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Shortest Path Routing Modelling, Infeasibility and Polyhedra Linköping Studies in Science and Technology. Dissertations. No. 1486 Shortest Path Routing Modelling, Infeasibility and Polyhedra Mikael Call Department of Mathematics Linköping University, SE–581 83 Linköping, Sweden Linköping 2012 Linköping Studies in Science and Technology. Dissertations. No. 1486 Shortest Path Routing – Modelling, Infeasibility and Polyhedra Mikael Call [email protected] www.mai.liu.se Division of Optimization Department of Mathematics Linköping University SE–581 83 Linköping Sweden ISBN 978-91-7519-751-7 ISSN 0345-7524 Copyright ⃝c 2012 Mikael Call Printed by LiU-Tryck, Linköping, Sweden 2012 In memory of my father. Abstract The Internet is constantly growing but the available resources, i.e. bandwidth, are limited. Using bandwidth efficiently to provide high quality of service to users is referred to as traffic engineering. This is of utmost importance. Traffic in IP networks is commonly routed along shortest paths with respect to auxiliary link weights, e.g. using the OSPF or IS-IS protocol. Here, shortest path routing is indirectly controlled via the link weights only, and it is therefore crucial to have a profound understanding of the shortest path routing mechanism to solve traffic engineering problems in IP networks. The theoretical aspects of such problems have received little attention. Traffic engineering in IP networks leads to very difficult optimization problems and the key element in exact methods for such problems is an inverse shortest path routing problem. It is used to answer the fundamental question of whether there exist link weights that reproduce a set of tentative paths. Path sets that cannot be obtained correspond to routing conflicts. Valid inequalities are instrumental to prohibit such routing conflicts. We analyze the inverse shortest path routing problem thoroughly. We show that the problem is NP-complete, contrary to what is claimed in the literature, and propose a stronger relaxation. Its Farkas system is used to develop a novel and compact formulation based on cycle bases, and to classify and characterize minimal infeasible subsystems. Valid inequalities that prevent routing conflicts are derived and separation algorithms are developed for such inequalities. We also consider several approaches to modelling traffic engineering problems, espe- cially Dantzig–Wolfe reformulations and extended formulations. We give characteriza- tions of facets for some relaxations of traffic engineering problems. Our results contribute to the theoretical understanding, modelling and solution of problems related to traffic engineering in IP networks. v Populärvetenskaplig sammanfattning Internet är ett världsomspännande nätverk som växer för var dag. Det består av tusentals mindre IP-nätverk. Den viktigaste uppgiften för en operatör av ett IP-nätverk är trafikpla- nering. Det innebär att bestämma vilka vägar som datatrafiken i nätverket skall använda. Detta kallas för ruttning och sker oftast via så kallade kortaste-väg-protokoll, såsom till exempel OSPF eller IS-IS. Det betyder att alla vägar som används måste vara kortas- te vägar med avseende på en uppsättning artificiella vikter på länkarna i nätverket. En fundamental implikation blir att dessa vikter är det enda tillgängliga medlet för att styra trafiken i sådana IP-nätverk. Således är det mycket viktigt att förstå hur denna typ av rutt- ning fungerar. Framförallt måste man kunna avgöra om en uppsättning önskade vägar går att realisera via någon uppsättning vikter. Detta problem kallas för det inversa kortaste- väg-problemet. I denna avhandling studeras trafikplanering i IP-nätverk där trafik ruttas längs kortaste vägar. De beskrivs som optimeringsproblem och angrips med metoder baserade på hel- talsoptimering och bivillkorsgenerering. Vi betraktar framförallt den teoretiska aspekten av problem där trafik ruttas längs kortaste vägar. Det viktigaste verktyget för att analysera sådana problem är via det inversa kortaste-väg-problemet. Den metod vi använder baseras på att karaktärisera när detta problem inte har några tillåtna lösningar. I sådana fall kan en så kallad ruttningskonflikt identifieras. Vår analys resulterar i en matematisk karaktärisering av ruttningskonflikter. Denna leder i sin tur till metoder för att hitta samt förbjuda sådana ruttningskonflikter. Dessa metoder kan sedan integreras med andra metoder som vanligen används för att lösa trafik- planeringsproblem utan den komplicerade aspekten att ruttning måste ske längs kortaste vägar. vii Acknowledgments I would like to start by expressing my gratitude towards my supervisor Professor Kaj Holmberg for his support and encouragement. I am very thankful for the opportunity to conduct research studies in combinatorial optimization, and have greatly