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Documents authored by Spooner, Jakob T.

Document
DOI: 10.4230/LIPIcs.SAND.2022.15
Parameterized Temporal Exploration Problems

Authors: Thomas Erlebach and Jakob T. Spooner

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

Abstract
In this paper we study the fixed-parameter tractability of the problem of deciding whether a given temporal graph 𝒢 admits a temporal walk that visits all vertices (temporal exploration) or, in some problem variants, a certain subset of the vertices. Formally, a temporal graph is a sequence 𝒢 = ⟨ G₁,..., G_L⟩ of graphs with V(G_t) = V(G) and E(G_t) ⊆ E(G) for all t ∈ [L] and some underlying graph G, and a temporal walk is a time-respecting sequence of edge-traversals. For the strict variant, in which edges must be traversed in strictly increasing timesteps, we give FPT algorithms for the problem of finding a temporal walk that visits a given set X of vertices, parameterized by |X|, and for the problem of finding a temporal walk that visits at least k distinct vertices in V, parameterized by k. For the non-strict variant, in which an arbitrary number of edges can be traversed in each timestep, we parameterize by the lifetime L of the input graph and provide an FPT algorithm for the temporal exploration problem. We also give additional FPT or W[2]-hardness results for further problem variants.
Cite as

Thomas Erlebach and Jakob T. Spooner. Parameterized Temporal Exploration Problems. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{erlebach_et_al:LIPIcs.SAND.2022.15,
  author =	{Erlebach, Thomas and Spooner, Jakob T.},
  title =	{{Parameterized Temporal Exploration Problems}},
  booktitle =	{1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)},
  pages =	{15:1--15:17},
  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.2022.15},
  URN =		{urn:nbn:de:0030-drops-159570},
  doi =		{10.4230/LIPIcs.SAND.2022.15},
  annote =	{Keywords: Temporal graphs, fixed-parameter tractability, parameterized complexity}
}
Document
Track C: Foundations of Networks and Multi-Agent Systems: Models, Algorithms and Information Management
DOI: 10.4230/LIPIcs.ICALP.2019.141
Two Moves per Time Step Make a Difference

Authors: Thomas Erlebach, Frank Kammer, Kelin Luo, Andrej Sajenko, and Jakob T. Spooner

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

Abstract
A temporal graph is a graph whose edge set can change over time. We only require that the edge set in each time step forms a connected graph. The temporal exploration problem asks for a temporal walk that starts at a given vertex, moves over at most one edge in each time step, visits all vertices, and reaches the last unvisited vertex as early as possible. We show in this paper that every temporal graph with n vertices can be explored in O(n^{1.75}) time steps provided that either the degree of the graph is bounded in each step or the temporal walk is allowed to make two moves per step. This result is interesting because it breaks the lower bound of Omega(n^2) steps that holds for the worst-case exploration time if only one move per time step is allowed and the graph in each step can have arbitrary degree. We complement this main result by a logarithmic inapproximability result and a proof that for sparse temporal graphs (i.e., temporal graphs with O(n) edges in the underlying graph) making O(1) moves per time step can improve the worst-case exploration time at most by a constant factor.
Cite as

Thomas Erlebach, Frank Kammer, Kelin Luo, Andrej Sajenko, and Jakob T. Spooner. Two Moves per Time Step Make a Difference. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 141:1-141:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{erlebach_et_al:LIPIcs.ICALP.2019.141,
  author =	{Erlebach, Thomas and Kammer, Frank and Luo, Kelin and Sajenko, Andrej and Spooner, Jakob T.},
  title =	{{Two Moves per Time Step Make a Difference}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{141:1--141:14},
  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.2019.141},
  URN =		{urn:nbn:de:0030-drops-107176},
  doi =		{10.4230/LIPIcs.ICALP.2019.141},
  annote =	{Keywords: Temporal Graph Exploration, Algorithmic Graph Theory, NP-Complete Problem}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.36
Faster Exploration of Degree-Bounded Temporal Graphs

Authors: Thomas Erlebach and Jakob T. Spooner

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
A temporal graph can be viewed as a sequence of static graphs indexed by discrete time steps. The vertex set of each graph in the sequence remains the same; however, the edge sets are allowed to differ. A natural problem on temporal graphs is the Temporal Exploration problem (TEXP): given, as input, a temporal graph G of order n, we are tasked with computing an exploration schedule (i.e., a temporal walk that visits all vertices in G), such that the time step at which the walk arrives at the last unvisited vertex is minimised (we refer to this time step as the arrival time). It can be easily shown that general temporal graphs admit exploration schedules with arrival time no greater than O(n^2). Moreover, it has been shown previously that there exists an infinite family of temporal graphs for which any exploration schedule has arrival time Omega(n^2), making these bounds tight for general TEXP instances. We consider restricted instances of TEXP, in which the temporal graph given as input is, in every time step, of maximum degree d; we show an O(n^2/log n) bound on the arrival time when d is constant, and an O(d log d * n^2/log n) bound when d is given as some function of n.
Cite as

Thomas Erlebach and Jakob T. Spooner. Faster Exploration of Degree-Bounded Temporal Graphs. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{erlebach_et_al:LIPIcs.MFCS.2018.36,
  author =	{Erlebach, Thomas and Spooner, Jakob T.},
  title =	{{Faster Exploration of Degree-Bounded Temporal Graphs}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{36:1--36:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.36},
  URN =		{urn:nbn:de:0030-drops-96181},
  doi =		{10.4230/LIPIcs.MFCS.2018.36},
  annote =	{Keywords: temporal graph exploration, algorithmic graph theory, NP-complete problem}
}
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