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Flows and Colorings in Oriented Matroids Flows and Colorings in Oriented Matroids Dissertation zur Erlangung des akademischen Grades DOKTOR RER. NAT. der Fakult¨atf¨ur Mathematik und Informatik der FernUniversit¨at in Hagen von Dipl. Math. Robert Nickel aus Berlin Hagen, 2012 Berichterstatter: Prof. Dr. Winfried Hochst¨attler Prof. Dr. Luis Goddyn Tag der m¨undlichen Pr¨ufung: 15. Mai 2012 Danksagungen Besonderer Dank geht an meinen Betreuer Professor Dr. Winfried Hochst¨attlerf¨ur eine fantastische Themenwahl, f¨urdie zahlreichen spannenden Diskussionen in seinem B¨uro und f¨urseine nicht enden wollende Motivation und das entgegengebrachte Vertrauen bis zur lang ersehnten Abgabe dieser Arbeit. Er hat mich angesteckt mit seiner Begeisterung f¨urOrientierte Matroide und mir in vielerlei Hinsicht einiges beigebracht. Desweiteren danke ich: • Professor Luis Goddyn f¨urdas Begutachten dieser Arbeit, • Dominique Andres f¨ur die zahlreichen spannenden Diskussionen zum Thema, • Britta Peis f¨urdas tapfere Korrekturlesen großer Teile dieser Arbeit und die zahlrei- chen Verbesserungsvorschl¨age, • Lukas Finschi f¨urdie Bereitstellung seiner Datenbank aller kleinen orientierten Matroide, • Oliver Klein, der mit seiner Software etliche Einelementerweiterungen f¨urmich berechnet hat, • Xun Dong, der mit seiner Arbeit [30] das gesamte dritte Kapitel motiviert hat, • meinen Kollegen bei der DCAM GmbH, allen voran Edmund Groß und Jens Franke, die mir den R¨ucken frei gehalten haben als ich es brauchte, • zahlreichen weiteren Personen, die es nie leid waren, mich in regelm¨aßigenAbst¨anden immer wieder nach dem Stand dieser Arbeit zu fragen und damit wesentlich zu deren Fertigstellung beigetragen haben, • Steve f¨ur ein besseres Englisch in der Einleitung, • meinen Eltern die mir w¨ahrend des Studiums, der Zeit an der Universit¨atund bis heute unsch¨atzbare Unterst¨utzungentgegenbringen • und meiner Frau und meiner Tochter f¨urderen Liebe, Geduld, Beistand und Ver- trauen w¨ahrendlanger Phasen k¨orperlicher sowie geistiger Abwesenheit. Zusammenfassung Wir werden uns in dieser Arbeit mit einer Struktur besch¨aftigen,deren St¨arke darin liegt, verschiedenste Gebiete der Mathematik zusammenzuf¨uhrenwie Lineare Optimierung, affine/projektive Geometrie, Algebra, Verbandstheorie, Komplexit¨atstheorie,Transver- salentheorie und Graphentheorie. Orientierte Matroide reduzieren die Eigenschaften von Punktkonfigurationen, Polyedern und (Pseudo)Hyperebenenarangements auf deren rein kombinatorischen Kern und wurden 1978 von Folkman und Lawrence [38] und Bland und Las Vergnas [12] mit der Motivation eingef¨uhrt,die praktisch belegte, theoretisch aber nie nachgewiesene gute Effizienz des Simplex-Algorithmus der linearen Optimierung zu beweisen. Außerdem weiß man von Optimierungsproblemen, denen eine Matroidstruk- tur zugrunde liegt, dass diese exakt mit einer \gierigen" (greedy) Strategie gel¨ostwer- den k¨onnen([10]). Die nicht-orientierten Pendants, die Matroide, wurden bereits 1935 von Whitney [111] eingef¨uhrt,der die gemeinsamen Eigenschaften von minimal linear abh¨angigenMengen von Vektoren und Kreisen in Graphen untersuchte. Weitere davon unabh¨angigeVerweise findet man in