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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

A temporal graph is a graph whose edges appear only at certain points in time. In these graphs, reachability among nodes relies on paths that traverse edges in chronological order (temporal paths). Unlike standard paths, temporal paths may not be composable, thus the reachability relation is not transitive and connected components (i.e., sets of pairwise temporally connected nodes) do not form equivalence classes, a fact with far-reaching consequences.
Recently, Casteigts et al. [FOCS 2021] proposed a natural temporal analog of the seminal Erdős-Rényi random graph model, based on the same parameters n and p. 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. Casteigts et al. showed that the well-known single threshold for connectivity in the Erdős-Rényi model fans out into multiple phase transitions for several distinct notions of reachability in the temporal setting.
The second most basic phenomenon studied by Erdős and Rényi in static (i.e., non-temporal) random graphs is the emergence of a giant connected component. However, the existence of a similar phase transition in the temporal model was left open until now. In this paper, we settle this question. We identify a sharp threshold at p = log n/n, where the size of the largest temporally connected component increases from o(n) to n-o(n) nodes. This transition occurs significantly later than in the static setting, where a giant component of size n-o(n) already exists for any p ∈ ω(1/n) (i.e., as soon as p is larger than a constant fraction of n). Interestingly, the threshold that we obtain holds for both open and closed connected components, i.e., components that allow, respectively forbid, their connecting paths to use external nodes - a distinction arising from the absence of transitivity.
We achieve these results by strengthening the tools from Casteigts et al. and developing new techniques that provide means to decouple dependencies between past and future events in temporal Erdős-Rényi graphs, which could be of general interest in future investigations of these objects.

Ruben Becker, Arnaud Casteigts, Pierluigi Crescenzi, Bojana Kodric, Malte Renken, Michael Raskin, and Viktor Zamaraev. Giant Components in Random Temporal Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 29:1-29:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{becker_et_al:LIPIcs.APPROX/RANDOM.2023.29, author = {Becker, Ruben and Casteigts, Arnaud and Crescenzi, Pierluigi and Kodric, Bojana and Renken, Malte and Raskin, Michael and Zamaraev, Viktor}, title = {{Giant Components in Random Temporal Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {29:1--29:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.29}, URN = {urn:nbn:de:0030-drops-188542}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.29}, annote = {Keywords: random temporal graph, Erd\H{o}s–R\'{e}nyi random graph, sharp threshold, temporal connectivity, temporal connected component, edge-ordered graph} }

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**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

A property of a graph G is robust if it remains unchanged in all connected spanning subgraphs of G. This form of robustness is motivated by networking contexts where some links eventually fail permanently, and the network keeps being used so long as it is connected. It is then natural to ask how certain properties of the network may be impacted as the network deteriorates. In this paper, we focus on two particular properties, which are the diameter, and pairwise distances among nodes. Surprisingly, the complexities of deciding whether these properties are robust are quite different: deciding the robustness of the diameter is coNP-complete, whereas deciding the robustness of the distance between two given nodes has a linear time complexity. This is counterintuitive, because the diameter consists of the maximum distance over all pairs of nodes, thus one may expect that the robustness of the diameter reduces to testing the robustness of pairwise distances. On the technical side, the difficulty of the diameter is established through a reduction from hamiltonian paths. The linear time algorithm for deciding robustness of the distance relies on a new characterization of two-terminal series-parallel graphs (TTSPs) in terms of excluded rooted minor, which may be of independent interest.

Arnaud Casteigts, Timothée Corsini, Hervé Hocquard, and Arnaud Labourel. Robustness of Distances and Diameter in a Fragile Network. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 9:1-9:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{casteigts_et_al:LIPIcs.SAND.2022.9, author = {Casteigts, Arnaud and Corsini, Timoth\'{e}e and Hocquard, Herv\'{e} and Labourel, Arnaud}, title = {{Robustness of Distances and Diameter in a Fragile Network}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.9}, URN = {urn:nbn:de:0030-drops-159514}, doi = {10.4230/LIPIcs.SAND.2022.9}, annote = {Keywords: Dynamic networks, Longest path, Series-parallel graphs, Rooted minors} }

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**Published in:** Dagstuhl Reports, Volume 11, Issue 3 (2021)

This report documents the program and the outcomes of Dagstuhl Seminar 121171 "Temporal Graphs: Structure, Algorithms, Applications". The seminar was organized around four core areas: models, concepts, classes; concrete algorithmic problems; distributed aspects; applications. Because of the ongoing pandemic crisis, the seminar had to be held fully online, with talk and open problems sessions focussing on afternoons. Besides 19 contributed talks and small-group discussions, there were lively open-problem sessions, and some of the problems and research directions proposed there are part of this document. Despite strongly missing the usual Dagstuhl atmosphere and personal interaction possibilities, the seminar helped to establish new contacts and to identify new research directions in a thriving research area between (algorithmic) graph theory and network science.

Arnaud Casteigts, Kitty Meeks, George B. Mertzios, and Rolf Niedermeier. Temporal Graphs: Structure, Algorithms, Applications (Dagstuhl Seminar 21171). In Dagstuhl Reports, Volume 11, Issue 3, pp. 16-46, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@Article{casteigts_et_al:DagRep.11.3.16, author = {Casteigts, Arnaud and Meeks, Kitty and Mertzios, George B. and Niedermeier, Rolf}, title = {{Temporal Graphs: Structure, Algorithms, Applications (Dagstuhl Seminar 21171)}}, pages = {16--46}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2021}, volume = {11}, number = {3}, editor = {Casteigts, Arnaud and Meeks, Kitty and Mertzios, George B. and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.11.3.16}, URN = {urn:nbn:de:0030-drops-146892}, doi = {10.4230/DagRep.11.3.16}, annote = {Keywords: algorithm engineering, complex network analysis, distributed computing, models and classes, parameterized complexity analysis} }

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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

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 feedback vertex number or the pathwidth of the underlying graph. The main question thus is 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.

