---
title: Vertex Interval Membership Width
url: https://www.emergentmind.com/topics/vertex-interval-membership-width
type: topic
---

# Vertex Interval Membership Width

Searching arXiv for the cited papers and recent work on vertex-interval-membership-width.
arXiv search query: 2103.05387 vertex-interval-membership-width temporal graphs
arXiv search query: 2505.15699 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 \(t\) [2103.05387]. 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 [2507.05760] [2505.15699] [2510.08124].

## 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 \(w\) need not be distinct. The vertex-interval-membership-width is then
\[
\vimw(G,\tau):=\max_{t\in[\Lambda]} |F_t|.
\]
Intuitively, a vertex \(v\) belongs to \(F_t\) if it is incident to some edge active at or before \(t\) and also to some edge active at or after \(t\); equivalently, \(t\) lies between the first and last times at which \(v\) is temporally relevant [2103.05387].

Later papers restate the same idea in model-specific notation. In the point model for restless temporal paths, a node \(u\) is active over time \(\tau\) if there is some incident timed arc appearing no later than \(\tau\) and some incident timed arc arriving no earlier than \(\tau\). Using
\[
\tau^{min}(u)=\min \{\tau \in \mathbb{N}\mid \exists v\in V,\delta\in\mathbb{N}_{>0}, (u,v,\tau,\delta)\in E \text{ or } (v,u,\tau,\delta)\in E\},
\]
and
\[
\tau^{max}(u)=\max \{\tau+\delta \in \mathbb{N}\mid \exists v\in V,\ (u,v,\tau,\delta)\in E \text{ or } (v,u,\tau,\delta)\in E\},
\]
the active set at time \(\tau\) is
\[
F_\tau=\{u\in V\mid \tau^{min}(u)\leq \tau\leq\tau^{max}(u)\},
\]
and
\[
vimw(G)=\max_{\tau\in \llbracket 0,\Lambda\rrbracket}|F_\tau|.
\]
In the snapshot-based timeline model, the bags are defined by
\[
F_i := \{v \in V(\mathcal{G}) : \exists p,q \in [T] \text{ such that } p \le i \le q,\ \deg_{G_p}(v)\ge 1,\ \deg_{G_q}(v)\ge 1\},
\]
with
\[
\vimw(\mathcal{G}) := \max_{i\in[T]} |F_i|.
\]
Across these formulations, the common content is that \(\vimw\) counts the maximum number of vertices that are within their active interval at any time [2507.05760] [2510.08124].

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

The edge-based interval-membership-width \(\imw\) is the older analogue in which bags contain edges rather than vertices. In the formulation used by Bumpus and Meeks,
\[
F_t^{\text{edge}}=\{e \in E(G): \min\tau(e)\le t \le \max\tau(e)\}, \quad \imw(G,\tau)=\max_t |F_t^{\text{edge}}|.
\]
The vertex version counts vertices that are temporally straddling time \(t\), rather than edges whose time span covers \(t\) [2103.05387].

The distinction is algorithmically significant. A concrete separation is given by taking a disjoint union of \(k\) two-edge paths \(P_1,\dots,P_k\), where the two edges of \(P_i\) appear at times \(i\) and \(i+k\). In this construction, every edge appears at exactly one time, so \(\imw(G,\tau)=1\), but the midpoint vertex of each path is incident to one edge before time \(k\) and one edge after time \(k\), so \(\vimw(G,\tau)\ge k\). Thus \(\vimw\) can be arbitrarily larger than its edge counterpart [2103.05387].

The snapshot-based timeline literature makes the same structural point in a different language. There, the paper explicitly states that \(\vimw = \Omega(\sqrt{\imw})\), and that there are instances with constant \(\imw\) but arbitrarily large \(\vimw\). A common misconception is therefore that bounded \(\imw\) and bounded \(\vimw\) 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 [2510.08124].

## 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 \(\minreachdelete\):

\[
\minreachdelete
\]

**Input:** A temporal graph \((G,\tau)\), a set of source vertices \(S\), and \(k,h\in \mathbb{N}\).

**Question:** Is there a set \(E'\) of time-edges with \(|E'|\le k\) such that the temporal reachability of \(S\) in \((G,\tau)\setminus E'\) is at most \(h\)?

