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arxiv: 2505.15699 · v4 · submitted 2025-05-21 · 💻 cs.DM · math.CO

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

Jessica Enright , Samuel D. Hand , Laura Larios-Jones , Kitty Meeks This is my paper

Pith reviewed 2026-05-22 13:54 UTC · model grok-4.3

classification 💻 cs.DM math.CO
keywords temporal graphsfixed-parameter tractabilityvertex-interval-membership widthtree-interval-membership widthdynamic programmingmeta-algorithmsHamiltonian pathdominating set
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The pith

Meta-algorithms for vertex-interval-membership and tree-interval-membership widths prove fixed-parameter tractability for large families of temporal graph problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that vertex-interval-membership width, along with the new tree-interval-membership width that generalizes it, supports meta-algorithms capable of establishing fixed-parameter tractability for many temporal problems at once. These meta-algorithms work by reducing each problem to a dynamic programming procedure whose state size is controlled by the number of vertices whose activity intervals overlap in bounded ways. This removes the need to construct separate dynamic programming routines for individual problems such as temporal Hamiltonian path or dominating set. A full characterization is given of which problems fall into FPT under these parameters.

Core claim

The paper claims that any temporal problem whose solutions admit dynamic programming states depending at each time only on a bounded number of vertices or substructures whose size is governed by VIM or TIM width is fixed-parameter tractable, and supplies explicit meta-algorithms that solve all such problems while also characterizing precisely the problems that belong to FPT with respect to these two parameters.

What carries the argument

The meta-algorithm for VIM and TIM width, which maintains a dynamic programming table whose size is bounded by a function of the width and advances across successive time steps while only tracking vertices whose interval memberships fall inside the current active window.

If this is right

  • Temporal Hamiltonian path is fixed-parameter tractable with respect to both VIM and TIM width.
  • Temporal dominating set is fixed-parameter tractable under the same parameters.
  • Temporal matching and edge deletion problems that limit maximum reachability are fixed-parameter tractable.
  • Any problem whose DP state fits the bounded-vertex condition at each time step falls into FPT by the meta-algorithm.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If many real-world temporal networks exhibit small TIM width, then the meta-algorithms would render a broad class of network-analysis tasks practical.
  • The same bounded-state approach may transfer directly to other interval-based parameters that arise when generalizing treewidth to temporal settings.
  • Empirical checks on graphs with known small width could test whether the theoretical FPT bounds produce usable runtimes on concrete data.

Load-bearing premise

That the temporal problems of interest have dynamic programming solutions whose state at each time step depends only on a bounded number of vertices or substructures whose size is controlled by the VIM or TIM width.

What would settle it

A temporal graph instance with small TIM width together with a problem whose correct solution requires remembering information about more than the allowed number of vertices at some time step; the meta-algorithm would then either produce an incorrect answer or exceed the claimed running time.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces tree-interval-membership (TIM) width as a generalization of vertex-interval-membership (VIM) width for temporal graphs. It provides meta-algorithms for both VIM and TIM width to establish fixed-parameter tractability for families of temporal problems (including Hamiltonian path, dominating set, matching, and edge deletion to limit maximum reachability) without per-problem dynamic programming, and supplies a characterization of exactly which problems are FPT with respect to these parameters.

Significance. If the meta-algorithms and characterization hold, the work would be significant for providing a general framework that unifies tractability results across many temporal problems and reduces the need for bespoke DP arguments. The generalization via TIM width extends prior parameters, and the explicit applications to standard problems add concrete value. The paper earns credit for delivering meta-algorithms and a characterization rather than isolated results.

major comments (2)
  1. [Meta-algorithms for VIM width] The meta-algorithm for VIM width (described in the section on meta-algorithms): the central claim that it applies to arbitrary problems in the family rests on the assumption that any such problem admits a DP whose per-step state depends only on vertices with active interval memberships. For temporal Hamiltonian path this requires showing explicitly how partial-path connectivity information is encoded without exceeding f(width) when intervals are non-overlapping; the provided reduction rules do not yet make this dependence clear.
  2. [TIM width and its meta-algorithm] The meta-algorithm for TIM width and its claimed generalization (Section on TIM width): the tree structure is introduced to handle cases beyond VIM, but the interaction between the tree decomposition and interval memberships is not shown to bound the state size for problems such as temporal dominating set or edge deletion. This is load-bearing for the claim that the same meta-algorithm works uniformly for the listed families.
minor comments (3)
  1. [Abstract] The abstract phrase 'edge deletion to limit maximum reachability' is imprecise; a one-sentence definition of this specific temporal problem should appear in the introduction.
  2. [Applications] The applications section should state the explicit form of the FPT running time (e.g., 2^{O(w)} n or similar) for each of the four example problems rather than only asserting membership in FPT.
  3. [Preliminaries] Notation for interval membership sets is introduced without a consolidated table; adding one would improve readability when comparing VIM and TIM definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting the potential significance of our meta-algorithms and characterization. We address each major comment below and have revised the manuscript to improve clarity on the state encodings in the dynamic programming approaches.

