Research Collection Doctoral Thesis Unique Sink Orientations: Complexity, Structure and Algorithms Author(s): Thomas, Antonis Publication Date: 2017 Permanent Link: https://doi.org/10.3929/ethz-b-000245658 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library DISS. ETH NR. 24722 Unique Sink Orientations: Complexity, Structure and Algorithms A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich) presented by Antonios Thomas M. Sc. in Computer Science, Utrecht University born on 26.02.1988 citizen of Greece accepted on the recommendation of Prof. Dr. Bernd G¨artner,examiner Prof. Dr. Emo Welzl, co-examiner Prof. Dr. Thomas Dueholm Hansen, co-examiner 2017 Contents Abstract v Zusammenfassung ix Acknowledgments xiii 1 Overview 1 1.1 Motivation . .1 1.2 Unique Sinks on the Hypercube . .5 1.3 Our Contributions . .9 2 Preliminaries 13 2.1 Cubes . 13 2.2 Unique Sink Orientations . 15 2.3 The constructive lemmata . 21 i ii CONTENTS 3 The Complexity of Recognizing USO 25 3.1 Introduction . 25 3.2 Recognizing USOs . 27 3.3 Long Cycles in USO . 28 3.4 Recognizing Acyclic USOs . 33 3.5 Remarks . 44 4 Niceness and related concepts 45 4.1 Introduction . 45 4.2 Reachmap and niceness . 47 4.2.1 Algorithmic properties of the reachmap . 48 4.3 Random Edge on i-nice USO . 50 4.3.1 A derandomization of Random Edge . 51 4.4 On the niceness of the known lower bound constructions 54 4.4.1 Niceness upper bounds . 56 4.5 Counting 1-nice USO . 57 4.6 Bounds on niceness . 61 4.6.1 An n-nice lower bound for cyclic USO . 61 4.6.2 An upper bound for AUSO . 63 4.6.3 A matching lower bound for AUSO . 66 4.7 Fibonacci Seesaw revisited . 69 4.8 Layerings . 72 4.8.1 The Construction . 75 4.8.2 Existence of paths . 79 4.8.3 The lower bound . 83 CONTENTS iii 5 Exponential paths for memory-based pivot rules 87 5.1 Introduction . 87 5.2 Exponential lower bound for Cunningham's and John- son's rule . 90 5.2.1 The rules and the lower bound strategy . 90 5.2.2 Proving the lower bound for two list based rules 92 5.2.3 About Cunningham's and Johnson's rules . 99 5.3 Exponential lower bound for Zadeh's Rule . 100 5.4 Remarks . 111 Bibliography 112 Curriculum Vitae iv CONTENTS Abstract In this thesis we study Unique Sink Orientations (USO). Those are useful combinatorial objects that serve as an abstraction to many opti- mization problems, such as Linear Programming. The concept of USO was originally introduced by Stickney and Watson [77], in the late 70s, in the context of mathematical programming. Afterwards, it was es- sentially forgotten until revived by Szab´oand Welzl [78] in 2001. Since then, USO have been extensively studied and this thesis contains our small contributions towards better understanding the concept and how to use it as a tool in other contexts. A USO is an orientation of the n-dimensional hypercube graph such that every non-empty face induces a subgraph with a unique sink (in the graph theory sense, i.e. a vertex with only incoming edges). In particular, this implies that the whole cube has a unique sink, called global, since it is a face of itself. The graph that corresponds to a USO can either be cyclic or acyclic (in the latter case we call it an AUSO). The algorithmic problem is to find the global sink. The computational model assumes the existence of an oracle which, given a vertex, returns the orientation of its incident edges. The goal then is to minimize how many calls to this oracle are needed until the global sink is found. The v vi Abstract computational complexity of this problem is currently unsettled, with the best known algorithms being superpolynomial. In the language of USO, a polynomial-time algorithm is one that needs a polynomial number of oracle calls. Our contributions in this thesis are split into three parts. In the first part, we settle the computational complexity of the problem of recog- nizing a (A)USO from a succinct description. Specifically, our input consists of a Boolean circuit of length polynomial in n and the question to answer is if this circuit represents the oracle of a USO. We prove that this problem is coNP-complete. Afterwards, we turn our attention to the acyclic case, which appears to be much more complicated. Firstly, we give a construction of a cyclic USO which contains one unique cycle of exponential length. This implies that the canonical representation of a cycle (i.e. by listing its vertices) cannot serve as a short certificate for cyclicity. Inspired by this fact, we eventually settle the complexity of recognizing if the input Boolean circuit represents an acyclic USO by proving that this problem is PSPACE-complete. In the second part, we study some concepts that are relevant