The paper proves that \(\minreachdelete\) 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 [2103.05387].

For \(\vimw\), the same paper gives an FPT-time algorithm. If \(w=\vimw(G,\tau)\), then the problem can be decided in time
\[
\mathcal{O}(w^2 h 2^{w^2}\Lambda).
\]
The algorithm is a dynamic program over times \(i=1,\dots,\Lambda\). For each time \(i\), it stores states
\[
(r,f)\in \{0,\dots,h\}\times F_i^{\{0,1\}},
\]
where \(r\) is an upper bound on the number of vertices reached so far and \(f\) records which vertices of the current bag \(F_i\) are currently reachable. The combinatorial bound comes from the fact that, since \(|F_i|\le w\), the number of edges active at time \(i\) is at most \(\binom{w}{2}\), there are at most \(2^{w^2}\) deletion choices per time step, and only \(h2^w\) reachability states. The paper also states a lemma that the vertex interval-membership sequence can be computed in time
\[
\mathcal{O}(\vimw(G,\tau)\Lambda).
\]
This is the original algorithmic payoff of \(\vimw\): a problem that remains hard when \(\imw=1\) becomes fixed-parameter tractable when parameterized by the number of temporally active vertices [2103.05387].

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

A later line of work studies \(\vimw\) 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 \(F_\tau\) as a temporal separator: if a temporal path starts at a vertex active before \(\tau\) and ends at one active after \(\tau\), then it must pass through some active vertex at time \(\tau\) [2507.05760].

The main results split sharply by model. In the point model with uniform delay one, \(\textsc{point-restless-temporal-path}\) can be solved in deterministic time
\[
\mathcal{O}(Mk2^k),
\]
where \(k=vimw(G)\). In the point model with arbitrary positive delays, the runtime becomes
\[
\mathcal{O}(Mk2^k(k+\log M)).
\]
By contrast, in the interval model,
\[
\textsc{interval-restless-temporal-path} \text{ is NP-hard even when } vimw^{int}(G^{int})=3.
\]
The same paper also proves that the arc analogue is too weak in the point model:
\[
\textsc{point-restless-temporal-path} \text{ is NP-hard even if the input has arc-IM-width }3 \text{ and uniform delay one.}
\]
This establishes that bounded \(\vimw\) is useful in the point model, but does not by itself control interval-model complexity [2507.05760].

The point-model FPT algorithm works by scanning timed arcs in increasing appearance time and maintaining, for each vertex \(u\), a collection of traces of partial restless paths. The trace of a path \(P\) at time \(\tau\) is
\[
V(P)\cap F_\tau.
\]
For each trace \(S\), the algorithm stores the last arrival time \(\sigma\) of a path realizing that trace. Extension by a future arc \((u,v,\tau)\) is controlled by two conditions recorded in the paper: \(v\notin S\), which preserves simplicity, and \(\tau-\sigma\leq \Delta\), which enforces the restless constraint. The update rule is
\[
S'=(S\cap F_\tau)\cup\{v\}.
\]
Because every cleaned-up trace is a subset of at most \(k\) active vertices, there are at most \(2^k\) possible traces. This is the state-compression mechanism behind the FPT bounds [2507.05760].

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

The meta-algorithmic development of \(\vimw\) formalizes a pattern already visible in the problem-specific dynamic programs. For locally temporally uniform problems, the paper proves a generic theorem: if \(x\) is an instance of a \((k,X)\)-locally temporally uniform problem \(P\) with transition routine \(Tr\), accepting routine \(Ac\), and initial states \(S_0\), then one can determine whether \(x\) is a yes-instance in time
\[
O(\Lambda p(n)b^{2k}|X|^{2\omega}),
\]
where \(\omega\) is the VIM width, \(b\) is the maximum size of any counter variable in a state, and \(p\) bounds the time for \(Tr\) and \(Ac\). 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 [2505.15699].

