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Investigating Polyhedra by Oracles and Analyzing Simple Extensions of Polytopes

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Investigating Polyhedra by Oracles and Analyzing Simple Extensions of Polytopes Investigating Polyhedra by Oracles and Analyzing Simple Extensions of Polytopes Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) von Dipl.-Comp.-Math. Matthias Walter geb. am 24. September 1986 in Wolmirstedt genehmigt durch die Fakultät für Mathematik der Otto-von-Guericke-Universität Magdeburg Gutachter: Prof. Dr. Volker Kaibel Otto-von-Guericke-Universität Magdeburg Prof. Dr. Marc Pfetsch Technische Universität Darmstadt eingereicht am: 16. Dezember 2015 Verteidigung am: 29. März 2016 Dedicated to Sarah. Zusammenfassung Ein wichtiges Hilfsmittel der Polyedrischen Kombinatorik ist die computer- gestützte Analyse von Polyedern, um deren strukturelle Eigenschaften zu untersuchen. Aufgrund des enumerativen Charakters der klas- sischen Algorithmen sind diese Ansätze bereits bei verhältnismäßig kleinen Dimensionen nur noch mit enormen Ressourcen zu bewältigen. Da oft die konvexen Hüllen der Lösungsmengen von Optimierungs- problemen von Interesse sind und es sehr effiziente Software (zum Beispiel im Falle gemischt-ganzzahliger linearer Optimierungsprobleme (MIPs)) zur Lösung dieser Probleme gibt, stellt sich die Frage, inwieweit man eine Analyse basierend auf solcher Software, oder formal einem Optimierungsorakel, durchführen kann. Dafür wurde im Rahmen der Arbeit die Softwarebibliothek IPO (Investigating Polyhedra by Oracles) entwickelt. Der erste Schritt einer Analyse ist in dem meisten Fällen die Bestimmung der affinen Hülle eines Polyeders P Rn, insbesondere der Dimension. Dazu wurde ein Algorithmus entwickelt,⊆ der mit 2n Orakel-Aufrufen auskommt und in der Praxis sehr effizient ist. Zudem wurde die Optimalität dieser Anzahl für den allgemeinen Fall bewiesen. Unter Nutzung einer entsprechen- den Implementation und des MIP-Lösers SCIP wurden Instanzen der MIPLIB 2.0 betrachtet und Dimensionen verschiedener Polyeder be- stimmt: Lineare Relaxierung und gemischt-ganzzahlige Hüllen vor und • nach Anwendung der Presolve-Routinen von SCIP. Optimale Seiten bezüglich der Zielfunktion der Instanz. • Durch gegebene Ungleichungen induzierte Seiten. • Die Hauptanwendung von IPO ist die Bestimmung von Facetten von P, da diese zu nicht dominierten Ungleichungen in einer Ungleichungs- beschreibung korrespondieren und daher ein großes Potential besitzen, bei der Lösung (mit Hilfe von MIP-Lösern) von Nutzen zu sein. Facetten- definierende Ungleichungen, welche von einem gegebenen Punkt maxi- mal verletzt werden, können mit Hilfe von Target-Cuts effektiv bestimmt werden. Dies wird zum Beispiel für Linearisierungen von Matching- Polytopen mit einem quadratischen Term demonstriert, indem mit Hilfe von IPO eine bisher unbekannte Klasse von Facetten entdeckt wurde. i Zusätzlich zur Facetten- und Affine-Hülle-Bestimmung hat IPO eine dritte Funktionalität, die die Überprüfung der Adjazenz zweier Ecken von P betrifft. Diese ist äquivalent dazu, dass der Normalenkegel an ihren Mittelpunkt (n 1)-dimensional ist. Da man mit Hilfe eines Optimierungsorakels für− P – ähnlich zu Target Cuts – ein Optimierungs- orakel für den betrachteten Normalenkegel konstruieren kann, reduziert sich das Adjazenzproblem ebenfalls auf das Affine-Hülle-Problem. Zur Demonstration wurden eine große Anzahl von Adjazenzen zufälliger Ecken von TSP-Polytopen überprüft, wobei als Orakel die TSP-Software concorde genutzt wurde. Damit IPO grundsätzlich korrekt arbeitet, die Probleme jedoch mög- lichst effizient löst, wird an vielen Stellen sowohl rationale als auch Gleitkomma-Arithmetik verwendet. Dies ist besonders bei Dimensio- nen größer als 500 enorm relevant, da hier gelegentlich exakte Zahlen mit Kodierungslängen von über 106 Bits auftreten. Um weiterhin die meist laufzeitintensiven Orakelaufrufe zu sparen, erlaubt IPO die Nut- zung von Heuristiken, von denen außer der Zulässigkeit der ange- gebenen Lösungen nichts verlangt wird. So können beispielsweise die gerundeten Lösungen von SCIP als heuristisch betrachtet und die exakte Version von SCIP zur “Verifikation” genutzt werden. Der zweite Teil der Arbeit fällt in das Forschungsgebiet der Erweiter- ten Formulierungen und basiert auf einem gemeinsam mit Volker Kaibel veröffentlichten Artikel. In diesem Gebiet geht es im Wesentlichen darum, ein gegebenes Polytop P als affine Projektion eines anderen Poly- tops Q mit möglichst wenigen Facetten darzustellen. Die größte Moti- vation hierfür ist die (in gewissem Sinne äquivalente) Frage, ob man ein lineares Optimierungsproblem über P mit Hilfe von weiteren Variablen als ein lineares Optimierungsproblem darstellen kann, welches mit sehr wenigen Ungleichungen auskommt. In der Arbeit wurde untersucht, inwieweit solche kompakten Dar- stellungen möglich sind, wenn man