arXiv subjects

Arnaud Casteigts

Publications and source records attributed to Arnaud Casteigts.

At least 19 recordsLinked to original sources

arXiv2026

Structural Lemmas on Temporal Connectivity

This paper presents several lemmas on the structure of temporal connectivity in temporal graphs. Some of these lemmas are adapted from the literature on gossip from the 70-90's and reformulated in the context of temporal graph theory. Many others are original. For ease of presentation and for avoiding lengthy case distinctions, we formulate all the lemmas in the setting of simple and proper (a.k.a, happy) temporal graphs, discussing occasionally their generality beyond this setting.

cs.DM↗

arXiv2026

On the Hardness of Finding Temporally Connected Subgraphs of Any Size

Temporal graphs are graphs whose edges are present only at certain points in time. Reachability in these graphs is defined via temporal paths, in which edges are traversed in chronological order. A temporal graph is temporally connected (or TC) if every ordered pair of vertices is connected by a temporal path. When the graph itself is not TC, a natural question is whether it admits a TC subgraph (a.k.a.~closed temporal component) of a given size $k$. This question was one of the earliest and most studied in the field, shown to be NP-hard by Bhadra and Ferreira in 2003. We strengthen this result dramatically, showing that deciding if a temporal graph admits a TC subgraph \emph{of any size} (beyond the trivial case of a single vertex in the directed and a single edge in the undirected case) is already NP-hard. Our result holds for all standard temporal graph settings, answering a series of open questions in the field and strengthening several existing results. This sharply separates closed components from open ones (where temporal paths can travel outside the component), for which the analogous problem is trivially solvable in polynomial time. More precisely, our reductions imply that the size of the largest TC subgraph cannot even be approximated within a factor of~$(1-ε)n$ in directed graphs, and within a factor of~$(1-ε)\frac{n}{2}$ in undirected graphs. They also complete the complexity landscape for TC subgraphs of size exactly $k$ when parameterized by $k$ (answering the missing ``non-strict'' case). Our results also have structural implications. In particular, they imply that there exist arbitrarily large TC graphs without nontrivial TC subgraphs, and that there exist TC graphs of arbitrary girth, both facts being of independent interest.

cs.CC↗

arXiv2026

Realization of Temporally Connected Graphs Based on Degree Sequences

Given an undirected graph $G$, the problem of deciding whether $G$ admits a simple and proper time-labeling that makes it temporally connected is known to be NP-hard (Göbel et al., 1991). In this article, we relax this problem and ask whether a given degree sequence can be realized as a temporally connected graph. Our main results are a complete characterization of the feasible cases, and a recognition algorithm that runs in $O(n)$ time for graphical degree sequences (realized as simple temporal graphs) and in $O(n+m)$ time for multigraphical degree sequences (realized as non-simple temporal graphs, where the number of time labels on an edge corresponds to the multiplicity of the edge in the multigraph). In fact, these algorithms can be made constructive at essentially no cost. Namely, we give a constructive $O(n+m)$ time algorithm that outputs, for a given (multi)graphical degree sequence $\mathbf{d}$, a temporally connected graph whose underlying (multi)graph is a realization of $\mathbf{d}$, if one exists.

cs.DS↗

arXiv2026

Minimum Temporal Spanners in Happy Graphs

Temporal graphs have edge sets that change over discrete time steps. Such graphs are temporally connected (TC) if all pairs of vertices can reach each other using paths that traverse the edges in a time-respecting way (temporal paths). Given a TC temporal graph it, a natural question is to find a minimum spanning subgraph of it that preserves temporal connectivity. These structures, known as temporal spanners, are fundamental and their properties (especially size) have been studied thoroughly in the past decade. In particular, the problem of minimizing the size of a temporal spanner is known to be hard. However, the existing results establish hardness for several incomparable settings and versions of the problem. In this article, we unify and strengthen these results by showing that this problem is NP-hard even on temporal graphs that are simple and proper (also known as "happy"), i.e., where every edge appears only one time, and a vertex cannot be incident to several edges simultaneously. Proving hardness in this extremely restricted setting implies, at once, that the problem is NP-hard for all the previously considered settings and versions of the problem, resolving Open Question 4 in [Casteigts et al. TCS, 2024]. We also initiate the parameterized study of this problem, showing that in the happy setting, the problem can be solved in polynomial time if the underlying graph has a constant-size vertex cover, this result being actually the first positive result on temporal spanners in general. We also show that in the non-happy setting, the problem is W[1]-hard when parameterized by the feedback vertex number of the underlying graph.

