Aspects of Set Packing Partitioning and Covering Ralf Borndorfer C L C L C L L C L C Aspects of Set Packing Partitioning and Covering vorgelegt von DiplomWirtschaftsmathematiker Ralf Borndorfer Vom Fachb ereich Mathematik der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften genehmigte Dissertation Promotionsausschu Vorsitzender Prof Dr Kurt Kutzler Berichter Prof Dr Martin Grotschel Berichter Prof Dr Rob ert Weismantel Tag der wissenschaftlichen Aussprache Marz Berlin D Zusammenfassung Diese Dissertation b efat sich mit ganzzahligen Programmen mit Systemen SetPacking Partitioning und CoveringProbleme Die drei Teile der Dissertation b ehandeln p olyedrische algorithmische und angewandte Asp ekte derartiger Mo delle Teil diskutiert p olyedrische Asp ekte Den Auftakt bildet ei ne Literaturubersicht in Kapitel In Kapitel untersuchen wir SetPackingRelaxierungen von kombinatorischen Optimierungspro blemen ub er Azyklische Digraphen und Lineare Ordnungen Schnitte und Multischnitte Ub erdeckungen von Mengen und ub er Packungen von Mengen Familien von Ungleichungen fur geeignete SetPacking Relaxierungen sowie deren zugehorige Separierungsalgorithmen sind auf diese Probleme ub ertragbar T eil ist algorithmischen und rechnerischen Asp ekten gewidmet Wir dokumentieren in Kapitel die wesentlichen Bestandteile ei nes BranchAndCut Algorithmus zur Losung von SetPartitioning Problemen Der Algorithmus implementiert einige der theoretischen Ergebnisse aus Teil Rechenergebnisse fur Standardtestprobleme der Literatur werden b erichtet Teil ist angewandt Wir untersuchen die Eignung von Set PartitioningMetho den zur Optimierung des Berliner Behinderten fahrdienstes Telebus der mit einer Flotte von Fahrzeugen taglich etwa Fahrwunsche b edient Der BranchAndCut Algorith mus aus Teil ist ein Bestandteil eines Systems zur Fahrzeugein satzplanung das seit dem Juni in Betrieb ist Dieses Sy stem ermoglichte Verb esserungen im Service und gleichzeitig erhebli che Kosteneinsparungen Schlusselb egrie Ganzzahlige Programmierung Polyedrische Kombinatorik Schnitteb enen BranchAndCut Anrufsammeltaxi systeme Fahrzeugeinsatzplanung Mathematics Sub ject Classication MSC C Abstract This thesis is ab out integer programs with constraint systems Set packing partitioning and covering problems The three parts of the thesis investigate p olyhedral algorithmic and application asp ects of such mo dels Part discusses p olyhedral asp ects Chapter is a prelude that sur veys results on integer programs from the literature In Chapter we investigate set packing relaxations of combinatorial optimization problems asso ciated with acyclic digraphs and linear orderings cuts and multicuts multiple knapsacks set coverings and no de packings themselves Families of inequalities that are valid for such a relaxation and the asso ciated separation routines carry over to the problems un der investigation Part is devoted to algorithmic and computational asp ects We do cu ment in Chapter the main features of a branchandcut algorithm for the solution of set partitioning problems The algorithm implements some of the results of the theoretical investigations of the preceding part Computational exp erience for a standard test set from the liter ature is rep orted Part deals with an application We consider in Chapter set par titioning metho ds for the optimization of Berlins Telebus for handi capp ed p eople that services requests p er day with a eet of mini busses Our branchandcut algorithm of Part is one mo dule of a scheduling system that is in use since June and resulted in improved service and signicant cost savings Keywords Integer Programming Polyhedral Combinatorics Cutting Planes BranchandCut Vehicle Scheduling DialARide Systems Mathematics Sub ject Classication MSC C Preface Asp ects of set packing partitioning and covering is the title of this thesis and it was chosen delib erately The idea of the thesis is to try to b end the b owfrom theory via algorithms to a practical application but the red thread is not always pursued conclusively This resulted in three parts that corresp ond to the three parts of the b ow and b elong together but that can also stand for themselves This selfcontainment is reected in separate indices and reference lists There is no explanation of notation or basic concepts of optimization Instead I have tried to resort to standards and in particular to the book Grotschel Lovasz Schrijver Geometric Algorithms and Combinatorial Optimization Springer Verlag Berlin It is p erhaps also useful to explain the system of emphasis that is at the bottom of the writing Namely emphasized words exhibit either the topic of the curren t paragraph andor they mark contents of the various indices or they sometimes just stress a thing I am grateful to the Senate of Berlins Departments for Science Re search and Culture and for So cial Aairs that supp orted the Tele bus pro ject and to Fridolin Klostermeier and Christian Kuttner for their co op eration in this pro ject Iamindebted to the KonradZuse Zentrum for its hospitality and for its supp ort in the publication of this thesis Iwould like to thank my sup ervisor Martin Grotschel for his example not only as a mathematician and esp ecially for his patience I also thank Andreas Schulz and Akiyoshi Shioura who have kindly p ointed out anumb er of errors in an earlier version of this thesis My friends and Alexander Martin havehelpedme Norb ert Ascheuer Bob Bixby with many discussions on asp ects of this thesis and I want to express my gratitude for this A sp ecial thanks go es to Andreas Lob el for his friendship and supp ort My last sp ecial thanks go es to my friend Rob ert Weismantel I simply want to say that without him not only this thesis would not b e as it is Ihopethat who ever reads this can prot a little from these notes and p erhaps even enjoy them Berlin August Ralf Borndorfer Contents Zusammenfassung v Abstract vii Preface ix I Polyhedral Asp ects Integer Programs Two Classical Theorems of Konig Intro duction The Set Packing Partitioning and Covering Problem Relations to Stable Sets and Indep endence Systems Blo cking and AntiBlo cking Pairs Perfect and Ideal Matrices Minor Characterizations Balanced Matrices The Set Packing Polytop e Facet Dening Graphs Comp osition Pro cedures Polyhedral Results on ClawFree Graphs Quadratic and Semidenite Relaxations Adjacency The Set Covering Polytop e Facet Dening Matrices Set Packing Relaxations Intro duction The Construction The Acyclic Sub digraph and the Linear Ordering Problem The Clique Partitioning Multi and Max Cut Problem The Set Packing Problem Wheel Inequalities A New Family of Facets for the Set Packing Polytop e Chain Inequalities Some Comp osition Pro cedures The Set Covering Problem The Multiple Knapsack Problem The Programming Problem with Nonnegative Data Bibliography of Part Index of Part xii Contents II Algorithmic Asp ects An Algorithm for Set Partitioning Intro duction Prepro cessing Reductions Data Structures Probabilistic Analyses Empty Columns EmptyRows and Row Singletons Duplicate and Dominated Columns Duplicate and Dominated Rows Row Cliques Parallel Columns Symmetric Dierences Column Singletons Reduced Cost Fixing Probing Pivoting The Prepro cessor Separation The Fractional Intersection Graph Clique Inequalities Cycle Inequalities Aggregated Cycle Inequalities Computational Results Bibliography of Part Index of Part III Application Asp ects Vehicle Scheduling at Telebus Intro duction Telebus The Vehicle Scheduling Problem Pieces of Work Requests Constraints Ob jectives Solution Approach Transition Network Decomp osition Set Partitioning AVehicle Scheduling Algorithm Related Literature