Birkhoff [7], van der Waerden [107] und Nakasawa [84]. Wir konzentrieren uns in dieser Arbeit auf Konzepte der Graphentheorie und deren Ver- allgemeinerung auf orientierte Matroide. Sowohl aus theoretischer, als auch aus praktis- cher Sicht, liefert die Graphentheorie eine schier endlose Quelle von einfach darzustel- lenden Problemen, deren L¨osung sich aber nicht selten als harte Nuss herausstellt. Das wahrscheinlich prominenteste Beispiel ist das Vier-Farben-Problem (Francis Guthrie um 1852, siehe Cayley [22]): Gegeben ist eine geographische Landkarte mit zusam- menh¨angendenL¨andern(also z. B. ohne Kolonien). Ist es immer m¨oglich, deren L¨ander so einzuf¨arben, dass man benachbarte L¨andermit unterschiedlichen Farben versieht und h¨ochstens vier Farben verwendet? Ein Graph besteht aus einer Menge von Knoten und einer Menge von Kanten, die einige Paare dieser Knoten miteinander verbinden. Identifiziert man L¨andermit Knoten und Grenzen mit Kanten, so stellt sich das Vier-Farben-Problem wie folgt dar: Gegeben ist ein planarer Graph, d. h. ein Graph, der so in der Ebene gezeichnet werden kann, dass sich Kanten nur in ihren Endpunkten kreuzen. Gibt es eine F¨arbungder Knoten, so dass durch Kanten verbundene Knoten verschieden gef¨arbt sind und h¨ochstens vier Farben ben¨otigt werden? Bereits 1852 erstmals erw¨ahnt, dauerte es fast anderthalb Jahrhunderte, bis ein weitestge- hend akzeptierter Beweis daf¨urgefunden wurde, dass eine solche F¨arbungimmer existiert (siehe Appel und Haken [4], Robertson, Sanders, Seymour und Thomas [89]). Bis heute f¨ulltdie Theorie der Graphenf¨arbungganze B¨ucher und deren zahlreiche Anwendungen ¨uberschreiten bei weitem den Rahmen dieser Einleitung ([25]). F¨uhrtman in einem Graphen f¨urjede Kante eine Richtung ein, so kann man durch Label an den Kanten Fluss zwischen den Knoten modellieren, die nun als Verzweigungspunkte in einem (Verkehrs-, Rohr- oder elektrischen) Netzwerk fungieren. Mit der Bedingung, dass alles, was in einen Knoten hineinfließt auch wieder herausfließen soll (Kirchhoff'sches Gesetz [72]), erh¨alt man ein Modell f¨urRundfl¨usse.Wenn zus¨atzlich zwei der Knoten die Rolle von Quelle und Senke ¨ubernehmen und jede gerichtete Kante mit einer Maximalka- pazit¨atversehen wird, endet man in einem Modell f¨urmaximale Fl¨ussezwischen zwei Knoten mit Kapazit¨atsbeschr¨ankung, was den Ausgangspunkt der Flusstheorie bildete (Ford und Fulkerson [39]). Im Falle eines planaren Graphen sind Rundfl¨usseund Knotenf¨arbungen\duale" Objekte (Tutte [101]). Um diese Dualit¨atf¨urnicht-planare Graphen und allgemeinere Strukturen zu erhalten, ben¨otigtman orientierte Matroide. Zwei verschiedene Verallgemeinerungen von Fl¨ussenin Graphen auf orienterte Matroide finden sich in Ver¨offentlichungen der letz- ten Jahre (siehe Hochst¨attlerund Neˇsetˇril[59] und Goddyn, Tarsi und Zhang [45]). Wir untersuchen Fl¨ussein orientierten Matroiden wie sie von Hochst¨attlerund Neˇsetˇril[59] eingef¨uhrtwurden und diskutieren Fragestellungen, die beschr¨anktauf Graphen, zu promi- nenten Problemen z¨ahlen. Die ersten beiden Kapitel dienen der Einf¨uhrungvon Theorie und Notation, die f¨urdie Aussagen und Beweise der folgenden Kapitel von Bedeutung sind. W¨ahrendKapitel 1 den Grundlagen der