Combinatorial optimization: polyhedra and efficiency Vol. A. Path, flows, matchings : chapters 1 - 38
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2003
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| Schriftenreihe: | Algorithms and combinatorics
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| Beschreibung: | XXXVII, 647 S. graph. Darst. |
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| 100 | 1 | |a Schrijver, Alexander |d 1948- |e Verfasser |0 (DE-588)13576355X |4 aut | |
| 245 | 1 | 0 | |a Combinatorial optimization |b polyhedra and efficiency |n Vol. A. |p Path, flows, matchings : chapters 1 - 38 |c Alexander Schrijver |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2003 | |
| 300 | |a XXXVII, 647 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Algorithms and combinatorics |v 24 | |
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TABLE
OF
CONTENTS
VOLUME
A
1
INTRODUCTION
.
1
1.1
INTRODUCTION
.
1
1.2
MATCHINGS
.
2
1.3
BUT
WHAT
ABOUT
NONBIPARTITE
GRAPHS?
.
4
1.4
HAMILTONIAN
CIRCUITS
AND
THE
TRAVELING
SALESMAN
PROBLEM
.
5
1.5
HISTORICAL
AND
FURTHER
NOTES
.
6
1.5A
HISTORICAL
SKETCH
ON
POLYHEDRAL
COMBINATORICS
.
6
1.5B
FURTHER
NOTES
.
8
2
GENERAL
PRELIMINARIES
.
9
2.1
SETS
.
9
2.2
ORDERS
.
11
2.3
NUMBERS
.
11
2.4
VECTORS,
MATRICES,
AND
FUNCTIONS
.
11
2.5
MAXIMA,
MINIMA,
AND
INFINITY
.
14
2.6
FEKETE
'
S
LEMMA
.
14
3
PRELIMINARIES
ON
GRAPHS
.
16
3.1
UNDIRECTED
GRAPHS
.
16
3.2
DIRECTED
GRAPHS
.
28
3.3
HYPER
GRAPHS
.
36
3.3A
BACKGROUND
REFERENCES
ON
GRAPH
THEORY
.
37
4
PRELIMINARIES
ON
ALGORITHMS
AND
COMPLEXITY
.
38
4.1
INTRODUCTION
.
38
4.2
THE
RANDOM
ACCESS
MACHINE
.
39
4.3
POLYNOMIAL-TIME
SOLVABILITY
.
39
4.4
P
.
40
4.5
NP
.
40
4.6
CO-NP
AND
GOOD
CHARACTERIZATIONS
.
42
4.7
OPTIMIZATION
PROBLEMS
.
42
4.8
NP-COMPLETE
PROBLEMS
.
43
4.9
THE
SATISFIABILITY
PROBLEM
.
44
X
TABLE
OF
CONTENTS
4.10
NP-COMPLETENESS
OF
THE
SATISFIABILITY
PROBLEM
.
44
4.11
NP-COMPLETENESS
OF
SOME
OTHER
PROBLEMS
.
46
4.12
STRONGLY
POLYNOMIAL-TIME
.
47
4.13
LISTS
AND
POINTERS
.
48
4.14
FURTHER
NOTES
.
49
4.14A
BACKGROUND
LITERATURE
ON
ALGORITHMS
AND
COMPLEXITY
.
49
.
4.14B
EFFICIENCY
AND
COMPLEXITY
HISTORICALLY
.
49
5
PRELIMINARIES
ON
POLYHEDRA
AND
LINEAR
AND
INTEGER
PROGRAMMING
.
59
5.1
CONVEXITY
AND
HALFSPACES
.
59
5.2
CONES
.
60
5.3
POLYHEDRA
AND
POLYTOPES
.
60
5.4
FARKAS
'
LEMMA
.
61
5.5
LINEAR
PROGRAMMING
.
61
5.6
FACES,
FACETS,
AND
VERTICES
.
63
5.7
POLARITY
.
65
5.8
BLOCKING
POLYHEDRA
.
65
5.9
ANTIBLOCKING
POLYHEDRA
.
67
5.10
METHODS
FOR
LINEAR
PROGRAMMING
.
67
5.11
THE
ELLIPSOID
METHOD
.
68
5.12
POLYHEDRA
AND
NP
AND
CO-NP
.
71
5.13
PRIMAL-DUAL
METHODS
.
72
5.14
INTEGER
LINEAR
PROGRAMMING
.
73
5.15
INTEGER
POLYHEDRA
.
74
5.16
TOTALLY
UNIMODULAR
MATRICES
.
75
5.17
TOTAL
DUAL
INTEGRALITY
.
76
5.18
HILBERT
BASES
AND
MINIMAL
TDI
SYSTEMS
.
81
5.19
THE
INTEGER
ROUNDING
AND
DECOMPOSITION
PROPERTIES
.
82
5.20
BOX-TOTAL
DUAL
INTEGRALITY
.
83
5.21
THE
INTEGER
HULL
AND
CUTTING
PLANES
.
83
5.21A
BACKGROUND
LITERATURE
.
