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Extended Formulations for Combinatorial Polytopes

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Extended Formulations for Combinatorial Polytopes Extended Formulations for Combinatorial Polytopes Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) von Kanstantsin Pashkovich geb. am 06.07.1985 in Bobruisk genehmigt durch die Fakultät für Mathematik der Otto-von-Guericke-Universität Magdeburg Gutachter: Prof. Dr. Volker Kaibel Prof. Dr. Michelangelo Conforti eingereicht am: 01.02.2012 Verteidigung am: 31.08.2012 1 ABSTRACT i Abstract Typically polytopes arising from real world problems have a lot of facets. In some cases even no linear descriptions for them are known. On the other hand many of these polytopes can be described much nicer and with less facets using extended formulations, i.e. as a projection of simpler higher dimensional polytopes. The presented work studies extended formulations for polytopes: the possibilities to construct extended formulations and limitations of them. In the first part, some known techniques for constructions of extended formulations are reviewed and the new framework of polyhedral relations (see Kaibel and Pashkovich [2011]) is presented. We in particular elaborate on the special case of reflection relations. Reflection relations provide extended formulations for several polytopes that can be con- structed by iteratively taking convex hulls of polytopes and their refelections at hyper- planes. Using this framework we are able to derive small extended formulations for the G-permutahedra of all finite reflection groups G. The second part deals with extended formulations which use special structures of graphs involved in combinatorial problems. Here we present some known extended for- mulations and apply a few changes to the extended formulation of Gerards for the perfect matching polytope in graphs with small genus in order to reduce its size. Furthermore a new compact proof of an extended formulation of Rivin for the subtour elimination poly- tope is provided. The third part (partly based on joint work with Volker Kaibel, Samuel Fiorini and Dirk Oliver Theis, see Fiorini, Kaibel, Pashkovich, and Theis [2011a]) involves general ques- tions on the extended formulations of polytopes. The primal interest here are lower bounds for extended formulations. We study different techniques to obtain these lower bounds, all of which could be derived from so called non-negative factorizations of the slack matrix of the initial polytope. The minimal such factorization provides the minimal number of in- equalities needed in an extended formulation. We compare different techniques, find their limitations and provide examples of the polytopes for which they give tight lower bounds on the complexity of extensions. The fourth part studies the impact of symmetry on the sizes of extended formulations. In joint work with Volker Kaibel and Dirk Oliver Theis we showed that for certain con- strained cardinality matching and cycle polytopes there exist no polynomial symmetric extended formulations, but there are polynomial non-symmetric ones (for further details see Kaibel, Pashkovich, and Theis [2010]). Beyond these results the thesis also contains a proof showing that the well known symmetric extended formulation for the permutahe- dron via the Birkhoff polytope is the best (up to a constant factor) one among symmetric extended formulations (see Pashkovich [2009]). ii Zusammenfassung Viele kombinatorische Polytope, die ihre Anwendung in praktischen Problemen fin- den, haben eine große Anzahl von Facetten. In manchen Beispielen ist nicht einmal ei- ne lineare Beschreibung dieser Polytope bekannt. Andererseits lassen viele Polytope eine kompakte und schönere Darstellung mit Hilfe von erweiterten Formulierungen (d.h. als Projektion einfacherer höher-dimensionaler Polytope) zu. Die vorgelegte Arbeit untersucht Erweiterungen von Polytopen: Möglichkeiten, eine kompakte Erweiterung zu finden, und Einschränkungen dieses Ansatzes. Im ersten Teil, werden einige bekannte Konstruktionen für Erweiterungen dargestellt und das neue Framework der Polyedrischen Relationen eingeführt, welches Teil einer ge- meinsamen Arbeit mit Volker Kaibel ist (siehe Kaibel and Pashkovich [2011]). Insbeson- dere arbeiten wir den Fall von Spiegelungsrelationen aus. Spiegelungsrelationen liefern erweiterte Formulierungen für Polytope, die durch iterierte Bildung konvexer Hüllen von Polytopen und ihrer Spiegelungen an Hyperebenen konstruiert werden können. Mit Hil- fe dieses Frameworks können wir kompakte Erweiterungen von G-Permutaedern für alle endliche Spiegelungsgruppen G konstruieren. Der zweite Teil beschäftigt sich mit Erweiterungen, welche spezielle Eigenschaften von den Graphen ausnutzen, die in kombinatorischen Problemen auftauchen. Hier präsen- tieren wir einige bekannte Erweiterungen, und nehmen kleine Änderungen in der erwei- terten Formulierung von Gerards für Perfekte Matching Polytope in Graphen mit kleinem Geschlecht vor, um die Grösse der Erweiterung zu reduzieren. Dazu präsentieren wir einen einfachen Beweis für die Erweiterung von Rivin des Subtour Elimination Polytops. Der dritte Teil untersucht prinzipielle