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Structural Graph Theory Meets Algorithms: Covering and Connectivity Problems in Graphs Saeed Akhoondian Amiri Fakult¨atIV { Elektrotechnik und Informatik der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Rolf Niedermeier Gutachter: Prof. Dr. Stephan Kreutzer Gutachter: Prof. Dr. Marcin Pilipczuk Gutachter: Prof. Dr. Dimitrios Thilikos Tag der wissenschaftlichen Aussprache: 13. October 2017 Berlin 2017 2 This thesis is dedicated to my family, especially to my beautiful wife Atefe and my lovely son Shervin. 3 Contents Abstract iii Acknowledgementsv I. Introduction and Preliminaries1 1. Introduction2 1.0.1. General Techniques and Models......................3 1.1. Covering Problems.................................6 1.1.1. Covering Problems in Distributed Models: Case of Dominating Sets.6 1.1.2. Covering Problems in Directed Graphs: Finding Similar Patterns, the Case of Erd}os-P´osaproperty.......................9 1.2. Routing Problems in Directed Graphs...................... 11 1.2.1. Routing Problems............................. 11 1.2.2. Rerouting Problems............................ 12 1.3. Structure of the Thesis and Declaration of Authorship............. 14 2. Preliminaries and Notations 16 2.1. Basic Notations and Defnitions.......................... 16 2.1.1. Sets..................................... 16 2.1.2. Graphs................................... 16 2.2. Complexity Classes................................. 18 2.2.1. Approximation Algorithms........................ 18 2.2.2. Parametrized Complexity......................... 19 II. Covering Problems 21 3. Distributed Approximation Algorithms for Dominating Set 22 3.1. Introduction..................................... 22 3.2. Defnitions and General Lemmas......................... 23 3.3. The Local MDS Approximation......................... 26 3.3.1. Local MDS Approximation: Ideas and Proof Sketch......... 26 3.3.2. MDS Local Approximation Algorithm................. 28 3.3.3. Analysis................................... 29 3.4. O(g) Approximation Factor in O(g) Communication Rounds......... 33 3.5. (1 + δ)-Approximation for dominating set in H-Minor Free Graphs...... 35 3.5.1. Ideas and Proof Sketch.......................... 35 3.5.2. Proof Details................................ 36 i Contents 3.6. On Connected r-Dominating Set Problem ................ 38 3.7. Conclusion and Future Works........................... 41 4. On Dualities in Digraphs: Erd}os-P´osaproperty in Digraphs 42 4.1. Introduction..................................... 42 4.2. Directed Minors, Directed Grids and Directed Tree-Width........... 45 4.3. The Erd}os-P´osaproperty for Strongly Connected Digraphs.......... 50 4.4. The EP-Property for Vertex-Cyclic Digraphs.................. 52 4.5. Positive Instances for the Erd}os-P´osaProperty on Weakly Connected Digraphs 65 4.5.1. Strong Star................................. 66 4.5.2. Two Cycles Connected by a Single Edge................. 68 4.6. Conclusion and Future Work........................... 75 III. Routing and Connectivity Problems 76 5. On Disjoint Paths Problem in Digraphs 77 5.1. Introduction..................................... 77 5.2. Preliminaries.................................... 80 5.3. NP-Completeness of UpPlan-VDPP ...................... 81 5.4. A Linear Time Algorithm for UpPlan-VDPP for Fixed k ........... 86 5.5. Disjoint Paths With Congestion on DAGs.................... 90 5.6. Lower Bounds on Disjoint Paths with Congestion on DAGs.......... 93 5.7. Induced Path Problem............................... 93 5.8. Conclusion and Future Work........................... 97 6. Rerouting Problem 98 6.1. Introduction..................................... 98 6.2. Model and Defnitions............................... 98 6.3. Preliminaries.................................... 102 6.4. Congestion-Free Rerouting of Flows on DAGs.................. 103 6.5. NP-Hardness of 2-Flow Update in General Graphs............... 103 6.6. Optimal Solution for k = 2 Flows......................... 103 6.7. Updating k-Flows in DAGs is NP-complete................... 