cs.DS↗

arXiv2026

In search of the lost tree: Hardness and relaxation of spanning trees in temporal graphs

A temporal graph is a graph whose edges appear at certain points in time. These graphs are temporally connected (in class TC) if all vertices can reach each other by temporal paths (traversing the edges in chronological order). Reachability based on temporal paths is not transitive, with important consequences. For instance, TC graphs do not always admit TC spanning trees. In this paper, we show that deciding if a given temporal graph admits a TC spanning tree is actually NP-complete. Then, we explore possible relaxations. A key feature of TC spanning trees is to support reachability along the same paths in both directions. We show that this property is not equivalent to TC spanning trees, it is more general and it can be tested in polynomial time. Still, minimizing the size of a spanner preserving this property -- a bidirectional spanner -- is \textsf{NP}-hard even more generally than TC spanning tree, including the setting of simple temporal graphs. Along the way, we show that deciding the existence of TC spanning tree is FPT when parameterized by the feedback edge set number (fes) of the underlying graph, and deciding bidirectional spanners of size $k$ is FPT when parameterized by fes + $\ell$ (the maximum number of labels per edge). On the structural side, we show that TC trees always admit a pivot vertex or a pivot edge -- reachable by all vertices by a certain time and able to reach all vertices afterward -- a fact that may be of independent interest.

cs.DM↗

arXiv2026

Optimal Trajectories in Discrete Space with Acceleration Constraints

In the racetrack acceleration model, proposed by Martin Gardner in 1973, each step consists of changing the position of the vehicle by a vector in $\mathbb{Z}^2$, with the constraints that two consecutive vectors differ by at most one unit in each dimension. We investigate three problems related to this model in arbitrary dimension in open space (no obstacles), where a configuration of the vehicle consists of its current position and the last-used vector. The three problems are the following. In Branching Cost (BC), given two configurations, the goal is to compute the minimum number of intermediate configurations (length of a trajectory) between the two configurations. Branching Trajectory (BT) has the same input and asks for a description of the corresponding trajectory. Multipoint Trajectory (MT) asks for an optimal trajectory that visits given points $p_1,\dots,p_n$ in a prescribed order, starting and ending with zero-speed configurations.\\ We revisit known approaches to solve BC in 2D, showing that this problem can be solved in constant time in any fixed number of dimensions $d$ (more generally, in $O(d \log d)$ time). We show that BT can also be solved in constant time for any fixed $d$, despite the fact that the length of the trajectory is not constant, by leveraging the fact that there always exists \emph{one} optimal trajectory compactly represented by $O(1)$ intermediate configurations. For MT, we collect theoretical and experimental evidence that the speed cannot be trivially bounded; local decisions may be impacted by points that are arbitrarily far in the visit order; and an optimal trajectory may require significant excursions out of the convex hull of the points. We still establish conservative speed bounds that a natural dynamic programming (DP) algorithm can exploit to solve reasonably large instances efficiently.

cs.CG↗

arXiv2025

Vector TSP: A Traveling Salesperson Problem with Racetrack-like Acceleration Constraints

We study a new version of the Traveling Salesperson Problem, called \VectorTSP, where the traveler is subject to discrete acceleration constraints, as defined in the paper-and-pencil game Racetrack (also known as Vector Racer). In this model, the degrees of freedom at a certain point in time depends on the current velocity, and the speed is not limited. The paper introduces this problem and initiates its study, discussing also the main differences with existing versions of TSP. Not surprisingly, the problem turns out to be NP-hard. A key feature of \VectorTSP is that it deals with acceleration in a discrete, combinatorial way, making the problem more amenable to algorithmic investigation. The problem involves two layers of trajectory planning: (1) the order in which cities are visited, and (2) the physical trajectory realizing such a visit, both interacting with each other. This interaction is formalized as an interactive protocol between a high-level tour algorithm and a trajectory oracle, the former calling the latter repeatedly. We present an exact implementation of the trajectory oracle, adapting the A* algorithm for paths over multiple checkpoints whose ordering is \emph{given} (this algorithm being possibly of independent interest). To motivate the problem further, we perform experiments showing that the naive approach consisting of solving the instance as an \EuclideanTSP first, then optimizing the trajectory of the resulting tour, is typically suboptimal and outperformed by simple (but dedicated) heuristics.