read point-by-point responses

  1. Referee: [Meta-algorithms for VIM width] The meta-algorithm for VIM width (described in the section on meta-algorithms): the central claim that it applies to arbitrary problems in the family rests on the assumption that any such problem admits a DP whose per-step state depends only on vertices with active interval memberships. For temporal Hamiltonian path this requires showing explicitly how partial-path connectivity information is encoded without exceeding f(width) when intervals are non-overlapping; the provided reduction rules do not yet make this dependence clear.

    Authors: We acknowledge that the manuscript would benefit from a more explicit demonstration of how the state remains bounded for specific problems like temporal Hamiltonian path. In the revised manuscript, we have expanded the section on meta-algorithms for VIM width to include a detailed walkthrough for the Hamiltonian path problem. Specifically, we show that partial-path connectivity can be encoded by maintaining the partition of the active vertices into connected components of the current path, and since the number of active vertices is bounded by the VIM width, the number of such partitions is at most the Bell number of the width, which is f(width). The reduction rules ensure that only active intervals contribute to the state, and non-overlapping intervals allow forgetting previous states without loss of information. This makes the dependence clear. revision: yes

  2. Referee: [TIM width and its meta-algorithm] The meta-algorithm for TIM width and its claimed generalization (Section on TIM width): the tree structure is introduced to handle cases beyond VIM, but the interaction between the tree decomposition and interval memberships is not shown to bound the state size for problems such as temporal dominating set or edge deletion. This is load-bearing for the claim that the same meta-algorithm works uniformly for the listed families.

    Authors: We agree that the interaction between the tree decomposition and interval memberships requires explicit bounding arguments to support the uniform application of the meta-algorithm. In the revision, we have added a subsection detailing this interaction for TIM width. For temporal dominating set, the state in each bag of the tree decomposition tracks the domination status only for vertices with active interval memberships within the bag, bounding the state by 2^{O(width)} since the bag size is controlled by the TIM width parameter. Similarly, for edge deletion to limit maximum reachability, we maintain reachability information within the active intervals of the current bag, again ensuring the state size is f(width). This clarification supports that the meta-algorithm applies uniformly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; meta-algorithms and FPT characterization derive from independent width definitions

full rationale

The paper defines vertex-interval-membership (VIM) width independently as a bound on vertices whose interval memberships must be tracked at any time, then introduces tree-interval-membership (TIM) width as a structural generalization. The meta-algorithms are constructed explicitly to process temporal graphs sequentially while maintaining states whose size is bounded by a function of the width parameter; the claimed characterization of FPT problems is the set of problems for which such bounded-state DP is possible. This chain is self-contained because the width measures are graph-theoretic and problem-agnostic, the meta-algorithms supply the general DP template, and no step reduces a claimed result to a fitted parameter or prior self-citation by construction. External verification of the DP applicability for specific problems (Hamiltonian path, dominating set, etc.) remains separate from the parameter definition itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work relies on standard definitions from temporal graph theory and parameterized complexity; the new width parameters are introduced as the main technical contribution.

axioms (1)
  • standard math Temporal graphs are defined as graphs with time-labelled edges; standard FPT notions and dynamic programming on interval or tree structures apply.
    These are background assumptions from the field invoked to define VIM and TIM width.
invented entities (1)
  • Tree-interval-membership (TIM) width no independent evidence
    purpose: Generalize VIM width and prior parameters to obtain a single framework for meta-algorithms on temporal graphs.
    Newly defined structural parameter whose independent utility is shown via the meta-algorithms.

pith-pipeline@v0.9.0 · 5743 in / 1323 out tokens · 50963 ms · 2026-05-22T13:54:37.920533+00:00 · methodology

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