84
PART
I:
PATHS
AND
FLOWS
85
6
SHORTEST
PATHS:
UNIT
LENGTHS
.
87
6.1
SHORTEST
PATHS
WITH
UNIT
LENGTHS
.
87
6.2
SHORTEST
PATHS
WITH
UNIT
LENGTHS
ALGORITHMICALLY:
BREADTH-FIRST
SEARCH
.
88
6.3
DEPTH-FIRST
SEARCH
.
89
6.4
FINDING
AN
EULERIAN
ORIENTATION
.
91
6.5
FURTHER
RESULTS
AND
NOTES
.
91
6.5A
ALL-PAIRS
SHORTEST
PATHS
IN
UNDIRECTED
GRAPHS
.
91
6.5B
COMPLEXITY
SURVEY
.
93
TABLE
OF
CONTENTS
XI
6.5C
EAR-DECOMPOSITION
OF
STRONGLY
CONNECTED
DIGRAPHS
.
93
6.5D
TRANSITIVE
CLOSURE
.
94
6.5E
FURTHER
NOTES
.
94
7
SHORTEST
PATHS:
NONNEGATIVE
LENGTHS
.
96
7.1
SHORTEST
PATHS
WITH
NONNEGATIVE
LENGTHS
.
96
7.2
DIJKSTRA
'
S
METHOD
.
97
7.3
SPEEDING
UP
DIJKSTRA
'
S
ALGORITHM
WITH
A:-HEAPS
.
98
7.4
SPEEDING
UP
DIJKSTRA
'
S
ALGORITHM
WITH
FIBONACCI
HEAPS
.
99
7.5
FURTHER
RESULTS
AND
NOTES
.
101
7.5A
WEAKLY
POLYNOMIAL-TIME
ALGORITHMS
.
101
7.5B
COMPLEXITY
SURVEY
FOR
SHORTEST
PATHS
WITH
NONNEGATIVE
LENGTHS
.
103
7.5C
FURTHER
NOTES
.
105
8
SHORTEST
PATHS:
ARBITRARY
LENGTHS
.
107
8.1
SHORTEST
PATHS
WITH
ARBITRARY
LENGTHS
BUT
NO
NEGATIVE
CIRCUITS
.
107
8.2
POTENTIALS
.
107
8.3
THE
BELLMAN-FORD
METHOD
.
109
8.4
ALL-PAIRS
SHORTEST
PATHS
.
110
8.5
FINDING
A
MINIMUM-MEAN
LENGTH
DIRECTED
CIRCUIT
.
ILL
8.6
FURTHER
RESULTS
AND
NOTES
.
112
8.6A
COMPLEXITY
SURVEY
FOR
SHORTEST
PATH
WITHOUT
NEGATIVE-LENGTH
CIRCUITS
.
112
8.6B
NP-COMPLETENESS
OF
THE
SHORTEST
PATH
PROBLEM
.
114
8.6C
NONPOLYNOMIALITY
OF
FORD
'
S
METHOD
.
115
8.6D
SHORTEST
AND
LONGEST
PATHS
IN
ACYCLIC
GRAPHS
.
116
8.6E
BOTTLENECK
SHORTEST
PATH
.
117
8.6F
FURTHER
NOTES
.
118
8.6G
HISTORICAL
NOTES
ON
SHORTEST
PATHS
.
119
9
DISJOINT
PATHS
.
131
9.1
MENGER
'
S
THEOREM
.
131
9.1A
OTHER
PROOFS
OF
MENGER
'
S
THEOREM
.
133
9.2
PATH
PACKING
ALGORITHMICALLY
.
134
9.3
SPEEDING
UP
BY
BLOCKING
PATH
PACKINGS
.
135
9.4
A
SOMETIMES
BETTER
BOUND
.
136
9.5
COMPLEXITY
OF
THE
VERTEX-DISJOINT
CASE
.
137
9.6
FURTHER
RESULTS
AND
NOTES
.
138
9.6A
COMPLEXITY
SURVEY
FOR
THE
DISJOINT
S
-
T
PATHS
PROBLEM
.
138
9.6B
PARTIALLY
DISJOINT
PATHS
.
140
9.6C
EXCHANGE
PROPERTIES
OF
DISJOINT
PATHS
.
140
9.6D
FURTHER
NOTES
.
141
XII
TABLE
OF
CONTENTS
9.6E
HISTORICAL
NOTES
ON
MONGER
'
S
THEOREM
.
142
10
MAXIMUM
FLOW
.
148
10.1
FLOWS:
CONCEPTS
.
148
10.2
THE
MAX-FLOW
MIN-CUT
THEOREM
.
150
10.3
PATHS
AND
FLOWS
.
151
10.4
FINDING
A
MAXIMUM
FLOW
.
151
10.4A
NONTERMINATION
FOR
IRRATIONAL
CAPACITIES
.
152
10.5
A
STRONGLY
POLYNOMIAL
BOUND
ON
THE
NUMBER
OF
ITERATIONS
.
153
10.6
DINITS
'
O(N
2
M)
ALGORITHM
.
154
10.6A
KARZANOV
'
S
O(N
3
)
ALGORITHM
.