cs.DS↗

arXiv2025

Dismountability in Temporal Cliques Revisited

A temporal graph is a graph whose edges are available only at certain points in time. It is temporally connected if the nodes can reach each other by paths that traverse the edges chronologically (temporal paths). In general, temporal graphs do not always admit small subsets of edges that preserve connectivity (temporal spanners). In the case of temporal cliques, spanners of size $O(n\log n)$ are guaranteed. The original proof by Casteigts et al. [ICALP 2019] combines a number of techniques, one of which is dismountability. In a recent work, Angrick et al. [ESA 2024] simplified the proof and showed, among other things, that a one-sided version of dismountability can be used to replace the second part of the proof. In this paper, we revisit the dismountability principle. We characterizing the structure that a temporal clique has if it is not 1-hop dismountable, then not {1,2}-hop dismountable, and finally not {1,2,3}-hop dismountable. It turns out that if a clique is k-hop dismountable for any other k, then it must also be {1,2,3}-hop dismountable. Interestingly, excluding only 1-hop and 2-hop dismountability is already sufficient for reducing the spanner problem from cliques to bi-cliques. Put together with the strategy of Angrick et al., the entire $O(n \log n)$ result can now be recovered using only dismountability. An interesting by-product of our analysis is that any minimal counter-example to the existence of $4n$ spanners must satisfy the properties of non {1,2,3}-hop dismountable cliques. In the second part, we discuss connections between dismountability and pivotability. We show that recursively k-hop dismountable cliques are pivotable (and thus admits $2n$ spanners, whatever k). We define a family of labelings (called full-range) which force both dismountability and pivotability and that gives some evidence that large lifetimes could be exploited more generally.

cs.DM↗

arXiv2024

Simple, strict, proper, happy: A study of reachability in temporal graphs

Dynamic networks are a complex subject. Not only do they inherit the complexity of static networks (as a particular case); they are also sensitive to definitional subtleties that are a frequent source of confusion and incomparability of results in the literature. In this paper, we take a step back and examine three such aspects in more details, exploring their impact in a systematic way; namely, whether the temporal paths are required to be \emph{strict} (i.e., the times along a path must increasing, not just be non-decreasing), whether the time labeling is \emph{proper} (two adjacent edges cannot be present at the same time) and whether the time labeling is \emph{simple} (an edge can have only one presence time). In particular, we investigate how different combinations of these features impact the expressivity of the graph in terms of reachability. Our results imply a hierarchy of expressivity for the resulting settings, shedding light on the loss of generality that one is making when considering either combination. Some settings are more general than expected; in particular, proper temporal graphs turn out to be as expressive as general temporal graphs where non-strict paths are allowed. Also, we show that the simplest setting, that of \emph{happy} temporal graphs (i.e., both proper and simple) remains expressive enough to emulate the reachability of general temporal graphs in a certain (restricted but useful) sense. Furthermore, this setting is advocated as a target of choice for proving negative results. We illustrates this by strengthening two known results to happy graphs (namely, the inexistence of sparse spanners, and the hardness of computing temporal components). Overall, we hope that this article can be seen as a guide for choosing between different settings of temporal graphs, while being aware of the way these choices affect generality.

cs.DM↗

arXiv2024

Distance to Transitivity: New Parameters for Taming Reachability in Temporal Graphs

A temporal graph is a graph whose edges only appear at certain points in time. Reachability in these graphs is defined in terms of paths that traverse the edges in chronological order (temporal paths). This form of reachability is neither symmetric nor transitive, the latter having important consequences on the computational complexity of even basic questions, such as computing temporal connected components. In this paper, we introduce several parameters that capture how far a temporal graph $\mathcal{G}$ is from being transitive, namely, \emph{vertex-deletion distance to transitivity} and \emph{arc-modification distance to transitivity}, both being applied to the reachability graph of $\mathcal{G}$. We illustrate the impact of these parameters on the temporal connected component problem, obtaining several tractability results in terms of fixed-parameter tractability and polynomial kernels. Significantly, these results are obtained without restrictions of the underlying graph, the snapshots, or the lifetime of the input graph. As such, our results isolate the impact of non-transitivity and confirm the key role that it plays in the hardness of temporal graph problems.