Arnaud Casteigts, Anne-Sophie Himmel, Hendrik Molter, and Philipp Zschoche. Finding Temporal Paths Under Waiting Time Constraints. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 30:1-30:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{casteigts_et_al:LIPIcs.ISAAC.2020.30, author = {Casteigts, Arnaud and Himmel, Anne-Sophie and Molter, Hendrik and Zschoche, Philipp}, title = {{Finding Temporal Paths Under Waiting Time Constraints}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {30:1--30:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.30}, URN = {urn:nbn:de:0030-drops-133745}, doi = {10.4230/LIPIcs.ISAAC.2020.30}, annote = {Keywords: Temporal graphs, disease spreading, waiting-time policies, restless temporal paths, timed feedback vertex set, NP-hard problems, parameterized algorithms} }

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Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Let G=(V,E) be an undirected graph on n vertices and lambda:E -> 2^{N} a mapping that assigns to every edge a non-empty set of positive integer labels. These labels can be seen as discrete times when the edge is present. Such a labeled graph {G}=(G,lambda) is said to be 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 (STOC 2000) asked whether, given such a temporal graph, a sparse subset of edges can always be found whose labels suffice to preserve temporal connectivity - a temporal spanner. Axiotis and Fotakis (ICALP 2016) answered negatively by exhibiting a family of Theta(n^2)-dense temporal graphs which admit no temporal spanner of density o(n^2). The natural question is then whether sparse temporal spanners always exist in some classes of dense graphs.
In this paper, we answer this question affirmatively, by showing that if the underlying graph G is a complete graph, then one can always find temporal spanners of density O(n log n). The best known result for complete graphs so far was that spanners of density binom{n}{2}- floor[n/4] = O(n^2) always exist. Our result is the first positive answer as to the existence of o(n^2) sparse spanners in adversarial instances of temporal graphs since the original question by Kempe et al., focusing here on complete graphs. The proofs are constructive and directly adaptable as an algorithm.

Arnaud Casteigts, Joseph G. Peters, and Jason Schoeters. Temporal Cliques Admit Sparse Spanners. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 134:1-134:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{casteigts_et_al:LIPIcs.ICALP.2019.134, author = {Casteigts, Arnaud and Peters, Joseph G. and Schoeters, Jason}, title = {{Temporal Cliques Admit Sparse Spanners}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {134:1--134:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.134}, URN = {urn:nbn:de:0030-drops-107108}, doi = {10.4230/LIPIcs.ICALP.2019.134}, annote = {Keywords: Dynamic networks, Temporal graphs, Temporal connectivity, Sparse spanners} }

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**Published in:** LIPIcs, Volume 70, 20th International Conference on Principles of Distributed Systems (OPODIS 2016)

We consider networks of processes which interact with beeps. In the basic model defined by Cornejo and Kuhn, which we refer to as the BL variant, processes can choose in each round either to beep or to listen. Those who beep are unable to detect simultaneous beeps. Those who listen can only distinguish between silence and the presence of at least one beep. Stronger variants exist where the nodes can also detect collision while they are beeping (B_{cd}L) or listening (BL_{cd}), or both (B_{cd}L_{cd}). Beeping models are weak in essence and even simple tasks are difficult or unfeasible with them.
This paper starts with a discussion on generic building blocks (design patterns) which seem to occur frequently in the design of beeping algorithms. They include multi-slot phases: the fact of dividing the main loop into a number of specialised slots; exclusive beeps: having a single node beep at a time in a neighbourhood (within one or two hops); adaptive probability: increasing or decreasing the probability of beeping to produce more exclusive beeps; internal (resp. peripheral) collision detection: for detecting collision while beeping (resp. listening); and emulation of collision detection: for enabling this feature when it is not available as a primitive.
We then provide algorithms for a number of basic problems, including colouring, 2-hop colouring, degree computation, 2-hop MIS, and collision detection (in BL). Using the patterns, we formulate these algorithms in a rather concise and elegant way. Their analyses (in the full version) are more technical, e.g. one of them relies on a Martingale technique with non-independent variables; another improves that of the MIS algorithm (P. Jeavons et al.) by getting rid of a gigantic constant (the asymptotic order was already optimal).
Finally, we study the relative power of several variants of beeping models. In particular, we explain how every Las Vegas algorithm with collision detection can be converted, through emulation, into a Monte Carlo algorithm without, at the cost of a logarithmic slowdown. We prove that this slowdown is optimal up to a constant factor by giving a matching lower bound.

Arnaud Casteigts, Yves Métivier, John Michael Robson, and Akka Zemmari. Design Patterns in Beeping Algorithms. In 20th International Conference on Principles of Distributed Systems (OPODIS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 70, pp. 15:1-15:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{casteigts_et_al:LIPIcs.OPODIS.2016.15, author = {Casteigts, Arnaud and M\'{e}tivier, Yves and Robson, John Michael and Zemmari, Akka}, title = {{Design Patterns in Beeping Algorithms}}, booktitle = {20th International Conference on Principles of Distributed Systems (OPODIS 2016)}, pages = {15:1--15:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-031-6}, ISSN = {1868-8969}, year = {2017}, volume = {70}, editor = {Fatourou, Panagiota and Jim\'{e}nez, Ernesto and Pedone, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2016.15}, URN = {urn:nbn:de:0030-drops-70840}, doi = {10.4230/LIPIcs.OPODIS.2016.15}, annote = {Keywords: Beeping models, Design patterns, Collision detection, Colouring, 2-hop colouring, Degree computation, Emulation} }

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