measures.

Abstract

Temporal graphs are graphs whose edges are labelled with times at which they are active. Their time-sensitivity provides a useful model of real networks, but renders many problems studied on temporal graphs more computationally complex than their static counterparts. To contend with this, there has been recent work devising parameters for which temporal problems become tractable. One such parameter is vertex-interval-membership (VIM) width. Broadly, this gives a bound on the number of vertices we need to keep track of at any given time to solve many problems. Our contributions are two-fold. Firstly, we introduce a new parameter, tree-interval-membership (TIM) width, that generalises both VIM width and several existing generalisations. Secondly, we provide meta-algorithms for both VIM and TIM width which can be used to prove fixed-parameter-tractability for large families of problems, bypassing the need to give involved dynamic programming arguments for every problem. In doing this, we provide a characterisation of problems in FPT with respect to both parameters. We apply these algorithms to temporal versions of Hamiltonian path, dominating set, matching, and edge deletion to limit maximum reachability.

Families of tractable problems with respect to vertex-interval-membership width and its generalisations

TL;DR

This work introduces tree-interval-membership (TIM) width as a unifying generalization of existing temporal-graph width measures and develops two meta-algorithms that yield fixed-parameter tractability results for broad families of temporal problems. TIM width, together with VIM width, is used to characterise when problems admit efficient FPT algorithms, with complete characterisations tying tractability to simple algorithmic properties (local temporally uniformity and component-exchangeability). The authors apply these meta-algorithms to temporal Hamiltonian Path, Temporal Dominating Set, Delta-Temporal Matching, and Temporal Reachability Edge Deletion, delivering explicit FPT runtimes that scale with width parameters. The results highlight the practical impact of TIM width as a powerful toolkit for proving tractability across diverse temporal problems, and establish theoretical links to the treewidth of static expansions and to prior width notions. They also outline promising directions, including computing TIM width in polynomial time and extending the framework to edge-based width measures.

Abstract

Temporal graphs are graphs whose edges are labelled with times at which they are active. Their time-sensitivity provides a useful model of real networks, but renders many problems studied on temporal graphs more computationally complex than their static counterparts. To contend with this, there has been recent work devising parameters for which temporal problems become tractable. One such parameter is vertex-interval-membership (VIM) width. Broadly, this gives a bound on the number of vertices we need to keep track of at any given time to solve many problems. Our contributions are two-fold. Firstly, we introduce a new parameter, tree-interval-membership (TIM) width, that generalises both VIM width and several existing generalisations. Secondly, we provide meta-algorithms for both VIM and TIM width which can be used to prove fixed-parameter-tractability for large families of problems, bypassing the need to give involved dynamic programming arguments for every problem. In doing this, we provide a characterisation of problems in FPT with respect to both parameters. We apply these algorithms to temporal versions of Hamiltonian path, dominating set, matching, and edge deletion to limit maximum reachability.

Paper Structure

This paper contains 17 sections, 45 theorems, 8 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. A hierarchy of parameters
  4. Algorithmic distinctions
  5. Meta-algorithms
  6. Meta-algorithm parameterised by VIM width
  7. Meta-algorithm parameterised by TIM width
  8. Applications of VIM meta-algorithm
  9. Temporal Hamiltonian Path
  10. Temporal Dominating Set
  11. Applications of TIM meta-algorithm
  12. Temporal Hamiltonian Path
  13. Temporal Dominating Set
  14. Delta-Temporal Matching
  15. Singleton Temporal Reachability Edge Deletion
  16. ...and 2 more sections

Key Result

Lemma 11

Let $\mathcal{G}$ be a temporal graph such that $\min\{\psi_{\leq}, \psi_{\geq}\}=k$, where $\psi_d$ is the $d$-connected-VIM width. Then, $\mathcal{G}$ has TIM width at most $k$.

Figures (8)

  • Figure 1: (A) An example temporal graph $\mathcal{G}$, where the number(s) on an edge indicates the time(s) at which it is active. Note that this graph has lifetime 5. The VIM sequence (B) of $\mathcal{G}$, and a TIM decomposition (C) of $\mathcal{G}$.
  • Figure 2: A hierarchy of parameters. There is an arc from parameter A to parameter B if bounding A implies that B is also bounded. The relationships are strict -- for every arc from A to B, there exists an infinite family of graphs for which B is bounded and A is unbounded. The parameters we are focussing on are highlighted with boxes.
  • Figure 3: A comparison of the bags of a VIM sequence (A), a bidirectional connected-VIM sequence (B), and a TIM decomposition (C). Dashed boxes group the bags of the decompositions which are labelled with the same time. The point here is that as the bags of the decompositions decrease in size, the structure of the decomposition graph becomes more unruly. In decomposition (B), there is a bag from which all bags branch out. If the image were to depict a $\leq$- or $\geq$-connected-VIM decomposition instead, the bag would be at either the start or end, respectively.
  • Figure 4: An example of a temporal graph $\mathcal{G}$ with TIM width 2 and $\psi_{\sim}(\mathcal{G})=k$. The dashed edge replaces a path consisting of $k-4$ edges labelled with consecutive times.
  • Figure 5: A path on $n$ vertices where all edges are only active at times 1 and $\Lambda$ and its static expansion. Dashed lines a portion of a (directed) path that is not pictured.
  • ...and 3 more figures

Theorems & Definitions (61)

  • Definition 1: Vertex-Interval-Membership Width (Bumpus and Meeks bumpus_edge_2023)
  • Definition 2: Connected-Vertex-Interval-Membership Width (Christodoulou et al. christodoulou_making_2024)
  • Definition 3: Bidirectional Connected-Vertex-Interval-Membership Width (Christodoulou et al. christodoulou_making_2024)
  • Definition 4: Tree-Interval-Membership Width
  • Lemma 11
  • Lemma 11
  • Definition 12: Static expansion fluschnik_as_2020, Definition 2
  • Definition 13: Tree Decomposition, Treewidth
  • Theorem 14
  • Definition 15
  • ...and 51 more