3. Circulation theorem

Chapters in this book

1. Frontmatter i
2. CONTENTS vii
3. Foreword to the 2010 edition ix
4. PREFACE xvii
5. ACKNOWLEDGMENTS xix
6. CHAPTER I STATIC MAXIMAL FLOW
7. Introduction 1
8. 1. Networks 2
9. 2. Flows in networks 4
10. 3. Notation 9
11. 4. Cuts 10
12. 5. Maximal flow 11
13. 6. Disconnecting sets and cuts 14
14. 7. Multiple sources and sinks 15
15. 8. The labeling method for solving maximal flow problems 17
16. 9. Lower bounds on arc flows 22
17. 10. Flows in undirected and mixed networks 23
18. 11. Node capacities and other extensions 23
19. 12. Linear programming and duality principles 26
20. 13. Maximal flow value as a function of two arc capacities 30
21. References 35
22. CHAPTER II FEASIBILITY THEOREMS AND COMBINATORIAL APPLICATIONS
23. Introduction 36
24. 1. A supply-demand theorem 36
25. 2. A symmetric supply-demand theorem 42
26. 3. Circulation theorem 50
27. 4. The König-Egerváry and Menger graph theorems 53
28. 5. Construction of a maximal independent set of admissible cells 55
29. 6. A bottleneck assignment problem 57
30. 7. Unicursal graphs 59
31. 8. Dilworth's chain decomposition theorem for partially ordered sets 61
32. 9. Minimal number of individuals to meet a fixed schedule of tasks 64
33. 10. Set representatives 67
34. 11. The subgraph problem for directed graphs 75
35. 12. Matrices composed of O's and l's 79
36. References 91
37. CHAPTER III MINIMAL COST FLOW PROBLEMS
38. Introduction 93
39. 1. The Hitchcock problem 95
40. 2. The optimal assignment problem [56, 57, 60, 61, 68, 69] 111
41. 3. The general minimal cost flow problem 113
42. 4. Equivalence of Hitchcock and minimal cost flow problems 127
43. 5. A shortest chain algorithm 130
44. 6. The minimal cost supply-demand problem: non-negative directed cycle costs 134
45. 7. The warehousing problem 137
46. 8. The caterer problem 140
47. 9. Maximal dynamic flow 142
48. 10. Project cost curves 151
49. 11. Constructing minimal cost circulations [28] 162
50. References 169
51. CHAPTER IV MULTI-TERMINAL MAXIMAL FLOWS
52. Introduction 173
53. 1. Forests, trees, and spanning subtrees 173
54. 2. Realization conditions 176
55. 3. Equivalent networks 177
56. 4. Network synthesis 187
57. References 191
58. INDEX 193