Vertex Interval Membership Width

• Vertex Interval Membership Width is a temporal graph parameter measuring the maximum number of vertices bridging past and future edge activations at any time.
• It underpins fixed-parameter tractability by enabling dynamic programming approaches on temporal problems like reachability, path, and domination.
• Generalizations such as tree-interval-membership width extend its utility, distinguishing it sharply from edge-based metrics with significant algorithmic implications.

Searching arXiv for the cited papers and recent work on vertex-interval-membership-width. arXiv search query: (Bumpus et al., 2021) vertex-interval-membership-width temporal graphs arXiv search query: (Enright et al., 21 May 2025) Families of tractable problems with respect to vertex-interval-membership width and its generalisations Vertex-interval-membership-width, usually denoted $\vimw$, is a temporal width parameter for temporal graphs that measures the maximum number of vertices whose active interval covers a given time. In the formulation introduced by Bumpus and Meeks, it is a vertex-based analogue of interval-membership-width and is defined from bags $F_t$ containing vertices that are incident to some edge active at or before $t$ and also to some edge active at or after $t$ (Bumpus et al., 2021). Subsequent work has treated $\vimw$ as a structural parameter for fixed-parameter algorithms, for meta-algorithmic frameworks, and for complexity separations between temporal models, especially for reachability, path, covering, and domination problems (Cauvi et al., 8 Jul 2025, Enright et al., 21 May 2025, Herrmann et al., 9 Oct 2025).

1. Formal definitions and equivalent viewpoints

For a temporal graph $(G,\tau)$ with lifetime $\Lambda$, the original vertex interval-membership sequence is the sequence $(F_t)_{t\in[\Lambda]}$ where

$$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$

Here $u$ and $F_t$0 need not be distinct. The vertex-interval-membership-width is then

$F_t$1

Intuitively, a vertex $F_t$2 belongs to $F_t$3 if it is incident to some edge active at or before $F_t$4 and also to some edge active at or after $F_t$5; equivalently, $F_t$6 lies between the first and last times at which $F_t$7 is temporally relevant (Bumpus et al., 2021).

Later papers restate the same idea in model-specific notation. In the point model for restless temporal paths, a node $F_t$8 is active over time $F_t$9 if there is some incident timed arc appearing no later than $t$0 and some incident timed arc arriving no earlier than $t$1. Using

$t$2

and

$t$3

the active set at time $t$4 is

$t$5

and

$t$6

In the snapshot-based timeline model, the bags are defined by

$t$7

with

$t$8

Across these formulations, the common content is that $t$9 counts the maximum number of vertices that are within their active interval at any time (Cauvi et al., 8 Jul 2025, Herrmann et al., 9 Oct 2025).

2. Relation to edge-based interval-membership-width

The edge-based interval-membership-width $t$0 is the older analogue in which bags contain edges rather than vertices. In the formulation used by Bumpus and Meeks,

$t$1

The vertex version counts vertices that are temporally straddling time $t$2, rather than edges whose time span covers $t$3 (Bumpus et al., 2021).

The distinction is algorithmically significant. A concrete separation is given by taking a disjoint union of $t$4 two-edge paths $t$5, where the two edges of $t$6 appear at times $t$7 and $t$8. In this construction, every edge appears at exactly one time, so $t$9, but the midpoint vertex of each path is incident to one edge before time $\vimw$0 and one edge after time $\vimw$1, so $\vimw$2. Thus $\vimw$3 can be arbitrarily larger than its edge counterpart (Bumpus et al., 2021).

The snapshot-based timeline literature makes the same structural point in a different language. There, the paper explicitly states that $\vimw$4, and that there are instances with constant $\vimw$5 but arbitrarily large $\vimw$6. A common misconception is therefore that bounded $\vimw$7 and bounded $\vimw$8 are interchangeable. The available separations show that they are not: edge persistence can remain sparse even when many vertices are simultaneously temporally active in a way that affects state-space size (Herrmann et al., 9 Oct 2025).

3. Reachability, edge exploration, and the original algorithmic motivation

The parameter was introduced in the context of edge exploration of temporal graphs and was immediately accompanied by a vertex-oriented variant tailored to reachability minimization. The central problem in that setting is $\vimw$9:

$(G,\tau)$0

Input: A temporal graph $(G,\tau)$1, a set of source vertices $(G,\tau)$2, and $(G,\tau)$3.

