Gems of Combinatorial Optimization and Graph Algorithms

153 Pages·2016·3.32 MB·English

by  Andreas S. Schulz

Andreas S. Schulz
Martin Skutella
Sebastian Stiller
Dorothea Wagner
Editors
Gems of
Combinatorial
Optimization
and Graph
Algorithms
Gems of Combinatorial Optimization
and Graph Algorithms
Andreas S. Schulz Martin Skutella
(cid:129)
Sebastian Stiller Dorothea Wagner
(cid:129)
Editors
Gems of Combinatorial
Optimization and
Graph Algorithms
123
Editors
Andreas S.Schulz Sebastian Stiller
Technische UniversitätMünchen Technische UniversitätBraunschweig
Munich Braunschweig
Germany Germany
Martin Skutella Dorothea Wagner
Technische UniversitätBerlin Karlsruher Institut für Technologie
Berlin Karlsruhe
Germany Germany
ISBN978-3-319-24970-4 ISBN978-3-319-24971-1 (eBook)
DOI 10.1007/978-3-319-24971-1
LibraryofCongressControlNumber:2015953249
SpringerChamHeidelbergNewYorkDordrechtLondon
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To the joint mentor of the contributors to this
book, Prof. Dr. Rolf H. Möhring, on the
occasion of his retirement.
Preface
This book is intended as an homage to Prof. Rolf H. Möhring, an accomplished
leader in the field of combinatorial optimization and graph algorithms. We con-
tacted his former advisees who have successfully launched their own academic
careers. We asked them each to pick a topic that they thought would be both
interesting and accessible to a wide audience with a basic knowledge of graphs,
algorithms, and optimization. The result is this collection of beautiful results.
The reader can learn about the connection between shortest paths and mecha-
nismdesign,abouttheinterplayofpriorityrulesinschedulingandtheexistenceof
pure-strategy Nash equilibria in weighted congestion games, about the critical role
played by matroids in the existence of pure-strategy Nash equilibria in
resource-buying games, about some geometric commonality between proportional
resource allocation and selfish flows, about using the gasoline puzzle and the
adjacency structure of the matching polytope to solve the budgeted matching
problem, about the relation between the knotting graph and the linear structure of
graphs, about convex programming relaxations and randomized rounding in
scheduling,aboutthesignificanceofmotifsinnetworkanalysis,abouttheanalogy
between contraction hierarchies used for fast shortest path computations and
(perfect)eliminationorderingsingraphs,aboutuniversallygoodalgorithmsforthe
knapsack problem with varying capacity and for a scheduling problem, about the
pivotal role of Hanan grids for the minimum Steiner tree problem for rectilinear
distances,aboutalinear-timealgorithmthatcomputesthelongesttourforpointsin
the plane under the taxicab distance, and about a characterization of certain rect-
angular dissections with surprising applications.
Eachchaptercaneasily beusedasthebasisfor alecture ortwoinanadvanced
undergraduate course or in a graduate course on graph algorithms, combinatorial
optimization, algorithmic game theory, or computational geometry. For improved
readability, citations within a chapter are kept to a minimum, but each chapter
concludes with a discussion of the relevant literature and provides pointers for
further reading.
vii
viii Preface
We are indebted to all the contributors for their enthusiasm about our idea and
the time and care they put into preparing their chapters. We also wish to thank
Martin Peters for his excitement about our proposal and for making this book
possible and Ruth Allewelt for all her support and helpful guidance during the
various phases of this project. Finally, on behalf of all his former advisees, we
would like to express our deep gratitude to Rolf Möhring. Learning from and
working with him has sharpened and broadened our minds. The topics covered in
thisbookarereflectiveofhiswideinterests.Hehasbeenawonderfulmentoranda
true source of inspiration in all these years.
Andreas S. Schulz
Martin Skutella
Sebastian Stiller
Dorothea Wagner
Contents
Shifting Segments to Optimality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Stefan Felsner
Linear Structure of Graphs and the Knotting Graph . . . . . . . . . . . . . . 13
Ekkehard Köhler
Finding Longest Geometric Tours . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Sándor P. Fekete
Generalized Hanan Grids for Geometric Steiner Trees in Uniform
Orientation Metrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Matthias Müller-Hannemann
Budgeted Matching via the Gasoline Puzzle . . . . . . . . . . . . . . . . . . . . . 49
Guido Schäfer
Motifs in Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Karsten Weihe
Graph Fill-In, Elimination Ordering, Nested Dissection
and Contraction Hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Ben Strasser and Dorothea Wagner
Shortest Path to Mechanism Design. . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Rudolf Müller and Marc Uetz
Selfish Routing and Proportional Resource Allocation. . . . . . . . . . . . . . 95
Andreas S. Schulz
Resource Buying Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Tobias Harks and Britta Peis
Linear, Exponential, but Nothing Else . . . . . . . . . . . . . . . . . . . . . . . . . 113
Max Klimm
ix
x Contents
Convex Quadratic Programming in Scheduling . . . . . . . . . . . . . . . . . . 125
Martin Skutella
Robustness and Approximation for Universal Sequencing. . . . . . . . . . . 133
Nicole Megow
A Short Note on Long Waiting Lists . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Sebastian Stiller
Shifting Segments to Optimality
StefanFelsner
Abstract Webeginwithtwoproblemswhichdonotappeartoberelated.Thenwe
usethe‘air-pressure’methodtoproveatheoremaboutrectangulardissectionswith
prescribedrectangleareas.Acorollarytothistheoremcharacterizesarea-universal
rectangulardissections.Thesedissectionshappentobecentraltothesolutionofthe
twoproblems.
1 IntroductionandTwoProblems
A rectangular dissection is a partition of a frame rectangle into rectangles, Fig.1
showsanexample.Rectangulardissectionsarestudiedinvariousfields,seeFig.2.
• Architectslookattheminthecontextoffloorplangeneration[11, 14].
• FloorplaningisrelevantformoduleplacementinVLSIdesign[4, 21].
• In graph drawing, rectangular dissections play a role in various representation
modelsforplanargraphs[8, 12].
• In cartography, rectilinear dissections are studied as a special class of car-
tograms[13, 17, 19].
Fig.1 Arectangulardissection
B
S.Felsner( )
InstitutfürMathematik,TechnischeUniversitätBerlin,Straßedes17.Juni136,
10623Berlin,Germany
e-mail:[email protected]
©SpringerInternationalPublishingSwitzerland2015 1
A.S.Schulzetal.(eds.),GemsofCombinatorialOptimization
andGraphAlgorithms,DOI10.1007/978-3-319-24971-1_1