CONJUNCTIVEQUERIES,ARITHMETICCIRCUITSAND COUNTINGCOMPLEXITY Dissertation zur Erlangung des Doktorgrades der Fakultät für Elektrotechnik, Informatik und Mathematik der Universität Paderborn vorgelegt von Stefan Mengel Paderborn, 21. Mai 2013 Stefan Mengel: Conjunctive Queries, Arithmetic Circuits and Counting Complexity, © May 21, 2012 "We can only see a short distance ahead, but we can see plenty there that needs to be done." —Alan Turing [Tur50] ABSTRACT This thesis deals with several subjects from counting complexity and arithmetic circuit complexity. The first part explores the complexity of counting solutions to con- junctive queries, which are a basic class of queries from database theory. We introduce a parameter, called the quantified star size of a query f, which measures how the free variables are spread in f. As usual in database theory, we associate a hypergraph to a query f. We show that for classes of queries for which these associated hyper- graphs have bounded generalized hypertree width, bounded quanti- fied star size exactly characterizes the subclasses of queries for which counting the number of solutions is tractable. In the case of bounded arity, this allows us to fully characterize the classes of conjunctive queries for which counting the solutions is tractable. Finally, we also analyze the complexity of computing the quantified star size of a con- junctive query. In the second part we characterize different classes from arithmetic circuit complexity by different means, including conjunctive queries and constraint satisfaction problems, graph polynomials on bounded treewidth graphs, and an extension of the classical arithmetic branch- ing program model by stack memory. In particular, this yields new characterizations of the arithmetic circuit class VP, a class that is cen- tral to the area but arguably not well understood. Finally, the third part studies the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of its monomials. We show that these problems are complete for different levels of the counting hier- archy, which had few or no known natural complete problems before. ZUSAMMENFASSUNG In dieser Arbeit geht es um verschiedene Themen aus der Zählkom- plexität und der arithmetischen Schaltkreiskomplexität. Der erst Teil untersucht die Komplexität des Zählens der Antworten auf Conjunctive Queries, eine grundlegende Klasse von Anfragen aus der Datenbanktheorie. Wir führen den Parameter quantified star size einer Query f ein, der misst, wie die freien Variablen in f verteilt sind. Wir ordnen der Query f einen Hypergraphen zu und zeigen, dass für Klassen von Queries mit beschränkter generalized hyper- tree width der Parameter quantified star size genau die Unterklassen charakterisiert, für die das Zählen von Antworten effizient möglich v ist. Dies erlaubt uns, im Fall beschränkter Arität die Klassen von Con- junctive Queries, für die effizientes Zählen von Antworten möglich ist, vollständig zu charakterisieren. Weiterhin betrachten wir auch die Kompexität der Berechnung der quantified star size von Conjunctive Queries. Im zweiten Teil der Arbeit charakterisieren wir unterschiedliche Klassen aus der arithmetischen Schaltkreiskomplexität auf verschie- dene Arten, und zwar durch Conjunctive Queries and Constraint Sat- isfaction Probleme, durch Graphpolynome auf Graphen beschränk- ter Baumweite und durch eine Erweiterung des klassischen Modells der arithmetischen Branchingprogramme durch Stack-Speicher. Ins- besondere zeigen wir neue Charakterisierungen der arithmetischen Schaltkreisklasse VP, einer Klasse, die zentral für den Bereich ist aber dennoch nicht gut verstanden. Der dritte Teil schließlich beschäftigt sich mit zwei Entscheidung- problemen zu Polynomen gegeben durch arithmetische Schaltkreise: Testen ob ein gegebenens Monom vorkommt und Zählen der vork- ommenden Monomen. Wir zeigen, dass diese Probleme vollständig sind für unterschiedliche Levels der sogenannten counting hierarchy, für die bisher wenige oder keine natürlichen vollständigen Probleme bekannt waren. vi ACKNOWLEDGMENTS After more than four years of work on this thesis there are many people I would like to thank. First and most of all I would like to thank my wife Hilke for letting me do this although it was not always easy for her. I am grateful to my advisor Peter Bürgisser for, on the hand, giving me the freedom to pursue my own research and, one the other hand, being supportive when I needed his opinion or help. In particular, I would also like to thank him for giving me the opportunity to meet many other people in the community. Also, I am thankful to him for relentlessly making me improve the presentation of this thesis (of course I take full responsibility for all shortcomings and mistakes