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Classic Graph Problems Made Temporal – a Parameterized Complexity Analysis Foundations of computing Volume 13 Hendrik Molter Classic Graph Problems Made Temporal – A Parameterized Complexity Analysis Universitätsverlag der TU Berlin Hendrik Molter Classic Graph Problems Made Temporal – A Parameterized Complexity Analysis The scientifc series Foundations of computing of the Technische Universität Berlin is edited by: Prof. Dr. Stephan Kreutzer, Prof. Dr. Uwe Nestmann, Prof. Dr. Rolf Niedermeier Foundations of computing j 13 Hendrik Molter Classic Graph Problems Made Temporal – A Parameterized Complexity Analysis Universitätsverlag der TU Berlin Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografe; detailed bibliographic data are available on the internet at http://dnb.dnb.de. Universitätsverlag der TU Berlin, 2020 http://www.verlag.tu-berlin.de Fasanenstr. 88, 10623 Berlin Tel.: +49 (0)30 314 76131 / Fax: -76133 E-Mail: [email protected] Zugl.: Berlin, Techn. Univ., Diss., 2019 Gutachter: Prof. Dr. Rolf Niedermeier Gutachter: Prof. Dr. Thomas Erlebach (University of Leicester) Gutachter: Dr. Ralf Klasing (Université de Bordeaux) Die Arbeit wurde am 19. Dezember 2019 an der Fakultät IV unter Vorsitz von Prof. Dr. Benjamin Blankertz erfolgreich verteidigt. This work – except for quotes and where otherwise noted – is licensed under the Creatice Commons Licence CC BY 4.0 http://creativecommons.org/licenses/by/4.0/ Cover image: Hendrik Molter, 2011 CC BY 4.0 j http://creativecommons.org/licenses/by/4.0/ Print: docupoint GmbH Layout/Typesetting: Hendrik Molter ORCID iD Hendrik Molter: 0000-0002-4590-798X http://orcid.org/0000-0002-4590-798X ISBN 978-3-7983-3172-3 (print) ISBN 978-3-7983-3173-0 (online) ISSN 2199-5249 (print) ISSN 2199-5257 (online) Published online on the institutional repository of the Technische Universität Berlin: DOI 10.14279/depositonce-10551 http://dx.doi.org/10.14279/depositonce-10551 Zusammenfassung In dieser Dissertation untersuchen wir die parametrisierte Berechnungskomplexi- tät von sechs klassischen Graphproblemen, die in einen temporalen Kontext gestellt werden. Formaler ausgedrückt betrachten wir Probleme, die auf temporalen Graphen defniert sind, wobei temporale Graphen aus einer unveränderlichen Knotenmenge bestehen, zusammen mit einer Kantenmenge, die sich über einen diskreten Zeit- raum hinweg verändern darf. Temporale Graphen eignen sich besonders gut zum Modellieren dynamischer Daten und sind daher in Situationen von Bedeutung, in denen dynamische Veränderungen oder zeitabhängige Interaktionen eine wichtige Rolle spielen. Beispiele hierfür sind das Betrachten von Kommunikationsnetzwerken, sozialen Netzwerken oder Netzwerken, deren Interaktionen räumliche Annäherun- gen modellieren. Das wichtigste Auswahlkriterium für unsere Problemstellungen war, dass sie in Kontexten der Analyse dynamischer Daten wohlmotiviert sind. Da temporale Graphen mathematisch gesehen komplexer sind als statische Gra- phen, ist es vielleicht nicht sehr überraschend, dass alle Probleme, die wir in dieser Dissertation betrachten, NP-schwer sind. Wir konzentrieren uns auf die Entwicklung exakter Algorithmen, wobei wir versuchen FPT-Resultate zu erzielen oder durch spezialisierte Reduktionen zu zeigen, dass das betrachtete Problem NP-schwer auf sehr eingeschränkten Instanzen ist, oder berechnungsschwer im parametrisierten Sinne bezüglich möglichst „großer“ Parameter ist. Im Kontext temporaler Graphen betrachten wir in erster Linie Strukturparameter des unterliegenden Graphen, das heißt des Graphen, den man erhält, wenn man alle Zeitinformationen ignoriert. Allerdings studieren wir auch andere Parameter, zum Beispiel solche, die das Aus- maß der zeitlichen Veränderung eines temporalen Graphen messen. Im Folgenden geben wir einen kurzen Überblick über unsere Problemstellungen und wichtigsten Ergebnisse. Restless Temporal Paths. Ein Pfad in einem temporalen Graph muss Kausalität oder Zeit respektieren. Dies bedeutet, dass die Kanten, die von dem temporalen Pfad benutzt werden, nicht zu abnehmenden Zeitpunkten erscheinen dürfen. Wir untersuchen temporale Pfade, die darüber hinaus eine maximal erlaubte Wartezeit in allen Knoten haben. Unsere Hauptresultate sind, dass Pfade dieser Art zu fnden NP- schwer ist, sogar in sehr restriktiven Instanzen, und dass das Problem W[1]-schwer bezüglich der kreiskritischen Knotenzahl des unterliegenden Graphen ist. Temporal Separators. Ein temporaler Separator ist eine Knotenmenge, die, wenn sie aus einem temporalen Graph entfernt wird, alle temporalen Pfade zwischen zwei i ausgewählten Knoten zerstört. Wir erzielen hier zwei Hauptresultate: Auf der einen Seite untersuchen wir die Berechnungskomplexität des Findens von temporalen Separatoren in temporalen