Graphentheorie, Geometrie und Verbandstheorie gewidmet ist, geben wir in Kapitel 2 eine recht ausf¨uhrliche Einf¨uhrungin die Theorie der orientierten Ma- troide. Dabei legen wir besonderen Wert auf die zahlreichen Axiomensysteme, mit de- nen ein orientiertes Matroid definiert werden kann. Diese Grundlagen sind unerl¨asslich f¨urKapitel 3, wo wir wohlbekannte Aussagen zur Komposition und Dekomposition von (nicht-orientierten) Matroiden betrachten und deren Korrektheit im Kontext orientierter Matroide beweisen. Der Begriff der Fl¨usseund Kofl¨usse(duale Fl¨ussebzw. F¨arbungen)wird in Kapitel 4 basierend auf der Arbeit von Hochst¨attlerund Neˇsetˇril[59] eingef¨uhrt.Wir untersuchen das sogenannte Flussgitter, die Menge aller Rundfl¨usse. Genauer gesagt bestimmen wir dieses exakt f¨urorientierte Matroide, die uniform sind oder deren Rang h¨ochstens drei ist. Bei der Betrachtung von Kofl¨ussenzeigen wir, dass das orientierte Matroid eines vollst¨andigenGraphen, bei dem jedes Paar von Knoten durch eine Kante verbunden ist, genau wie in der Graphentheorie auch im allgemeineren Kontext orientierter Matroide die einzige Worst-Case-Instanz darstellt. Nach einigen weiteren Resultaten f¨ur allgemeine orientierte Matroide beschließen wir das Ende des Kapitels mit Berechnungen auf der Datenbank aller kleinen orientierten Matroide von Finschi [37] und zusammenfassenden Abschnitten ¨uber den Ansatz von [45], gruppenwertige Fl¨usseund das Tutte-Polynom. Wie bereits erw¨ahnt, bildet das Problem der maximalen Fl¨usse mit Kapazit¨atsbe- schr¨ankungenden Ausgangspunkt der Flusstheorie in Graphen ([39]). In Kapitel 5 werden wir eine neue Verallgemeinerung von maximalen Fl¨ussenauf orientierte Matroide basierend auf dem Flussgitter pr¨asentieren. Die aus der Graphentheorie bekannte Dualit¨atvon maxi- malen Fl¨ussenund minimalen Schnitten verallgemeinern wir zu einer kompatiblen Aussage f¨urorientierte Matroide und leiten notwendige sowie hinreichende Bedingungen mit Hilfe von Eigenschaften des Flussgitters her. Contents Introduction1 1 Preliminaries3 1.1 Basics.......................................3 1.2 An Oriented Matroid Expedition........................ 16 2 Oriented Matroids 23 2.1 Circuit Axioms of an Oriented Matroid.................... 23 2.2 Topological Representation and Minors..................... 24 2.3 Duality...................................... 27 2.4 The Underlying Matroid............................. 31 2.5 The Convex Closure Operator.......................... 36 2.6 Basis Orientation and Chirotopes........................ 37 2.7 Classes of Matroids................................ 38 2.8 Guide through the next chapters........................ 41 3 Composition and Decomposition 43 3.1 Basic Connection Operators........................... 47 3.2 Generalized Parallel Connection, Modular Join, and Modular Sum........................ 56 3.3 A Decomposition Theorem........................... 60 3.4 Concluding Remarks and Open Problems................... 63 ix x Contents 4 Flows and Colorings 65 4.1 Regular Oriented Matroids........................... 70 4.2 Uniform Oriented Matroids........................... 72 4.3 Small Rank...................................
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