Mathematical Optimization and Public Transportation Dr. Ralf Bornd¨orfer Kumulative Habilitationsschrift an der Fakult¨at II – Mathematik und Naturwissenschaften der Technischen Universit¨at Berlin Lehrgebiet: Mathematik Er¨offnung des Verfahrens: 30.09.2009 Verleihung der Lehrbef¨ahigung: 20.06.2010 Gutachter Prof. Dr. Dr. h.c. mult. Martin Gr¨otschel Prof. Dr. Thomas Liebling Prof. Dr. Rolf H. M¨ohring Prof. Dr. Uwe Zimmermann Berlin 2010 D 83 Table of Contents Titlepage i Table of Contents i List of Figures iii List of Tables v List of Publications vii Introduction viii I Set Packing Relaxations of Some Integer Programs 1 1 Introduction .......................... 1 2 Terminology .......................... 3 3 The Acyclic Subdigraph and the Linear Ordering Problem 7 4 The Clique Partitioning Problem ............... 16 5 The Set Packing Problem ................... 20 II Combinatorial Packing Problems 32 1 Introduction .......................... 32 2 Combinatorial Packing .................... 34 3 Dantzig-Wolfe Set Packing Formulations ........... 39 III Decomposing Matrices into Blocks 49 1 Introduction .......................... 50 2 Integer Programming Formulation and Related Problems . 52 3 Polyhedral Investigations ................... 55 4 A Branch-And-Cut Algorithm ................ 67 5 Computational Results .................... 78 IV A Bundle Method for Integrated Multi-Depot Vehicle and Duty Scheduling in Public Transit 92 1 Introduction .......................... 93 2 Notation ............................ 95 3 Integrated Vehicle and Duty Scheduling ........... 95 4 A Bundle Method ....................... 98 5 Computational Results .................... 109 6 Conclusions ........................... 115 i Table of Contents ii V Models for Railway Track Allocation 118 1 Introduction .......................... 119 2 The Optimal Track Allocation Problem ........... 120 3 Integer Programming Models ................. 122 4 Computational Results .................... 129 VI A Column Generation Approach to Line Planning in Pub- lic Transport 135 1 Introduction .......................... 135 2 Related Work .......................... 137 3 Line Planning Model ...................... 138 4 Column Generation ...................... 143 5 Computational Results .................... 149 6 Conclusions ........................... 154 Index 157 Curriculum Vitae 174 List of Figures I Set Packing Relaxations of Some Integer Programs 1 1 A 4-Fence. ........................... 9 2 A M¨obius Ladder of 5 Dicycles. .............. 9 3 A Fence Clique of Forks. ................... 10 4 A M¨obius Cycle of Dipaths. .................. 10 5 A 2-Chorded Cycle. ...................... 17 6 Labeling Lower Triangles. ................... 18 7 An Odd Cycle of Lower Triangle Inequalities. ........ 19 8 A 5-Wheel. ........................... 22 9 A Cycle of Nodes and Edges. ................. 22 10 Two Generalizations of Odd Wheel Inequalities. ...... 23 11 A 5-Wheel of Type I and a 5-Cycle of Paths. ........ 24 12 A 5-Wheel of Type II and a 5-Cycle of Paths. ........ 25 13 A 5-Cycle of 5-Cycles. ..................... 27 II Combinatorial Packing Problems 32 1 Bipartite 2-Coloring. ...................... 38 2 Intersection Graph of a Combinatorial 2-Packing Problem. 42 III Decomposing Matrices into Blocks 49 1 Decomposing a Matrix into Bordered Block Diagonal Form. 50 IV A Bundle Method for Integrated Multi-Depot Vehicle and Duty Scheduling in Public Transit 92 1 IS-OPT Runtime Chart. .................... 107 2 Scheduling Graph for Scenario Fulda. ............ 113 V Models for Railway Track Allocation 118 1 Optimal Track Allocation Problem: Infrastructure Network (left), and Train Routing Digraph (right); Individual Train Routing Digraphs Bear Different Colors. ........... 121 2 Block Conflicts on a Single Track: Trips for a Slow (blue) and a Fast (red) Train (left), a Conflict-Free Configuration of Four Trips on this Track (middle), and the Block Conflict Graph Associated With the Track (right). .......... 123 iii List of Figures iv 3 Configuration Routing Digraph for a Single Track: Train Routing Digraph (left), Configuration (half-left), Configu- ration Routing Digraph (half-right), and the Corresponding Path (right). .......................... 127 4 Solving Models APP′, ACP, and PCP. ............ 131 VI A Column Generation Approach to Line Planning in Pub- lic Transport 135 1 Multi-Modal Transportation Network in Potsdam. Black: Tram, Lightgray: Bus, Darkgray: Ferry, Large Nodes: Ter- minals, Small Nodes: Stations, Grey: Rivers and Lakes. 139 2 The Node Splitting Gadget in the Proof of Proposition 4.1 145 3 Total Traveling Time (solid, left axis) and Total Line Cost (dashed, right axis) in Dependence on λ (x-axis in logscale). 