Einheitsintervallgraphen, einer Verallgemeinerung von Einheitsintervallgraphen im temporalen Kontext. Wir zeigen, dass das Problem auf temporalen Einheitsintervallgraphen NP-schwer ist, aber identifzieren eine weitere Einschränkung, die es erlaubt, das Problem in Polynomzeit zu lösen. Auf Letzterem aufbauend entwickeln wir einen FPT-Algorithmus, der eine „Distanz-zur-Trivialität“- Parametrisierung nutzt. Auf der anderen Seite zeigen wir, dass das Finden temporaler §P Separatoren, die alle ruhelosen temporalen Pfade zerstören, 2 -schwer ist. Temporal Matchings. Wir führen ein Modell für Matchings in temporalen Gra- phen ein, bei dem sich zwei Knoten „erholen“ müssen, nachdem sie zu einem bestimmten Zeitpunkt einander zugeordnet wurden. Das heißt, für eine gewisse Zeit können diese Knoten nicht wieder einander zugeordnet werden. Wir nutzen das Konzept von temporalen Kantengraphen, um zu zeigen, dass das Finden von temporalen Matchings NP-schwer ist, selbst wenn der unterliegende Graph ein Pfad ist. Temporal Coloring. Wir übertragen das klassische Knotenfärbungsproblem in den temporalen Rahmen. In unserem Modell muss jede Kante in jedem Zeitfenster einer bestimmten Größe mindestens einmal gültig gefärbt sein, also beide End- punkte eine andere Farbe haben. Wir zeigen, dass dieses Problem schon auf sehr eingeschränkten Instanzen NP-schwer ist – sogar für zwei Farben. Wir beschreiben einfache Exponentialzeitalgorithmen für dieses Problem. Eines unserer Hauptre- sultate ist, dass diese Algorithmen vermutlich nicht signifkant verbessert werden können. Temporal Cliques and s-Plexes. Wir stellen ein Modell für temporale s-Plexe vor, das eine kanonische Verallgemeinerung eines existierenden Modells für temporale Cliquen ist. Unser Hauptresultat ist ein FPT-Algorithmus, der alle maximalen tempo- ralen s-Plexe aufzählt, wobei wir eine temporale Variante von Degeneriertheit von Graphen als Parameter benutzen. Temporal Cluster Editing. Wir stellen ein Modell für Clustereditierung in tem- poralen Graphen vor, bei dem wir alle „Schichten“ eines temporalen Graphens in hinreichend ähnliche Cluster überführen wollen. Unsere Hauptergebnisse sind zum einen ein FPT-Algorithmus bezüglich des Parameters „Anzahl der Editierungen plus Ähnlichkeit der Cluster“. Zum anderen geben wir eine effziente Vorverarbeitungs- methode an, welche die Größe der Eingabeinstanz beweisbar so reduziert, dass sie unabhängig von der Anzahl der Knoten der ursprünglichen Instanz ist. ii Abstract This thesis investigates the parameterized computational complexity of six classic graph problems lifted to a temporal setting. More specifcally, we consider problems defned on temporal graphs, that is, a graph where the edge set may change over a discrete time interval, while the vertex set remains unchanged. Temporal graphs are well-suited to model dynamic data and hence they are naturally motivated in contexts where dynamic changes or time-dependent interactions play an important role, such as, for example, communication networks, social networks, or physical proximity networks. The most important selection criteria for our problems was that they are well-motivated in the context of dynamic data analysis. Since temporal graphs are mathematically more complex than static graphs, it is maybe not surprising that all problems we consider in this thesis are NP-hard. We focus on the development of exact algorithms, where our goal is to obtain fxed- parameter tractability results, and refned computational hardness reductions that either show NP-hardness for very restricted input instances or parameterized hard- ness with respect to “large” parameters. In the context of temporal graphs, we mostly consider structural parameters of the underlying graph, that is, the graph obtained by ignoring all time information. However, we also consider parameters of other types, such as ones trying to measure how fast the temporal graph changes over time. In the following we briefy discuss the problem setting and the main results. Restless Temporal Paths. A path in a temporal graph has to respect causality, or time, which means that the edges used by a temporal path have to appear at non- decreasing times. We investigate temporal paths that additionally have a maximum waiting time in every vertex of the temporal graph. Our main contributions are establishing NP-hardness for the problem of fnding restless temporal paths even in very restricted cases, and showing W[1]-hardness with respect to the feedback vertex number of the underlying graph. Temporal Separators. A temporal separator is a vertex set that, when removed from the temporal graph, destroys all temporal paths between two dedicated ver- tices. Our contribution here is twofold: Firstly, we investigate the computational
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