cs.CC↗

arXiv2024

Freeze-Tag in $L_1$ has Wake-up Time Five

The Freeze-Tag Problem, introduced in Arkin et al. (SODA'02) consists of waking up a swarm of $n$ robots, starting from a single active robot. In the basic geometric version, every robot is given coordinates in the plane. As soon as a robot is awakened, it can move towards inactive robots to wake them up. The goal is to minimize the wake-up time of the last robot, the makespan. Despite significant progress on the computational complexity of this problem and on approximation algorithms, the characterization of exact bounds on the makespan remains one of the main open questions. In this paper, we settle this question for the $\ell_1$-norm, showing that a makespan of at most $5r$ can always be achieved, where $r$ is the maximum distance between the initial active robot and any sleeping robot. Moreover, a schedule achieving a makespan of at most $5r$ can be computed in optimal time $O(n)$. Both bounds, the time and the makespan are optimal. This implies a new upper bound of $5\sqrt{2}r \approx 7.07r$ on the makespan in the $\ell_2$-norm, improving the best known bound so far $(5+2\sqrt{2}+\sqrt{5})r \approx 10.06r$.

cs.DS↗

arXiv2023

Sharp Thresholds in Random Simple Temporal Graphs

A graph whose edges only appear at certain points in time is called a temporal graph (among other names). Such a graph is temporally connected if each ordered pair of vertices is connected by a path which traverses edges in chronological order (i.e., a temporal path). In this paper, we consider a simple model of random temporal graph, obtained from an Erdős-Rényi random graph $G~G_{n,p}$ by considering a random permutation $π$ of the edges and interpreting the ranks in $π$ as presence times. Temporal reachability in this model exhibits a surprisingly regular sequence of thresholds. In particular, we show that at $p=\log n/n$ any fixed pair of vertices can a.a.s. reach each other; at $2\log n/n$ at least one vertex (and in fact, any fixed vertex) can a.a.s. reach all others; and at $3\log n/n$ all the vertices can a.a.s. reach each other, i.e., the graph is temporally connected. Furthermore, the graph admits a temporal spanner of size $2n+o(n)$ as soon as it becomes temporally connected, which is nearly optimal as $2n-4$ is a lower bound. This result is significant because temporal graphs do not admit spanners of size $O(n)$ in general (Kempe et al, STOC 2000). In fact, they do not even admit spanners of size $o(n^2)$ (Axiotis et al, ICALP 2016). Thus, our result implies that the obstructions found in these works, and more generally, all non-negligible obstructions, must be statistically insignificant: nearly optimal spanners always exist in random temporal graphs. All the above thresholds are sharp. Carrying the study of temporal spanners further, we show that pivotal spanners -- i.e., spanners of size $2n-2$ made of two spanning trees glued at a single vertex (one descending in time, the other ascending subsequently) -- exist a.a.s. at $4\log n/n$, this threshold being also sharp. Finally, we show that optimal spanners (of size $2n-4$) also exist a.a.s. at $p = 4\log n/n$.

cs.DM↗

arXiv2023

Giant Components in Random Temporal Graphs

A temporal graph is a graph whose edges appear only at certain points in time. Recently, the second and the last three authors proposed a natural temporal analog of the Erdős-Rényi random graph model. The proposed model is obtained by randomly permuting the edges of an Erdős-Rényi random graph and interpreting this permutation as an ordering of presence times. It was shown that the connectivity threshold in the Erdős-Rényi model fans out into multiple phase transitions for several distinct notions of reachability in the temporal setting. In the present paper, we identify a sharp threshold for the emergence of a giant temporally connected component. We show that at $p = \log n/n$ the size of the largest temporally connected component increases from $o(n)$ to~$n-o(n)$. This threshold holds for both open and closed connected components, i.e. components that allow, respectively forbid, their connecting paths to use external nodes.

cs.DM↗

arXiv2021

The Computational Complexity of Finding Temporal Paths under Waiting Time Constraints

Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration $Δ$, referred to as $Δ$-restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the "restless variant" of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the distance to disjoint path of the underlying graph, which implies W[1]-hardness for many other parameters like feedback vertex number and pathwidth. A natural question is thus whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

cs.DM↗

arXiv2021

Temporal Cliques Admit Sparse Spanners

Let $G=(V,E)$ be an undirected graph on $n$ vertices and $λ:E\to 2^{\mathbb{N}}$ a mapping that assigns to every edge a non-empty set of integer labels (times). Such a graph is {\em temporally connected} if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, Kempe, Kleinberg, and Kumar \cite{KKK02} asked whether, given such a temporal graph, a {\em sparse} subset of edges always exists whose labels suffice to preserve temporal connectivity -- a {\em temporal spanner}. Axiotis and Fotakis \cite{AF16} answered negatively by exhibiting a family of $Θ(n^2)$-dense temporal graphs which admit no temporal spanner of density $o(n^2)$. In this paper, we give the first positive answer as to the existence of $o(n^2)$-sparse spanners in a dense class of temporal graphs, by showing (constructively) that if $G$ is a complete graph, then one can always find a temporal spanner of density $O(n \log n)$.

cs.DM↗

arXiv2021

A Dynamic Data Structure for Temporal Reachability with Unsorted Contact Insertions

Temporal graphs represent interactions between entities over the time. These interactions may be direct (a contact between two nodes at some time instant), or indirect, through sequences of contacts called temporal paths (journeys). Deciding whether an entity can reach another through a journey is useful for various applications in communication networks and epidemiology, among other fields. In this paper, we present a data structure which maintains temporal reachability information under the addition of new contacts (i.e., triplets $(u,v,t)$ indicating that node $u$ and node $v$ interacted at time $t$). In contrast to previous works, the contacts can be inserted in arbitrary order -- in particular, non-chronologically -- which corresponds to systems where the information is collected a posteriori (e.g. when trying to reconstruct contamination chains among people). The main component of our data structure is a generalization of transitive closure called timed transitive closure (TTC), which allows us to maintain reachability information relative to all nested time intervals, without storing all these intervals, nor the journeys themselves. TTCs are of independent interest and we study a number of their general properties. Let $n$ be the number of nodes and $τ$ be the number of timestamps in the lifetime of the temporal graph. Our data structure answers reachability queries regarding the existence of a journey from a given node to another within given time interval in time $O(\logτ)$; it has an amortized insertion time of $O(n^2\logτ)$; and it can reconstruct a valid journey that witnesses reachability in time $O(k\logτ)$, where $k<n$ is the maximum number of edges of this journey. Finally, the space complexity of our reachability data structure is $O(n^2τ)$, which remains within the worst-case size of the temporal graph itself.

cs.DS↗

arXiv2021

Revisiting the Role of Coverings in Anonymous Networks: Spanning Tree Construction and Topology Recognition

This paper revisits two classical distributed problems in anonymous networks, namely spanning tree construction and topology recognition, from the point of view of graph covering theory. For both problems, we characterize necessary and sufficient conditions on the communication graph in terms of directed symmetric coverings. These characterizations answer along-standing open question posed by Yamashita and Kameda [YK96], and shed new light on the connection between coverings and the concepts of views and quotient graphs developed by the same authors. Characterizing conditions in terms of coverings is significant because it connects the field with a vast body of classical literature in graph theory and algebraic topology. In particular, it gives access to powerful tools such as Reidemeister's theorem and Mazurkiewicz's algorithm. Combined together, these tools allow us to present elegant proofs of otherwise intricate results, and their constructive nature makes them effectively usable in the algorithms. This paper also gives us the opportunity to present the field of covering theory in a pedagogical way, with a focus on the two aforementioned tools, whose potential impact goes beyond the specific problems considered in this work.

cs.DC↗

arXiv2019

JBotSim, a Tool for Fast Prototyping of Distributed Algorithms in Dynamic Networks

JBotSim is a java library that offers basic primitives for prototyping, running, and visualizing distributed algorithms in dynamic networks. With JBotSim, one can implement an idea in minutes and interact with it ({\it e.g. }, add, move, or delete nodes) while it is running. JBotSim is well suited to prepare live demonstrations of your algorithms to colleagues or students; it can also be used to evaluate performance at the algorithmic level (number of messages, number of rounds, etc.). Unlike most tools, JBotSim is not an integrated environment. It is a lightweight library to be used in your program. In this paper, we present an overview of its distinctive features and architecture.

cs.MS↗