108 6.8. Linear Time Algorithm for Constant Number of Flows on DAGs....... 112 6.9. Conclusion and Future Work........................... 123 ii Abstract Structural graph theory proved itself a valuable tool for designing efcient algorithms for hard problems over recent decades. We exploit structural graph theory to provide novel techniques and algorithms for covering and connectivity problems. First, we focus on the Local model of distributed computing. In the Local model, minimizing the number of communication rounds is the main goal. We exploit the local properties of bounded genus graphs. We provide the frst constant factor approximation algorithm that solves the dominating set problem in constant rounds on bounded genus graphs. Then, we arbitrarily well approximate it in O(log∗ jGj) rounds. We also introduce a simple technique for graphs of bounded expansion which turns any constant factor approximation of r-dominating sets to a constant factor approximation of connected r-dominating sets. Finding similar patterns in graphs is one of the main challenges in graph theory. One such problem is either to fnd many disjoint instances of a particular pattern or to fnd a small set of vertices such that deleting them destroys all instances of that pattern. This question frst raised by Erd}osand P´osa.We provide an algorithmic classifcation for strongly connected digraphs analogous to the classical results of Robertson and Seymour on undirected graphs. Furthermore, a good characterization for vertex cyclic digraphs is provided. In the latter, we generalize Younger's conjecture to weakly connected digraphs. Next, we focus on routing and connectivity problems. The frst routing problem we consider is the Vertex Disjoint Paths Problem (VDPP) and its descendant problems. We provide an efcient algorithm for solving k-VDPP on upward planar digraphs in linear time for a fxed k. Then we allow the vertices to have some congestion and we solve the disjoint paths problem with congestion in acyclic digraphs. On the complexity side, we show the hardness of those problems when the corresponding parameter (e.g. k) is part of the input. It follows that the time complexity of our algorithms are almost optimal. We also show that induced path problem is hard even on digraphs of bounded directed tree-width. Finally, we consider the problem of rerouting. In a computer network, we may need to reroute packets from their old paths to new paths. The rerouting procedure should satisfy some consistency rules. E.g., the capacity of links should be respected, the fow of the packets cannot be interrupted, etc. Elements of the network are asynchronous. Thus, it is not possible to do the rerouting instantly. We show that it is NP-hard to fnd a feasible rerouting algorithm even on DAGs. In contrast, we provide a linear time rerouting algorithm on acyclic graphs for a fxed number of paths. iii Abstract Kurzfassung Die strukturelle Graphentheorie hat in den letzten Jahrzehnten einen wichtigen Beitrag zur Theorie der efzienten Graph-Algorithmen geleistet. Wir entwickeln weitere struk- turelle Ergebnisse und neue Techniken um Uberdeckungs-¨ und Zusammenhangsprobleme auf Graphen zu efzient zu l¨osen. Zun¨achst konzentrieren wir uns auf das Local Modell, das in verteilten Algorithmen Anwendung fndet. In diesem Modell geht es darum, die Kommunikation zwischen Klien- ten in einem Netzwerk zu minimieren, die ein globales Problem l¨osenm¨ochten. Wenn der Netzwerkgraph speziellen topologischen Einschr¨ankungen gen¨ugt,k¨onnenProbleme wesentlich schneller und besser gel¨ostwerden als in allgemeinen Graphen. Wir zeigen die Existenz einer constant-factor Approximation f¨urdas dominating set Problem, die in einer konstanten Zahl von Kommunikationsrunden im lokalen Modell ausgef¨uhrtwerden kann. Wir zeigen, dass in O(log∗ jGj) Runden sogar eine beliebig gute Approximation f¨urdas Problem berechnet werden kann. Weiterhin zeigen wir eine einfache Technik, mit Hilfe derer aus einem r-dominating set ein nur konstant gr¨oßeresverbundenes r-dominating set auf Graphen mit beschr¨ankterExpansion (bounded expansion) berechnet werden kann. Eine wichtige Anwendung der algorithmischen Graphentheorie ist die Suche nach gewissen Patterngraphen in einem großen Eingabegraphen. Eine wichtige Problemstellung ist es, eine kleine Zahl von Knoten zu fnden, so dass jedes Vorkommen eines festen Patterngraphen einen der gew¨ahltenKnoten enth¨alt.So sagt zum Beispiel der Satz von Erd}osand P´osa dass f¨urjede feste Zahl k eine Zahl f(k) existiert, so dass in jedem Graphen, der keine k knotendisjunkten Kreise enth¨alt,alle Kreise mit f(k) Knoten ¨uberdeckt werden k¨onnen.Ein entsprechender Dualit¨atssatzf¨urMinoren wurde von Robertson und Seymour gefunden. Wir beweisen einen analogen Charakterisierungssatz f¨urgerichtete Graphen. Weiterhin liefern wir einen Charakterisierungssatz f¨urgerichtete vertex cyclic Graphen. Mit diesem Satz verallgemeinern wir die Vermutung von Younger auf schwach zusammenh¨angendegerichtete Graphen. Im n¨achsten Teil der Arbeit konzentrieren wir uns auf Routing und Zusammenhangsprob- leme. Zun¨achst betrachten wir das Vertex Disjoint Paths Problem (VDPP) und verwandte Probleme. Wir liefern einen efzienten Algorithmus f¨urdas k-VDPP Problem auf upward planaren Graphen, dessen Laufzeit f¨urjedes feste k linear in der Eingabegr¨oßeist.
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