Graphs, Networks and Algorithms

From the reviews of the previous editions ".... The book is a first class textbook and seems to be indispensable for everybody who has to teach combinatorial optimization. It is very helpful for students, teachers, and researchers in this area. The author finds a striking synthesis of nice and interesting mathematical results and practical applications. ... the author pays much attention to the inclusion of well-chosen exercises. The reader does not remain helpless; solutions or at least hints are given in the appendix. Except for some small basic mathematical and algorithmic knowledge the book is self-contained. ..." K.Engel, Mathematical Reviews 2002 The substantial development effort of this text, involving multiple editions and trailing in the context of various workshops, university courses and seminar series, clearly shows through in this new edition with its clear writing, good organisation, comprehensive coverage of essential theory, and well-chosen applications. The proofs of important results and the representation of key algorithms in a Pascal-like notation allow this book to be used in a high-level undergraduate or low-level graduate course on graph theory, combinatorial optimization or computer science algorithms. The well-worked solutions to exercises are a real bonus for self study by students. The book is highly recommended. P .B. Gibbons, Zentralblatt für Mathematik 2005 Once again, the new edition has been thoroughly revised. In particular, some further material has been added: more on NP-completeness (especially on dominating sets), a section on the Gallai-Edmonds structure theory for matchings, and about a dozen additional exercises – as always, with solutions. Moreover, the section on the 1-factor theorem has been completely rewritten: it now presents a short direct proof for the more general Berge-Tutte formula. Several recent research developments are discussed and quite a few references have been added. Table of Contents Cover Graphs, Networks and Algorithms, Fourth Edition ISBN 9783642322778 ISBN 9783642322785 Preface to the Fourth Edition Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Contents Chapter 1: Basic Graph Theory 1.1 Graphs, Subgraphs and Factors 1.2 Paths, Cycles, Connectedness, Trees 1.3 Euler Tours 1.4 Hamiltonian Cycles 1.5 Planar Graphs 1.6 Digraphs 1.7 An Application: Tournaments and Leagues Chapter 2: Algorithms and Complexity 2.1 Algorithms 2.2 Representing Graphs 2.3 The Algorithm of Hierholzer 2.4 How to Write Down Algorithms 2.5 The Complexity of Algorithms 2.6 Directed Acyclic Graphs 2.7 An Introduction to NP-completeness 2.8 Five NP-complete Problems Chapter 3: Shortest Paths 3.1 Shortest Paths 3.2 Finite Metric Spaces 3.3 Breadth First Search and Bipartite Graphs 3.4 Shortest Path Trees 3.5 Bellman's Equations and Acyclic Networks 3.6 An Application: Scheduling Projects 3.7 The Algorithm of Dijkstra 3.8 An Application: Train Schedules 3.9 The Algorithm of Floyd and Warshall 3.10 Cycles of Negative Length 3.11 Path Algebras Chapter 4: Spanning Trees 4.1 Trees and Forests 4.2 Incidence Matrices 4.3 Minimal Spanning Trees 4.4 The Algorithms of Prim, Kruskal and Boruvka 4.5 Maximal Spanning Trees 4.6 Steiner Trees 4.7 Spanning Trees with Restrictions 4.8 Arborescences and Directed Euler Tours Chapter 5: The Greedy Algorithm 5.1 The Greedy Algorithm and Matroids 5.2 Characterizations of Matroids 5.3 Matroid Duality 5.4 The Greedy Algorithm as an Approximation