The same work introduces tree-interval-membership-width (TIM width) as a generalisation of \(\vimw\) and of the connected variants introduced by Christodoulou et al. A TIM decomposition is a triple \((T,B,\tau)\), where \(T\) is a labelled directed tree, \(B=\{B(s): s\in V(T)\}\) is a collection of bags, and \(\tau:B\to[\Lambda]\) 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
\[
\max\{|B(s)|: s\in V(T)\},
\]
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 \(\le\)-connected-VIM, \(\ge\)-connected-VIM, and bidirectional connected-VIM parameters, and gives a polynomial-time algorithm to find a minimum-width TIM decomposition in time
\[
O(n^4\Lambda^2\phi),
\]
where \(\phi\) is the TIM width [2505.15699].

For the broader class of component-exchangeable temporally uniform problems, the TIM-width meta-theorem gives an algorithm with running time
\[
(n\Lambda)^{O(k)} (b|X|)^{O(k\phi^3)}.
\]
Applications listed in the paper include Temporal Hamiltonian Path, which is solvable in time
\[
O(\Lambda^2 n^3 3^{2\omega})
\]
parameterized by VIM width \(\omega\), and in time
\[
(n\Lambda)^{O(1)}3^{O(\phi^3)}
\]
parameterized by TIM width \(\phi\); \(\Delta\)-Temporal Matching, solvable in time
\[
(n\Lambda)^{O(1)}(\phi\Delta)^{O(\phi^3)};
\]
and Temporal Reachability Edge Deletion, solvable in time
\[
(n\Lambda)^{O(1)} \phi^{O(\phi^3)}.
\]
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 \(\geq\)-connected VIM width are at most \(3\). This suggests that \(\vimw\) and its generalisations support broad but not universal tractability phenomena [2505.15699].

## 6. Timeline covering and domination problems

In the timeline model, a temporal graph is a sequence of snapshots
\[
\mathcal{G}=(G_1,\dots,G_T)
\]
over a common vertex set. An activity interval of a vertex \(v\) is a triple \((v,a,b)\) with \(a\le b\), and a \(k\)-activity \(\ell\)-timeline allows at most \(k\) activity intervals per vertex, each of length at most \(\ell\). 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 \(\vimw+k+\ell\) [2510.08124].

The paper gives explicit bounds for the partial versions. Timeline PVC is solvable in
\[
\big((k+1)(\ell+2)\big)^{2\cdot \vimw}\cdot (n+T)^{O(1)},
\]
and Timeline PDS is solvable in
\[
\big((k+1)(\ell+2)\big)^{2\cdot \vimw}\cdot (n+T)^{O(1)}.
\]
The same dynamic-programming framework yields FPT for Timeline VC as well, and the paper’s table records all four problems as FPT for \(\vimw+k+\ell\). The DP processes the bags \(F_i\) 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 [2510.08124].

The same paper also introduces a ranked variant
\[
\vimw[x](\mathcal{G}) := |F_{i_x}|,
\]
where \(F_{i_1},\dots,F_{i_T}\) are sorted by non-increasing size. This yields refinements: Timeline PVC is solvable in
\[
\big((k+1)(\ell+2)\big)^{4\cdot \vimw[x]}\cdot (n+T)^{O(1)}
\]
for
\[
x = k(\ell+1)+1,
\]
and Timeline DS is solvable in
\[
\big((k+1)(\ell+2)\big)^{4\cdot \vimw[x]}\cdot (n+T)^{O(1)}
\]
for
\[
x=\ell+2.
\]
For Timeline PDS, however, the reduction cannot be pushed in the same way: the paper shows NP-hardness already for parameterization by \(\vimw[2]+k+\ell\). This is one of several separations demonstrating that the effect of \(\vimw\) depends materially on the exact optimization objective [2510.08124].

A further separation concerns comparison with \(\imw\). For the parameter \(\imw+k+\ell\), Timeline Dominating Set is FPT, whereas Timeline Vertex Cover, Timeline PVC, and Timeline PDS are NP-hard even when \(\imw+k+\ell\) is constant. In this setting, \(\vimw\) is therefore the stronger parameter for the activity-interval formulations of covering and domination. More generally, the accumulated literature treats \(\vimw\) 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 [2510.08124].

Source: https://www.emergentmind.com/topics/vertex-interval-membership-width