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Computational Complexity Aspects of Implicit Graph Representations Computational Complexity Aspects of Implicit Graph Representations Von der Fakultät für Elektrotechnik und Informatik der Gottfried Willhelm Leibniz Universität Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von M. Sc. Maurice Chandoo geboren am 27. Juni 1991 in Hannover 2017 Referent: Heribert Vollmer, Leibniz Universität Hannover Korreferent: Johannes Köbler, Humboldt-Universität zu Berlin Tag der Promotion: 20.07.2018 La clarté orne les pensées profondes. Luc de Clapiers, marquis de Vauvenargues v Acknowledgments I thank my doctoral advisor Heribert Vollmer for not only having introduced me to the field of theoretical computer science but also for having done so with inspiring clarity and great emphasis on intuition. Moreover, I deeply thank him for granting me the liberty to pursue my own research. vii Abstract Implicit graph representations are immutable data structures for restricted classes of graphs such as planar graphs. A graph class has an implicit representation if the vertices of every graph in this class can be assigned short labels such that the adjacency of two vertices can be decided by an algorithm which gets the two labels of these vertices as input. A representation of a graph in that class is then given by the set of labels of its vertices. The algorithm which determines adjacency is only allowed to depend on the graph class. Such representations are attractive because they are space-efficient and in many cases also allow for constant-time edge queries. Therefore they outperform less specialized representations such as adjacency matrices or lists and are even optimal in an asymptotic sense. In the first part of this thesis we investigate the limitations of such representations when constraining the complexity of an algorithm which decodes adjacency. First, we prove that imposing such computational constraints does indeed affect what graph classes have an implicit representation. Then we observe that the adjacency structure of almost all graph classes that are known to have an implicit representation can be described by formulas of first-order logic. The quantifier-free fragment of this logic can be characterized in terms of RAMs: a graph class can be expressed by a quantifier-free formula if and only if it has an implicit representation where edges can be queried in constant-time on a RAM without division. We provide two reduction notions for graph classes which reveal that trees and interval graphs are representative for certain fragments of this logic. We conclude this part by providing a big picture of the newly introduced classes and point out viable research directions. In the second part we consider the tractability of algorithmic problems on graph classes with implicit representations. Intuitively, if a graph class has an implicit representation with very low complexity then it should have a simple adjacency structure. Therefore it seems plausible to expect certain algorithmic problems to be tractable on such graph classes. We consider how realistic it is to expect an algorithmic meta-theorem of the form “if a graph class X has an implicit representation with complexity Y then problem Z is tractable on X”. Our considerations quickly reveal that even for the most humble choices of Y and various Z this is either impossible or leads to the frontiers of algorithmic research. We show that the complexity classes of graph classes introduced in the previous chapter can be interpreted as graph parameters and therefore can be considered within the framework of parameterized complexity. We embark on a case study where Z is the graph isomorphism problem and Y is the quantifier-free, four-variable fragment of first order logic with only the order predicate on the universe. This leads to a problem that has been studied independently and resisted classification for over two decades: the isomorphism problem for circular-arc (CA) graphs. We examine how a certain method, which we call flip trick, can be applied to this problem. We show that for a broad class of CA graphs the isomorphism problem reduces to the representation problem and as a consequence can be solved in polynomial-time. Keywords: adjacency labeling schemes, descriptive complexity of graph properties, re- ductions for graph classes, CA graph isomorphism, pointer numbers ix Zusammenfassung Implizite Graphrepräsentationen sind statische Datenstrukturen für beschränkte Graphk- lassen wie zum Beispiel planare Graphen. Eine Graphklasse hat eine implizite Repräsenta- tion, falls die Knoten jedes Graphen dieser Klasse mit kurzen Labels beschriftet werden können, sodass die Adjazenz zweier Knoten von einem Algorithmus entschieden wer- den kann, welcher die zwei Labels der Knoten als Eingabe