enjoyed the freedom to investigate my own research ideas. The collaboration with Professor Andreas Bley and Daniel Karch at TU Berlin have meant a lot to me. Andreas has given me feedback and insights about IP routing in practice when acting as the opponent on my Licentiate thesis and during my short visits at TU Berlin. Daniel has been an invaluable source of discussion and motivation. He also provided me with a nice computational framework in SCIP for experimenting. I believe that our continued collaboration will be very fruitful. I also want to thank my present and former colleagues. My roommate Åsa Holm has been a great asset, socially and scientifically. I enjoy our discussions and you make me realize when I am wrong — not only when proofreading the hardest parts of this thesis. Professor Torbjörn Larsson have contributed to my understanding of various topics in optimization by many valuable discussions, thank you. Finally, I want to thank my family and friends for all your love and support. Especially my wife Anna, who has stood by me through the entire process, and given me the best gift imaginable in our daughter Alice. Linköping, October 31, 2012 Mikael Call ix Contents 1 Introduction 1 1.1 Thesis Outline ................................ 3 1.2 Contributions ................................ 5 1.2.1 Contributions Related to the Client Problem ............ 6 1.2.2 Contributions Related to the Master Problem ........... 6 I Background 9 2 Bilevel Shortest Path Problems 11 2.1 The Shortest Path Problem ......................... 12 2.2 The Inverse Shortest Path Routing Problem ................ 14 2.3 Bilevel Shortest Path Problems ....................... 16 2.4 Examples of Bilevel Shortest Path Applications .............. 19 2.4.1 IP Network Routing Problems ................... 19 2.4.2 Tariff Optimization Problems ................... 22 2.5 Solving Bilevel Shortest Path Problems .................. 27 2.5.1 A Heuristic Approach: Search in the Arc Cost Space ....... 27 2.5.2 An Exact Approach: Solve the MILP Formulation ........ 31 3 Introduction to Inverse Shortest Path Routing Problems 33 3.1 Inverse Shortest Path Problems ....................... 33 3.1.1 A Polynomial Model for the Inverse Shortest Path Problem .... 34 3.2 Inverse Shortest Path Routing Problems .................. 35 3.2.1 A Polynomial Model for the Inverse Shortest Path Routing Problem 36 3.3 Remarks on Inverse Shortest Path Routing ................. 38 xi xii Contents II Inverse Shortest Path Routing 41 4 Complexity of Realizability 43 4.1 Background ................................. 43 4.2 Models for Realizability and Compatibility ................ 45 4.2.1 Incompatible and Unrealizable SP-graphs ............. 47 4.3 Complexity of ISPR Problems ....................... 49 4.4 Conclusion ................................. 55 5 Relaxations of Realizability 57 5.1 Background ................................. 57 5.1.1 Inverse Shortest Path Routing Problems .............. 58 5.2 Partial Realizability ............................. 60 5.2.1 Generalized Shortest Path Graphs ................. 62 5.2.2 Relation Between ISPR Formulations ............... 63 5.3 Valid Inequalities from Partial Realizability ................ 64 6 Classes of Infeasible Structures 69 6.1 Problem Formulation ............................ 69 6.1.1 The ISPR Compatibility Model .................. 70 6.1.2 The ISPR Partial Realizability Model ............... 71 6.2 Classes of Infeasible Routing Pattern Structures .............. 72 6.2.1 The General Infeasible Routing Pattern Structure ......... 73 6.2.2 The Binary, Unitary and Simplicial Structures ........... 77 6.3 Extreme Rays and Generators ....................... 79 6.3.1 Representation of Polyhedral Cones ................ 79 6.3.2 Extreme Rays and Generators of the Closure of Θ ......... 79 6.3.3 Irreducible Solutions in the Closure of Θ ............. 82 6.4 The Hierarchy of Infeasible Structures ................... 83 7 The Simplicial Structure 91 7.1 The Valid Cycle Structure .......................... 91 7.1.1 Valid Cycles with a Single SP-graph ................ 92 7.1.2 Saturating Solutions with Two SP-Graphs ............. 93 7.1.3 Non-Saturating Solutions with Two SP-Graphs .......... 94 7.1.4 Algorithms to Find Generalized Valid Cycles ........... 96 7.2 A Characterization of the Simplicial Structure ............... 97 7.2.1 Generating Simplicial Cycle Families ............... 98 7.3 Dependency Graphs ............................. 102 7.4 Characterization of Irreducible Generators ................. 106 8 Cycle Basis Formulations 111 8.1 Modelling with Cycle Bases ........................ 112 8.1.1 Oriented Circuits, Circulations and Cycle Bases .......... 112 8.1.2 Fundamental Cycle Bases ..................... 113 8.1.3 Modelling Circulations with Cycle Bases
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