Method 5.5 Minimization in Independence Systems 5.6 Accessible Set Systems Chapter 6: Flows 6.1 The Theorems of Ford and Fulkerson 6.2 The Algorithm of Edmonds and Karp 6.3 Auxiliary Networks and Phases 6.4 Constructing Blocking Flows 6.5 Zero-One Flows 6.6 The Algorithm of Goldberg and Tarjan 6.7 Further Reading Chapter 7: Combinatorial Applications 7.1 Disjoint Paths: Menger's Theorem 7.2 Matchings: K�nig's Theorem 7.3 Partial Transversals: The Marriage Theorem 7.4 Combinatorics of Matrices 7.5 Dissections: Dilworth's Theorem 7.6 Parallelisms: Baranyai's Theorem 7.7 Supply and Demand: The Gale-Ryser Theorem Chapter 8: Connectivity and Depth First Search 8.1 k-connected Graphs 8.2 Depth First Search 8.3 2-connected Graphs 8.4 Depth First Search for Digraphs 8.5 Strongly Connected Digraphs 8.6 Edge Connectivity Chapter 9: Colorings 9.1 Vertex Colorings 9.2 Comparability Graphs and Interval Graphs 9.3 Edge Colorings 9.4 Cayley Graphs 9.5 The Five Color Theorem Chapter 10: Circulations 10.1 Circulations and Flows 10.2 Feasible Circulations 10.3 Elementary Circulations 10.4 The Algorithm of Klein 10.5 The Algorithm of Busacker and Gowen 10.6 Potentials and epsilon-optimality 10.7 Optimal Circulations by Successive Approximation 10.8 A Polynomial Procedure REFINE 10.9 The Minimum Mean Cycle Cancelling Algorithm 10.10 Some Further Problems 10.11 An Application: Graphical Codes Chapter 11: The Network Simplex Algorithm 11.1 The Minimum Cost Flow Problem 11.2 Tree Solutions 11.3 Constructing an Admissible Tree Structure 11.4 The Algorithm Rule of the last blocking arc: 11.5 Efficient Implementations Chapter 12: Synthesis of Networks 12.1 Symmetric Networks 12.2 Synthesis of Equivalent Flow Trees 12.3 Synthesizing Minimal Networks 12.4 Cut Trees 12.5 Increasing the Capacities Chapter 13: Matchings 13.1 The Berge-Tutte Formula 13.2 Augmenting Paths 13.3 Alternating Trees and Blossoms 13.4 The Algorithm of Edmonds 13.5 The Gallai-Edmonds Structure Theorem 13.6 Matching Matroids Chapter 14: Weighted Matchings 14.1 The Bipartite Case 14.2 The Hungarian Algorithm 14.3 Matchings, Linear Programs, and Polytopes 14.4 The General Case 14.5 The Chinese Postman 14.6 Matchings and Shortest Paths 14.7 Some Further Problems 14.8 An Application: Decoding Graphical Codes Chapter 15: A Hard Problem: The TSP 15.1 Basic Definitions 15.2 Lower Bounds: Relaxations The Assignment Relaxation The MST Relaxation The s-tree Relaxation The LP Relaxation 15.3 Lower Bounds: Subgradient Optimization 15.4 Approximation Algorithms 15.5 Upper Bounds: Heuristics 15.6 Upper Bounds: Local Search 15.7 Exact Neighborhoods and Suboptimality 15.8 Optimal Solutions: Branch and Bound 15.9 Concluding Remarks Appendix A: Some NP-Complete Problems Appendix B: Solutions B.1 Solutions for Chap. 1 B.2 Solutions for Chap. 2 B.3 Solutions for Chap. 3 B.4 Solutions for Chap. 4 B.5 Solutions for Chap. 5 B.6 Solutions for Chap. 6 B.7 Solutions for Chap. 7 B.8 Solutions for Chap. 8 B.9 Solutions for Chap. 9 B.10 Solutions for Chap. 10 B.11 Solutions for Chap. 11 B.12 Solutions for Chap. 12 B.13 Solutions for Chap. 13 B.14 Solutions for Chap. 14 B.15 Solutions for Chap. 15 Appendix C: List of Symbols C.1 General Symbols Sets Mappings Numbers Matrices Sets of numbers and algebraic structures Miscellaneous C.2 Special Symbols Graphs and networks Objects in graphs Parameters for graphs Mappings on graphs and networks Matroids and independence systems Matrices Codes Miscellaneous References Index