Grenzen des Konzepts der erweiterten Formulie- rungen. Das Hauptziel des dritten Teils ist es, untere Schranken für die Größe erweiterter Formulierung herzustellen. Hier stellen wir einige gemeinsame Ergebnisse mit Volker Kai- bel, Samuel Fiorini und Dirk Oliver Theis dar (siehe Fiorini, Kaibel, Pashkovich, and Theis [2011a]). Wir vergleichen verschiedene Methoden, um untere Schranken zu bekommen, und schauen uns verschiedene Beispiele von Polytopen an (für manche Polytope sind die Schranken optimal). Im vierten Teil präsentieren wir eine weitere gemeinsame Arbeit mit Volker Kai- bel und Dirk Oliver Theis, die sich mit Symmetrien in Erweiterungen beschäftigt (se- he Kaibel, Pashkovich, and Theis [2010]). Hier haben wir Matching und Cycle Polytope gefunden, welche keine symmetrischen Erweiterungen haben, aber sich trotzdem mit Hilfe von Erweiterungen kompakt darstellen lassen. Über gemeinsame Arbeit hinaus, beweisen wir Ergebnisse bezüglich Erweiterungen von quadratischer Grösse, welche zum Beispiel zeigen, dass das Birkhoff Polytop eine asymptotisch minimale symmetrische Erweiterung des Permutaeders ist (sehe Pashkovich [2009]). Contents Abstract i Zusammenfassung ii Chapter 1. Introduction 1 Acknowledgments 2 1.1. Preliminaries 3 1.1.1. Polytopes 3 1.1.2. Extended Formulations, Extensions 4 1.1.3. Combinatorial Polytopes 5 1.1.3.1. Spanning Tree Polytope 5 1.1.3.2. Matching Polytope 5 1.1.3.3. Cycle Polytope 6 1.2. Extensions of Combinatorial Polytopes 6 Chapter 2. Balas Extensions, Flow Extensions and Polyhedral Relations 9 2.1. Balas Extensions 9 2.2. Dynamic Programming Extensions 11 2.3. Flow Extensions 11 2.4. Cardinality Indicating Polytope 12 2.5. Parity Polytope 14 2.6. Birkhoff Polytope and Perfect Matchings in Bipartite Graphs 15 2.7. Permutahedron 16 2.8. Edge Polytope 16 2.9. Cardinality Restricted Matching Polytopes 17 2.10. Cardinality Restricted Cycle Polytope 18 2.11. Polyhedral Relations 18 2.12. Sequential Polyhedral Relations 19 2.13. Affinely Generated Polyhedral Relations 20 2.14. Affine Generators and Domains from Polyhedral Relation 21 2.15. Polyhedral Relations from Affine Generators and Domains 22 2.16. Reflection Relations 23 2.17. Sequential Reflection Relations 24 2.18. Signing of Polytopes 24 2.19. Reflection Groups 25 2.20. Reflection group I2(m) 26 2.21. Reflection group An−1 27 2.22. Reflection group Bn 28 2.23. Reflection group Dn 29 2.24. Huffman Polytopes 30 Chapter 3. Planar Graphs 35 3.1. Graph Embeddings 35 3.2. Extended Formulation of T -join Polyhedron 36 3.2.1. Vector Spaces 36 iii iv CONTENTS 3.2.2. Extended Formulation of T -join Polyhedron 37 3.2.3. Construction of Extended Formulation 39 3.3. Extended Formulation of Cut Polytope in Planar Graphs 39 3.3.1. Projecting Linear System 40 3.3.2. Redundant Inequalities 41 3.3.3. Extended Formulation of Cut Polytope 41 3.3.4. Extended Formulation for T -join Polytope and Perfect Matching Polytope 42 3.3.5. Construction of Extended Formulation 42 3.4. Spanning Tree Polytope 42 3.5. Subtour Elimination Polytope 44 3.5.1. Redundant Inequalities 44 3.5.2. Extended Formulation via Spanning Tree Polytope 44 3.5.3. Extended Formulation for Subtour Elimination Polytope 45 Chapter 4. Bounds on General Extended Formulations for Polytopes 47 4.1. Minimal Extended Relaxation 47 4.2. Slack Matrices of Polyhedra 48 4.3. Non-Negative Factorization, Non-Negative Rank 49 4.4. Extended Relaxations from Non-Negative Factorizations 49 4.5. Non-Negative Factorizations from Extended Relaxations 50 4.6. Non-Negative Factorizations, Extensions of Polytopes 51 4.7. Extended Relaxation Problem from Non-Negative Rank Problem 52 4.8. Lattice Embedding 52 4.9. Relaxations of Lattice Embeddings 53 4.10. Rectangle Coverings from Lattice Embeddings 54 4.11. Lattice Embeddings from Rectangle Coverings 55 4.12. Communication Complexity 56 4.13. Upper Bounds on Rectangle Covering Number 57 4.13.1. Matching Polytope 57 4.13.2. Polytopes with Few Vertices on Every Facet 57 4.13.3. Edge Polytopes 58 4.14. Lower Bound on Rectangle Covering 58 4.14.1. Fooling Sets 59 4.14.2. Linear Relaxation 59 4.14.3. Measure of Rectangles 60 4.14.4. Number of Different Sign Patterns in Columns 60 4.15. Rectangle Covering: Graph Point of View 60 4.16. Lower Bounds on Rectangle Covering Number: Rectangle Measures 61 4.17. Lower Bounds on Rectangle Covering Number: Fooling Set 62 4.17.1. Combinatorial Cube 62 4.17.2. Birkhoff Polytope 62 4.17.3. Matching Polytope in Full Bipartite Graph 63 4.18. Lower Bounds on Rectangle Covering Number: Face Counting 63 4.18.1. Permutahedron 63 4.18.2. Huffman Polytope 63 4.18.3. Cardinality Indicating Polytope 64 4.19. Lower Bounds on Rectangle Covering Number: Direct Application 64 Chapter 5. Bounds on Symmetric Extended Formulations of Polytopes 65 5.1. Symmetric Extensions 65 5.2. Symmetric Extended Formulations 68 5.3. Symmetric Section 69 5.4. Examples: Symmetric Extension, Symmetric Section 71 CONTENTS v 5.5. Faces of a Symmetric Extensions 71 5.6. Yannakakis’ Method 72 5.6.1. Action of Group G 72 5.6.2. Action of Group G = S(n) 73 5.6.3. Section Slack Covectors 73 5.7. Matching Polytope 74 5.7.1. Action of Group G = S(n) 75 5.7.2.
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