153 List of Tables I Set Packing Relaxations of Some Integer Programs 1 II Combinatorial Packing Problems 32 III Decomposing Matrices into Blocks 49 1 Decomposing Linear Programming Basis Matrices. ..... 81 2 Decomposing Matrices of Mixed Integer Programs. ..... 83 3 Decomposing Transp. Matrices of Mixed Integer Programs. 85 4 Comparing Integer Programming Branching Rules. ..... 86 5 Decomposing Steiner-Tree Packing Problems. ........ 88 6 Equipartitioning Matrices. .................. 89 IV A Bundle Method for Integrated Multi-Depot Vehicle and Duty Scheduling in Public Transit 92 1 Statistics on the RVB Instances. ............... 110 2 Results for the RVB Sunday Scenario. ............ 111 3 Results for the RVB Workday Scenario. ........... 112 4 RKH Scenarios Marburg and Fulda. ............. 113 5 Solutions for Scenarios Marburg and Fulda. ......... 114 6 Results for ECOPT-Instances with 2 Depots Variant A. 114 7 Results for ECOPT-Instances with 4 Depots Variant A. 115 V Models for Railway Track Allocation 118 1 Notation for the Optimal Track Allocation Problem. 120 2 Sizes of Packing Formulations for the Track Alloc. Problem. 124 3 Sizes of Packing Formulations for the Track Alloc. Problem. 126 4 Test Scenarios. ......................... 131 5 Solving Model APP′ for Scenario 570. ............ 132 6 Solving Model ACP for Scenario 570. ............ 132 7 Solving Model PCP for Scenario 570. ............ 133 VI A Column Generation Approach to Line Planning in Pub- lic Transport 135 1 Notation and Terminology. .................. 140 2 Experimental Line Planning Results for λ = 0.9978. 151 v List of Algorithms I Set Packing Relaxations of Some Integer Programs 1 II Combinatorial Packing Problems 32 III Decomposing Matrices into Blocks 49 1 Separating z-Cover Inequalities With a Greedy Heuristic. 68 IV A Bundle Method for Integrated Multi-Depot Vehicle and Duty Scheduling in Public Transit 92 1 Generic Proximal Bundle Method (PBM). .......... 101 2 Inexact Proximal Bundle Method (PBM) with Column Gen- eration. ............................. 106 V Models for Railway Track Allocation 118 VI A Column Generation Approach to Line Planning in Pub- lic Transport 135 vi List of Publications [1] R. Borndorfer¨ & R. Weismantel. Set packing relaxations of some integer programs. Mathematical Programming 88:425–450, 2000. [2] R. Borndorfer¨ . Combinatorial packing problems. In M. Grotschel¨ , (Ed.), The Sharpest Cut – The Impact of Manfred Padberg and His Work, pp. 19–32. SIAM, Philadelphia, 2004. [3] R. Borndorfer,¨ C. E. Ferreira & A. Martin. Decomposing matrices into blocks. SIAM Journal on Optimization 9(1):236–269, 1998. [4] R. Borndorfer,¨ A. Lobel¨ & S. Weider. A bundle method for integrated multi-depot vehicle and duty scheduling in public transit. In M. Hickman, P. Mirchandani & S. Voß, (Eds.), Computer-aided Systems in Public Transport, vol. 600 of Lecture Notes in Economics and Mathematical Systems, pp. 3–24. Springer-Verlag, 2008. [5] R. Borndorfer¨ & T. Schlechte. Models for railway track allocation. In C. Liebchen, R. K. Ahuja & J. A. Mesa, (Eds.), Proceeding of the 7th Workshop on Algorithmic Approaches for Transportation Model- ing, Optimization, and Systems (ATMOS 2007), Dagstuhl, Germany, 2007. Internationales Begegbnungs- und Forschungszentrum f¨ur In- formatik (IBFI), Schloss Dagstuhl, Germany. [6] R. Borndorfer,¨ M. Grotschel¨ & M. E. Pfetsch. A column-generation approach to line planning in public transport. Transportation Science 41(1):123–132, 2007. vii Introduction I submit in this cumulative thesis the following six papers for obtaining the habilitation at the Technische Universit¨at Berlin, Fakult¨at II – Mathematik und Naturwissenschaften: (1) Set packing relaxations of some integer programs. (2) Combinatorial packing problems. (3) Decomposing matrices into blocks. (4) A bundle method for integrated multi-depot vehicle and duty scheduling in public transit. (5) Models for railway track allocation. (6) A column-generation approach to line planning in public transport. I have made some changes to the papers compared to the published versions. These pertain to layout unifications, i.e., common numbering, figure, table, and chapter head layout. I have not changed any notation or symbols, but I have eliminated some typos, updated the references, and added links and an index. The mathematical content is identical. The papers are about the optimization of public transportation systems, i.e., bus networks, railways, and airlines, and its mathematical foundations, i.e., the theory of packing problems. The papers discuss mathematical models, theoretical analyses, algorithmic approaches, and computational aspects of and to problems in this area. Papers 1, 2, and 3 are theoretical. They aim at establishing a theory of packing problems as a general