still in it). I would like to thank Friedhelm Meyer auf der Heide and Luc Segoufin who kindly agreed to act as reviewers of this thesis. Arnaud Durand has played a crucial role in the creation of this thesis. He patiently explained to me everything I know about logic, introduced me to several areas of computer science, and made key contributions to my research. He also gave me professional and per- sonal advice whenever I needed it. Finally, he made my stays in Paris possible by getting the necessary funding for me. I am very thankful for all of his support and I hope to pay back his kindness one day. I am very grateful to Guillaume Malod for several reasons. First, without his invitation to Paris to work together this thesis would not be what it is now. Also, he let me share some of his deep understand- ing of arithmetic circuit complexity. Last but not least I am grateful for his support in my (often hopeless) struggles with French bureau- cracy. I value also a lot the support on a professional and personal level that I received from Hervé Fournier during my several stays in Paris. My colleagues Dennis Amelunxen, Jesko Hüttenhain and Christian Ikenmeyer have been great companions during the last four years. Our discussions on math, computer science and life in general have sometimes been heated, but I value their support, their friendship and their opinions a lot. Moreover, I am thankful for the TikZ-support by Dennis and the LATEX-support by Jesko. I would like to thank the numerous people who I have shared of- fices with during the last few years for making my time at work so enjoyable. In particular, I will miss Maik Ringkamp’s cheerful com- pany when our common time is over. I am thankful to Sandra Pelster and Inga Gill for their support and their friendliness. vii I would like to thank Yann Strozecki for getting me and my fam- ily an apartment in Paris for one of my stays there. Moreover, I am grateful to Yanis Langeraert for letting me stay at his place for several months. I would like to thank the members of the Équipe de Logique Math- ématique of the Institut de Mathématiques de Jussieu at Université Paris 7 and the numerous people associated to this group for making me feel very welcome during my several stays in Paris. I would like to thank Hubie Chen for his hospitality during my short stay in San Sebastián. My stays in Paris would have been far less enjoyable if I had not learnt French before. I would like to thank my French teacher Sigrid Behrent for making this such a pleasure. I am also grateful to Barbara and Jens-Peter Kempkes for taking care of our son Jakob on several days during the final phase of writing this thesis. I would like to thank the organizers of the Dagstuhl Seminars 10481 and 13031 on Computational Counting. Some of the results in this thesis were conceived during these workshops. Moreover, meeting several people there has proved invaluable. Sébastien Tavenas pointed out an error in an earlier version of the proof of Lemma 12.3.7. Later, he and Pascal Koiran helped me find the proof presented in this thesis. I would like to thank both of them for their contribution. I would like to thank Friedhelm Meyer auf der Heide and his work- ing group for their hospitality during the first phase of my doctoral studies. The research in this thesis would not have been possible without the generous financial support by the Research Training Group GK- 693 of the Paderborn Institute for Scientific Computation (PaSCo) and by the Deutsche Forschungsgemeinschaft (DFG-grants BU 1371/2-2 and BU 1371/3-1). Furthermore, the research leading to the results presented in this thesis has received funding from the [European Community’s] Seventh Framework Programme [FP7/2007-2013] un- der grant agreement n° 238381. I am very grateful for this support. viii CONTENTS 1 introduction1 1.1 Part i: Counting solutions to conjunctive queries 2 1.1.1 Structural restrictions for tractable #CQ 5 1.2 Part ii: Understanding arithmetic circuit classes 7 1.2.1 Conjunctive queries and arithmetic circuits 10 1.2.2 Graph polynomials on bounded treewidth graphs 10 1.2.3 Modifying arithmetic branching programs 11 1.3 Part iii: Monomials in arithmetic circuits 11 1.4 Overview over the thesis 12 i counting solutions to conjunctive queries 15 2 preliminaries 17 2.1 Conjunctive queries 17 2.1.1 Model of computation and encoding of instances 20 2.1.2 Query problems 21 2.2 Parameterized complexity 22 2.3 Graph and hypergraph decompositions 24 2.3.1 Treewidth 24 2.3.2 Hypergraph decomposition techniques 27 3 thecomplexityof #CQ and quantified star size 37 3.1 The complexity of #CQ 37 3.2 Quantified star size 39 3.3 Formulation of main results 44 3.4 Digression: Unions of acyclic queries 46 4 computing