to un- derstanding the behavior of Random Edge on AUSO. This is arguably the most natural randomized pivot rule for the simplex algorithm: at every vertex choose an outgoing edge uniformly at random from the set of outgoing edges and proceed to the other endpoint of this edge. We study this algorithm from a novel angle, by exploring some concepts that were introduced by Welzl back in 2001 but remained unexplored ever since. Those are the concepts of reachmap and niceness of a USO. They pro- vide natural upper bounds for the behavior of Random Edge on AUSO. We settle the questions raised by Welzl by providing matching upper and lower bounds for the niceness of AUSO. Furthermore, we make use of the concept to show that the upper bounds obtained for Random Edge through niceness are tight or almost tight in many interesting cases. In addition and among other things, we make use of these con- n cepts to show that Random Edge is polynomial on at least nΩ(2 ) many (possibly cyclic) USO. We also give a derandomization of the algorithm which achieves asymptotically the same upper bounds. vii The last contribution of the second part is a study of the concept of layeredness which was introduced recently by Hansen and Zwick [44]. This is a generalization of niceness. We explain the exact relationship to niceness. Moreover, we translate the aforementioned derandom- ization to exploit the concept of layeredness. Our main result about layeredness is a lower bound on the number of layers needed for an AUSO. In the third and final part of the thesis, we study the behavior of some history-based pivot rules for the simplex algorithm on AUSO. These algorithms decide their next step based on the actions they have taken in the past. Specifically, we study Zadeh's least entered, Cunningham's least-recently considered and Johnson's least-recently basic rules. For all three of them we give exponential lower bounds on AUSO, thus set- tling the corresponding questions that were posed in earlier literature. Our results are the first superpolynomial lower bounds for Johnson's rule in any context and the first ones for Zadeh on AUSO. Our lower bound for Cunningham's serves as a simplification and small improve- ment of the previously known exponential lower bounds for AUSO. The ideas and techniques we use are quite general and we believe they can be applied in designing lower bounds for other deterministic simplex pivot rules (history-based or not). viii Abstract Zusammenfassung Gegenstand dieser Arbeit sind Unique Sink Orientations (USOs). Dabei handelt es sich um kombinatorische Objekte, die viele Optimierungsprob- leme abstrahieren, zum Beispiel das lineare Programmieren. Das Konzept der USO entspringt einer Arbeit ¨uber mathematisches Programmieren von Stickney und Watson [77] aus den sp¨aten70er Jahren. Grossen- teils in Vergessenheit geraten, wurden die USOs erst im Jahre 2001 von Szab´ound Welzl [78] wiederentdeckt. Heutzutage sind USOs gut un- tersuchte Objekte, und die vorliegende Arbeit enth¨altunseren kleinen Beitrag fr ein besseres abstraktes Verst¨andnisund zeigt auf, wie man USOs als Werkzeuge anderweitig verwenden kann. Eine USO ist eine Orientierung des n-dimensionalen Hyperw¨urfels,so dass jede nicht-leere Seite einen Teilgraph mit eindeutiger Senke in- duziert (im Sinne der Graphentheorie, d.h. einen Knoten ohne ausge- hende Kanten). Im speziellen bedeutet dies, dass der gesamte W¨urfel eine eindeutige Senke besitzt, die sogenannte globale Senke. Der resul- tierende gerichtete Graph kann entweder Kreise enthalten oder kreis- frei sein. Im zweiten Falle spricht man dann auch von einer azyklischen USO (AUSO). Das algorithmische Problem besteht darin, die globale Senke zu finden. ix x Zusammenfassung Das verwendete Rechenmodell stellt dabei ein Orakel bereit, das die Orientierung aller indzidenten Kanten zu einem gegebenen Knoten zur¨uckgibt. Das Ziel besteht darin, die globale Senke zu finden und gleichzeitig die Anzahl der Anfragen an das Orakel zu minimieren. Die Komplexit¨atdes eben genannten Problems ist ungekl¨art,und die besten bekannten Algorithmen sind superpolynomiell. In der Sprache der USOs ist ein polynomieller Algorithmus ein Algorithmus, der lediglich eine polynomielle Anzahl von Anfragen an das Orakel ben¨otigt. Unser Beitrag in dieser Arbeit besteht aus drei Teilen. Im ersten Teil untersuchen wir die Frage nach der algorithmischen Komplexit¨at der Erkennung einer (A)USO, die durch eine kompakte Beschreibung gegeben ist. Bei dieser Beschreibung handelt es sich um einen boo- leschen Schaltkreis polynomieller Gr¨osse, und die zu beantwortende Frage ist, ob der gegebene Schaltkreis dem Orakel einer USO entspricht.