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Parametrised Enumeration Arne Meier Februar 2020 color guide (the dark blue is the same with the Gottfried Wilhelm Leibnizuniversity Universität logo) Hannover c: 100 m: 70 Institut für y: 0 k: 0 Theoretische c: 35 m: 0 y: 10 Informatik k: 0 c: 0 m: 0 y: 0 k: 35 Fakultät für Elektrotechnik und Informatik Institut für Theoretische Informatik Fachgebiet TheoretischeWednesday, February 18, 15 Informatik Habilitationsschrift Parametrised Enumeration Arne Meier geboren am 6. Mai 1982 in Hannover Februar 2020 Gutachter Till Tantau Institut für Theoretische Informatik Universität zu Lübeck Gutachter Stefan Woltran Institut für Logic and Computation 192-02 Technische Universität Wien Arne Meier Parametrised Enumeration Habilitationsschrift, Datum der Annahme: 20.11.2019 Gutachter: Till Tantau, Stefan Woltran Gottfried Wilhelm Leibniz Universität Hannover Fachgebiet Theoretische Informatik Institut für Theoretische Informatik Fakultät für Elektrotechnik und Informatik Appelstrasse 4 30167 Hannover Dieses Werk ist lizenziert unter einer Creative Commons “Namensnennung-Nicht kommerziell 3.0 Deutschland” Lizenz. Für Julia, Jonas Heinrich und Leonie Anna. Ihr seid mein größtes Glück auf Erden. Danke für eure Geduld, euer Verständnis und eure Unterstützung. Euer Rückhalt bedeutet mir sehr viel. ♥ If I had an hour to solve a problem, I’d spend 55 minutes thinking about the problem and 5 min- “ utes thinking about solutions. ” — Albert Einstein Abstract In this thesis, we develop a framework of parametrised enumeration complexity. At first, we provide the reader with preliminary notions such as machine models and complexity classes besides proving them to be well-chosen. Then, we study the interplay and the landscape of these classes and present connections to classical enumeration classes. Afterwards, we translate the fundamental methods of kerneli- sation and self-reducibility into equivalent techniques in the setting of parametrised enumeration. Subsequently, we illustrate the introduced classes by investigating the parametrised enumeration complexity of Max-Ones-SAT and strong backdoor sets as well as sharpen the first result by presenting a dichotomy theorem for Max-Ones-SAT. After this, we extend the definitions of parametrised enumeration algorithms by allowing orders on the solution space. In this context, we study the relations “order by size” and “lexicographic order” for graph modification problems and observe a trade-off between enumeration delay and space requirements of enumeration algorithms. These results then yield an enumeration technique for generalised modi- fication problems that is illustrated by applying this method to the problems closest string, weak and strong backdoor sets, and weighted satisfiability. Eventually, we consider the enumeration of satisfying teams of formulas of poor man’s propositional dependence logic. There, we present an enumeration algorithm with FPT delay and exponential space which is one of the first enumeration complexity result of a problem in a team logic. Finally, we show how this algorithm can be modified such that only polynomial space is required, however, by increasing the delay to incremental FPT time. Keywords: Parametrised Complexity, Enumeration Complexity, Parametrised Enu- meration vi Zusammenfassung In diesem Werk begründen wir die Theorie der parametrisierten Enumeration, präsentieren die grundlegenden Definitionen und prüfen ihre Sinnhaftigkeit. Im nächsten Schritt, untersuchen wir das Zusammenspiel der eingeführten Komplex- itätsklassen und zeigen Verbindungen zur klassischen Enumerationskomplexität auf. Anschließend übertragen wir die zwei fundamentalen Techniken der Kernelisierung und Selbstreduzierbarkeit in Entsprechungen in dem Gebiet der parametrisierten Enumeration. Schließlich untersuchen wir das Problem Max-Ones-SAT und das Problem der Aufzählung starker Backdoor-Mengen als typische Probleme in diesen Klassen. Die vorherigen Resultate zu Max-Ones-SAT werden anschließend in einem Dichotomie-Satz vervollständigt. Im nächsten Abschnitt erweitern wir die neuen Def- initionen auf Ordnungen (auf dem Lösungsraum) und erforschen insbesondere die zwei Relationen „Größenordnung“ und „lexikographische Reihenfolge“ im Kontext von Graphen-Modifikationsproblemen. Hierbei scheint es, als müsste man zwischen Delay und Speicheranforderungen von Aufzählungsalgorithmen abwägen, wobei dies jedoch nicht abschließend gelöst werden kann. Aus den vorherigen Überlegungen wird schließlich ein generisches Enumerationsverfahren für allgemeine Modifika- tionsprobleme entwickelt und anhand der Probleme Closest String, schwacher und starker Backdoor-Mengen sowie gewichteter Erfüllbarkeit veranschaulicht. Im let- zten Abschnitt betrachten wir die parametrisierte Enumerationskomplexität von