Question: Is there a set $(G,\tau)$4 of time-edges with $(G,\tau)$5 such that the temporal reachability of $(G,\tau)$6 in $(G,\tau)$7 is at most $(G,\tau)$8?

The paper proves that $(G,\tau)$9 is NP-hard even if the input temporal graph has interval-membership-width one. This shows that bounded edge-based interval-membership-width does not suffice for tractability in this vertex-reachability setting (Bumpus et al., 2021).

For $\Lambda$0, the same paper gives an FPT-time algorithm. If $\Lambda$1, then the problem can be decided in time

$\Lambda$2

The algorithm is a dynamic program over times $\Lambda$3. For each time $\Lambda$4, it stores states

$\Lambda$5

where $\Lambda$6 is an upper bound on the number of vertices reached so far and $\Lambda$7 records which vertices of the current bag $\Lambda$8 are currently reachable. The combinatorial bound comes from the fact that, since $\Lambda$9, the number of edges active at time $(F_t)_{t\in[\Lambda]}$0 is at most $(F_t)_{t\in[\Lambda]}$1, there are at most $(F_t)_{t\in[\Lambda]}$2 deletion choices per time step, and only $(F_t)_{t\in[\Lambda]}$3 reachability states. The paper also states a lemma that the vertex interval-membership sequence can be computed in time

$(F_t)_{t\in[\Lambda]}$4

This is the original algorithmic payoff of $(F_t)_{t\in[\Lambda]}$5: a problem that remains hard when $(F_t)_{t\in[\Lambda]}$6 becomes fixed-parameter tractable when parameterized by the number of temporally active vertices (Bumpus et al., 2021).

4. Restless temporal paths and the dependence on the temporal model

A later line of work studies $(F_t)_{t\in[\Lambda]}$7 for restless temporal paths, where waiting time at each node is restricted. In that setting, the parameter is treated as a purely temporal width notion that captures the maximum number of vertices that are simultaneously relevant in time. The paper interprets the set $(F_t)_{t\in[\Lambda]}$8 as a temporal separator: if a temporal path starts at a vertex active before $(F_t)_{t\in[\Lambda]}$9 and ends at one active after $$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$0, then it must pass through some active vertex at time $$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$1 (Cauvi et al., 8 Jul 2025).

The main results split sharply by model. In the point model with uniform delay one, $$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$2 can be solved in deterministic time

$$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$3

where $$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$4. In the point model with arbitrary positive delays, the runtime becomes

$$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$5

By contrast, in the interval model,

$$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$6

The same paper also proves that the arc analogue is too weak in the point model: $$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$7 This establishes that bounded $$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$8 is useful in the point model, but does not by itself control interval-model complexity (Cauvi et al., 8 Jul 2025).

The point-model FPT algorithm works by scanning timed arcs in increasing appearance time and maintaining, for each vertex $$F_t:=\{v\in V(G): \exists i\le t\le j \text{ and } u,w\in V(G) \text{ such that } i\in \tau(uv) \text{ and } j\in \tau(wv)\}.$$9, a collection of traces of partial restless paths. The trace of a path $u$0 at time $u$1 is

$u$2

For each trace $u$3, the algorithm stores the last arrival time $u$4 of a path realizing that trace. Extension by a future arc $u$5 is controlled by two conditions recorded in the paper: $u$6, which preserves simplicity, and $u$7, which enforces the restless constraint. The update rule is

$u$8

Because every cleaned-up trace is a subset of at most $u$9 active vertices, there are at most $F_t$00 possible traces. This is the state-compression mechanism behind the FPT bounds (Cauvi et al., 8 Jul 2025).

5. Meta-algorithms and the generalisation to tree-interval-membership-width

The meta-algorithmic development of $F_t$01 formalizes a pattern already visible in the problem-specific dynamic programs. For locally temporally uniform problems, the paper proves a generic theorem: if $F_t$02 is an instance of a $F_t$03-locally temporally uniform problem $F_t$04 with transition routine $F_t$05, accepting routine $F_t$06, and initial states $F_t$07, then one can determine whether $F_t$08 is a yes-instance in time

$F_t$09

where $F_t$10 is the VIM width, $F_t$11 is the maximum size of any counter variable in a state, and $F_t$12 bounds the time for $F_t$13 and $F_t$14. The key locality condition is that labels on vertices outside their active interval never need to change. This packages many VIM-based dynamic programs into a single theorem (Enright et al., 21 May 2025).