erhält. Eine Repräsentation eines Graphen aus dieser Klasse besteht aus der Menge der Labels seiner Knoten. Der Algorithmus, welcher die Adjazenz entscheidet, darf nur von der Graphklasse abhängen. Solche Repräsentationen sind attraktiv, da sie speichereffizient sind und oftmals auch Kantenabfragen in konstanter Zeit zulassen. Deshalb übertreffen sie weniger spezialisierte Repräsentationen wie Adjazenzmatrizen oder -listen und sind sogar asymptotisch optimal. Im ersten Teil dieser Arbeit untersuchen wir, welche Auswirkungen das Einschränken der Komplexität von Algorithmen, welche die Adjazenz decodieren, auf die Menge von Graphklassen, die eine implizite Repräsentation haben, hat. Es stellt sich heraus, dass die Menge der Graphklassen mit so einer Repräsentation tatsächlich von der gewählten Komplexität abhängt. Anschließend beobachten wir, dass die Adjazenzstruktur von fast allen Graphklassen, von denen man weiß, dass sie eine implizite Repräsentation haben, in Prädikatenlogik erster Stufe ausgedrückt werden kann. Das quantorenfreie Fragment dieser Logik kann wie folgt charaktersiert werden: eine Graphklasse kann genau dann durch eine quantorenfreie Formel erster Stufe ausgedrückt werden, wenn sie eine implizite Repräsentation hat, in der Kantenabfragen in konstanter Zeit auf einer RAM ohne Division durchgeführt werden können. Wir führen zwei Reduktionsbegriffe für Graphklassen ein, welche es uns ermöglichen zu zeigen, dass Bäume und Intervalgraphen für bestimmte Fragmente dieser Logik repräsentativ sind. Im letzten Teil fassen wir unsere Ergebnisse zusammen und stellen die verschiedenen, neu eingeführten Klassen und deren Beziehungen in einem Schaubild dar. Im zweiten Teil beschäftigen wir uns mit der Komplexität algorithmischer Probleme auf Klassen von Graphen mit impliziten Repräsentationen. Intuitiv gesehen sollte eine Graphklasse mit einer impliziten Repräsentation von geringer Komplexität eine ebenso simple Adjazenzstruktur haben. Daher erscheint es plausibel zu erwarten, dass bestimmte algorithmische Probleme effizient auf solchen Graphklassen lösbar sind. Wir untersuchen die Frage, ob sich ein algorithmisches Metatheorem der Form „wenn eine Graphklasse X eine implizite Repräsentation mit Komplexität Y hat, dann ist Problem Z effizient auf X lösbar“ beweisen lässt. Es stellt sich schnell heraus, dass selbst für die bescheidenste Wahl von Y und verschiedene Z dies entweder unmöglich ist oder uns an die Grenzen der Forschung in der Algorithmik führt. Daher führen wir eine Fallstudie durch, wobei Z das Graphenisomorphieproblem ist und Y ein spezielles Fragment der Prädikatenlogik. Dies führt uns zum Isomorphieproblem für Kreisbogengraphen, welches seit mehr als zwei Jahrzenten trotz beachtlicher Anstrengungen nicht klassifiziert werden konnte. Wir schauen uns an, wie eine bestimmte Methode (Flip Trick) auf dieses Problem angewandt werden kann. Es stellt sich heraus, dass für eine große Klasse von Kreisbogengraphen das Isomorphieproblem auf das Repräsentationsproblem reduzierbar ist und somit in Polynomialzeit gelöst werden kann. xi Contents 1 Introduction1 2 Preliminaries5 2.1 General Notation and Terminology........................5 2.2 Complexity Theory.................................5 2.3 First-Order Logic..................................7 2.4 Graph Theory....................................7 2.4.1 Neighborhoods...............................8 2.4.2 Isomorphism and Invariance.......................8 2.4.3 Graph Classes, Graph Parameters and Graph Class Properties....8 2.4.4 CA Models, Representations and Matrices............... 10 2.5 Labeling Schemes.................................. 12 3 A Complexity Theory for Implicit Representations 15 3.1 Hierarchy of Labeling Schemes.......................... 16 3.2 Expressiveness of Primitive Labeling Schemes................. 20 3.3 Reductions Between Graph Classes........................ 25 3.3.1 Algebraic Reductions........................... 26 3.3.2 Subgraph Reductions........................... 29 3.4 Logical Labeling Schemes............................. 32 3.4.1 Definition and Basic Properties...................... 33 3.4.2 Complete Graph Classes.......................... 39 3.4.3 Polynomial-Boolean Systems....................... 42 3.4.4 Constant-Time RAMs........................... 47 3.5 Summary and Open Questions.......................... 50 4 Algorithmic Properties of Graph Classes with Implicit Representations 55 4.1 Parameterized Complexity of Sets of Graph Classes.............. 56 4.2 The Isomorphism Problem for CA Graphs.................... 59 4.3 Normalized Representations............................ 60 4.4 Flip Trick....................................... 62 4.5 Uniform CA Graphs................................ 65 4.6 Non-Uniform CA Graphs and Restricted CA
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