Erfüllbarkeitsproblemen im Bereich der Poor Man’s Propositional Dependence Logic und stellen einen Aufzählungsalgorithmus mit FPT Delay vor, der mit exponentiellem Platz arbeitet. Dies ist einer der ersten Aufzählungsalgorithmen im Bereich der Team- logiken. Abschließend zeigen wir, wie dieser Algorithmus so modifiziert werden kann, dass nur polynomieller Speicherplatz benötigt wird, bezahlen jedoch diese Einsparung mit einem Anstieg des Delays auf inkrementelle FPT Zeit (IncFPT). Schlagwörter: Parametrisierte Algorithmen, Aufzählungskomplexität, Parametri- sierte Aufzählungsalgorithmen vii Acknowledgement At first, I wish to deeply thank my family for always being sympathetic and support- ive. Secondly, I want to express my gratitude to all my coauthors of publications which have been incorporated in this thesis. In particular, I show appreciation to Nadia Creignou, Raïda Ktari, Julian-Steffen Müller, Fréderic Olive, Johannes Schmidt (thanks, Johannes, for the discussion on Theorem 3.2), and Heribert Vollmer who have been working with me at the initial steps into this exciting new field. Fur- thermore, I thank Maurice Chandoo and Anselm Haak for their valuable comments on various parts of this thesis. Particularly, I would like to thank Martin Lück for his feedback on Section 3.2, and Johannes Fichte for proof reading Chapter 1. I want to give thanks to Christian Reinbold for discussions on Chapter 6. Eventually, I appreciate Anselm’s feedback to my teaching demonstration and Fabian’s on my scientific talk. Finally, I wish to thank Stefan Woltran, and Till Tantau for reviewing this thesis. Danksagung Zuerst möchte ich meine tiefste Dankbarkeit bei meiner Familie ausdrücken. Euer Verständnis und eure Unterstützung in allen Lebenslagen sind das Fundament meiner Forschung. An zweiter Stelle bedanke ich mich bei allen meinen Mitautoren von Veröffentlichungen, die in diese Arbeit eingeflossen sind. Besonderer Dank gehört hierbei Nadia Creignou, Raïda Ktari, Julian-Steffen Müller, Fréderic Olive, Johannes Schmidt (besonderer Dank geht an Dich für die Diskussion zu Theorem 3.2) und Heribert Vollmer, die bei den wichtigen ersten Publikationen zur parametrisierten Enumeration mitgewirkt haben. Schließlich richtet sich meine Dankbarkeit an Maurice Chandoo und Anselm Haak für ihre wertvollen Rückmeldungen zu unter- schiedlichen Abschnitten dieser Arbeit. Insbesondere möchte ich Martin Lück für seine Hinweise zum Abschnitt 3.2 sowie Johannes Fichte für das Korrekturlesen von Kapitel 1 danken. Christian Reinbold bin ich für die Diskussionen zu Kapitel 6 zu Dank verpflichtet. Schließlich danke ich Anselm für seine Rückmeldung zu meiner Lehrprobe und Fabian für seine Tipps zu meinem wissenschaftlichen Vortrag. Abschließend möchte ich mich noch bei Stefan Woltran und Till Tantau für das begutachten dieser Arbeit bedanken. viii Contents 1 Introduction 1 1.1 Parametrised Complexity . 3 1.2 Enumeration . 6 1.3 Team Logics . 10 1.4 Results . 13 1.5 Publications . 13 2 Preliminaries 15 2.1 Complexity Theory . 15 2.2 Parametrised Complexity Theory . 15 2.3 Enumeration . 17 2.4 Parametrised Enumeration . 19 2.5 Orders . 21 3 Enumeration Complexity Landscape 23 3.1 Incremental FPT . 25 3.2 Connections to Classical Enumeration . 27 3.3 CardinalitySAT . 34 4 Principles of Parametrised Enumeration 37 4.1 Kernelisation . 37 4.2 Self-Reducibility and Bounded-Search-Trees . 42 4.2.1 Enumeration Complexity of Max-Ones-SAT . 42 4.2.2 Enumeration of Strong HORN-Backdoor Sets . 47 4.3 A Dichotomy for the Enumerability of Max-Ones-SAT . 48 5 Parametrised Enumeration with Orders 57 5.1 Graph Modification Problems . 57 5.1.1 Lexicographic Order . 61 5.1.2 Order by Size . 62 5.2 Generalised Modification Problems . 65 5.2.1 Closest String . 68 5.2.2 Backdoors . 69 5.2.3 Weighted Satisfiability Problems . 71 ix 6 Enumeration in Poor Man’s Propositional Dependence Logic 73 6.1 Team-based Propositional Logic . 74 6.2 Group Theory . 75 6.3 Enumeration Complexity . 76 6.3.1 The Group Action of Flipping Bits . 79 6.3.2 Limiting Memory Space . 89 7 Conclusion 95 8 Outlook 99 Bibliography 101 x Introduction 1 [::: ]; les charmes enchanteurs de cette sublime science ne se decelent dans toute leur beaute qu’a “ ceux qui ont le courage de l’approfondir. ” [::: ]; the enchanting charms of this sublime sci- ence reveal themselves in all their beauty only to those who have the courage to go deeply into it. — Carl Friedrich Gauß, 30. April 1807. [CF12] In the last decade, implicitly using modern technologies silently became a common procedure in copious areas of living. Before leaving home, we check the rain forecast on our smartphone. In the morning, we buy groceries or various other things over the internet, and they are delivered even the same day. Driving a car to a new place
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