The same work introduces tree-interval-membership-width (TIM width) as a generalisation of $F_t$15 and of the connected variants introduced by Christodoulou et al. A TIM decomposition is a triple $F_t$16, where $F_t$17 is a labelled directed tree, $F_t$18 is a collection of bags, and $F_t$19 labels each bag with a time. The bags of a TIM decomposition are indexed by an arbitrary directed tree, and there can be multiple bags associated with every timestep. The width is

$F_t$20

and TIM width is the minimum such width over all TIM decompositions. The paper states that the VIM width of a temporal graph is always at least the TIM width, proves that TIM width subsumes the $F_t$21-connected-VIM, $F_t$22-connected-VIM, and bidirectional connected-VIM parameters, and gives a polynomial-time algorithm to find a minimum-width TIM decomposition in time

$F_t$23

where $F_t$24 is the TIM width (Enright et al., 21 May 2025).

For the broader class of component-exchangeable temporally uniform problems, the TIM-width meta-theorem gives an algorithm with running time

$F_t$25

Applications listed in the paper include Temporal Hamiltonian Path, which is solvable in time

$F_t$26

parameterized by VIM width $F_t$27, and in time

$F_t$28

parameterized by TIM width $F_t$29; $F_t$30-Temporal Matching, solvable in time

$F_t$31

and Temporal Reachability Edge Deletion, solvable in time

$F_t$32

The same paper also shows a limitation: Temporal Firefighter is FPT with respect to VIM width via the local-temporal meta-algorithm, but remains NP-complete even on temporal graphs whose TIM width and $F_t$33-connected VIM width are at most $F_t$34. This suggests that $F_t$35 and its generalisations support broad but not universal tractability phenomena (Enright et al., 21 May 2025).

6. Timeline covering and domination problems

In the timeline model, a temporal graph is a sequence of snapshots

$F_t$36

over a common vertex set. An activity interval of a vertex $F_t$37 is a triple $F_t$38 with $F_t$39, and a $F_t$40-activity $F_t$41-timeline allows at most $F_t$42 activity intervals per vertex, each of length at most $F_t$43. Within this framework, the paper studies Timeline Vertex Cover, Timeline Dominating Set, and their partial versions. The central parameterized-complexity result is that all four considered problems admit FPT-algorithms when parameterized by $F_t$44 (Herrmann et al., 9 Oct 2025).

The paper gives explicit bounds for the partial versions. Timeline PVC is solvable in

$F_t$45

and Timeline PDS is solvable in

$F_t$46

The same dynamic-programming framework yields FPT for Timeline VC as well, and the paper’s table records all four problems as FPT for $F_t$47. The DP processes the bags $F_t$48 chronologically and tracks, for each vertex in the current bag, which activity interval number is currently relevant and where inside the interval the current time lies. As in earlier VIM-width algorithms, the exponential part of the state space depends on the current bag size, not on the full number of vertices (Herrmann et al., 9 Oct 2025).

The same paper also introduces a ranked variant

$F_t$49

where $F_t$50 are sorted by non-increasing size. This yields refinements: Timeline PVC is solvable in

$F_t$51

for

$F_t$52

and Timeline DS is solvable in

$F_t$53

for

$F_t$54

For Timeline PDS, however, the reduction cannot be pushed in the same way: the paper shows NP-hardness already for parameterization by $F_t$55. This is one of several separations demonstrating that the effect of $F_t$56 depends materially on the exact optimization objective (Herrmann et al., 9 Oct 2025).

A further separation concerns comparison with $F_t$57. For the parameter $F_t$58, Timeline Dominating Set is FPT, whereas Timeline Vertex Cover, Timeline PVC, and Timeline PDS are NP-hard even when $F_t$59 is constant. In this setting, $F_t$60 is therefore the stronger parameter for the activity-interval formulations of covering and domination. More generally, the accumulated literature treats $F_t$61 as a genuinely temporal vertex-width measure: it captures how many vertices are simultaneously in play across time, supports chronological dynamic programming and generic meta-theorems, and also marks clear boundaries of tractability when the temporal model or problem family changes (Herrmann et al., 9 Oct 2025).