noch zusätzlich fordert, dass das Erweiterungspolytop Q ein einfaches Polytop ist, das gesuchte lineare Optimierungsproblem also nicht (primal) degeneriert ist. Einerseits haben bereits bekannte Klassen von Erweiterungspolytopen diese Eigen- schaft, andererseits kann man für jedes Polytop zunächst ein einfaches Erweiterungspolytop konstruieren. Die charakteristische Größe ist die sogenannte simple extension complexity, die die minimale Facettenzahl ii eines solchen einfachen Erweiterungspolytops angibt. Hierzu wurde eine Technik entwickelt, die es ermöglicht, aufgrund von Eigenschaften des Graphen von P untere Schranken an diese Größe zu bestimmen. Trotz der genannten positiven Beispiele stellt sich her- aus, dass die Eigenschaft sehr selten ist: Bereits wenig komplizierte Polytope, wie Hypersimplizes, die zudem nur schwach degeneriert sind, haben eine simple extension complexity in der Größenordnung der Anzahl der Ecken des Polytops, einer trivialen oberen Schranke. Wenig überraschend ist da, dass dies auch für zahlreiche kombina- torische Polytope der Fall ist, unter ihnen perfekte Matching-Polytope vollständiger und vollständiger bipartiter Graphen, Flusspolytope un- zerlegbarer azyklischer Netzwerke, Spannbaumpolytope vollständiger Graphen, sowie zufällige 0=1-Polytope mit gewissen Eckenanzahlen. Um die untere Schranke im Falle perfekter Matchings auf vollständigen Graphen zu beweisen, wurde ein Resultat von Padberg & Rao über Adjazenzbeziehungen entsprechender Polytope verbessert. In einem kurzen Abschnitt dieses Teils wird charakterisiert, wann zwei der (wenigen) bekannten Methoden zur Konstruktion von Er- weiterten Formulierungen Einfachheit herstellen. iii Summary An important tool of Polyhedral Combinatorics is the computer-aided analysis of polyhedra for understanding their structural properties, which are often related to an underlying optimization problem. Due to the enumerative nature of the classical algorithms for polyhedral analysis this approach can often only be pursued for relatively small dimensions or using an enormous amount of resources. Since many polyhedra of interest are convex hulls of feasible sets of optimization problems and since there exists efficient software (e.g., in case of mixed-integer linear optimization problems (MIPs)) for solving these problems even in higher dimensions, it is a reasonable question how to carry out an analysis based on such software, or more formally, on an optimization oracle. This thesis is accompanied by a software library IPO (Investigating Polyhedra by Oracles) specifically designed for this approach. A first step in such an analysis is typically the computation of the affine hull of a polyhedron P Rn, in particular its dimension. For this we developed an algorithm that⊆ needs at most 2n oracle calls and is very efficient in practice. We also prove that this number is the minimum in the worst case. Using our implementation and the MIP solver SCIP we investigated instances of the MIPLIB 2.0 and determined dimensions of several polyhedra: Linear relaxations and mixed-integer hulls before and after apply- • ing presolve routines of SCIP. Optimal faces with respect to the objective of the instance. • Faces induced by given inequalities. • The main application of IPO is the detection of some of P’s facets since those correspond to undominated inequalities in an inequality descrip- tion, and hence usually have great potential of being useful during the solving process (using a MIP solver). Facet-defining inequalities that are maximally violated by a given point can be found efficiently us- ing Target Cuts. In particular, this is demonstrated for linearizations of matching polytopes with one quadratic term, for which we determined a new class of facets using IPO. iv In addition to facet- and affine-hull computations, IPO has a third component that can check adjacency of two vertices of P. The latter is equivalent to the property that the dimension of the normal cone at their barycenter is equal to n 1. Since we can – similar to Target Cuts – construct an optimization oracle− for this normal cone, this problem reduces to the affine-hull problem. Again for demonstration purposes we tested a huge number of vertex pairs of TSP polytopes for adjacency, using the software concorde as an optimization oracle. Since we strive for efficiency, but still want to ensure correct behavior of IPO, we often use floating-point- and rational arithmetic. This is par- ticularly important for dimensions greater than 500 as then sometimes the occurring exact numbers have encoding lengths of more than 106 bits. In order to save costly oracle calls, IPO allows to use heuristics from which we do not require optimality of returned solutions. This way we can use, for instance, the rounded (potentially incorrect) solutions of SCIP and use the exact version of SCIP for verification purposes
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