155
10.7
GOLDBERG
'
S
PUSH-RELABEL
METHOD
.
156
10.8
FURTHER
RESULTS
AND
NOTES
.
159
10.8A
A
WEAKLY
POLYNOMIAL
BOUND
.
159
10.8B
COMPLEXITY
SURVEY
FOR
THE
MAXIMUM
FLOW
PROBLEM
.
160
10.8C
AN
EXCHANGE
PROPERTY
.
162
10.
8D
FURTHER
NOTES
.
162
10.
8E
HISTORICAL
NOTES
ON
MAXIMUM
FLOW
.
164
11
CIRCULATIONS
AND
TRANSSHIPMENTS
.
170
11.1
A
USEFUL
FACT
ON
ARC
FUNCTIONS
.
170
11.2
CIRCULATIONS
.
171
11.3
FLOWS
WITH
UPPER
AND
LOWER
BOUNDS
.
172
11.4
6-TRANSSHIPMENTS
.
173
11.5
UPPER
AND
LOWER
BOUNDS
ON
EXCESS/
.
174
11.6
FINDING
CIRCULATIONS
AND
TRANSSHIPMENTS
ALGORITHMICALLY
.
175
11.6A
FURTHER
NOTES
.
176
12
MINIMUM-COST
FLOWS
AND
CIRCULATIONS
.
177
12.1
MINIMUM-COST
FLOWS
AND
CIRCULATIONS
.
177
12.2
MINIMUM-COST
CIRCULATIONS
AND
THE
RESIDUAL
GRAPH
DF
.
178
12.3
STRONGLY
POLYNOMIAL-TIME
ALGORITHM
.
179
12.4
RELATED
PROBLEMS
.
182
12.4A
A
DUAL
APPROACH
.
183
12.4B
A
STRONGLY
POLYNOMIAL-TIME
ALGORITHM
USING
CAPACITY-SCALING
.
186
12.5
FURTHER
RESULTS
AND
NOTES
.
190
12.5A
COMPLEXITY
SURVEY
FOR
MINIMUM-COST
CIRCULATION
.
190
12.5B
MIN-MAX
RELATIONS
FOR
MINIMUM-COST
FLOWS
AND
CIRCULATIONS
.
191
12.5C
DYNAMIC
FLOWS
.
192
12.5D
FURTHER
NOTES
.
195
TABLE
OF
CONTENTS
XIII
13
PATH
AND
FLOW
POLYHEDRA
AND
TOTAL
UNIMODULARITY
.
198
13.1
PATH
POLYHEDRA
.
198
13.1A
VERTICES,
ADJACENCY,
AND
FACETS
.
202
13.1B
THE
S
-
T
CONNECTOR
POLYTOPE
.
203
13.2
TOTAL
UNIMODULARITY
.
204
13.2A
CONSEQUENCES
FOR
FLOWS
.
205
13.2B
CONSEQUENCES
FOR
CIRCULATIONS
.
207
13.2C
CONSEQUENCES
FOR
TRANSSHIPMENTS
.
207
13.2D
UNIONS
OF
DISJOINT
PATHS
AND
CUTS
.
210
13.3
NETWORK
MATRICES
.
213
13.4
CROSS-FREE
AND
LAMINAR
FAMILIES
.
214
14
PARTIALLY
ORDERED
SETS
AND
PATH
COVERINGS
.
217
14.1
PARTIALLY
ORDERED
SETS
.
217
14.2
DILWORTH
'
S
DECOMPOSITION
THEOREM
.
218
14.3
PATH
COVERINGS
.
219
14.4
THE
WEIGHTED
CASE
.
220
14.5
THE
CHAIN
AND
ANTICHAIN
POLYTOPES
.
221
14.5A
PATH
COVERINGS
ALGORITHMICALLY
.
222
14.6
UNIONS
OF
DIRECTED
CUTS
AND
ANTICHAINS
.
224
14.6A
COMMON
SATURATING
COLLECTIONS
OF
CHAINS
.
226
14.7
UNIONS
OF
DIRECTED
PATHS
AND
CHAINS
.
227
14.7A
COMMON
SATURATING
COLLECTIONS
OF
ANTICHAINS
.
229
14.7B
CONJUGACY
OF
PARTITIONS
.
230
14.8
FURTHER
RESULTS
AND
NOTES
.
232
14.8A
THE
GALLAI-MILGRAM
THEOREM
.
232
14.8B
PARTIALLY
ORDERED
SETS
AND
DISTRIBUTIVE
LATTICES
.
233
14.8C
MAXIMAL
CHAINS
.
235
14.8D
FURTHER
NOTES
.
236
15
CONNECTIVITY
AND
GOMORY-HU
TREES
.
237
15.1
VERTEX-,
EDGE-,
AND
ARC-CONNECTIVITY
.
237
15.2
VERTEX-CONNECTIVITY
ALGORITHMICALLY
.
239
15.2A
COMPLEXITY
SURVEY
FOR
VERTEX-CONNECTIVITY
.
241
15.2B
FINDING
THE
2-CONNECTED
COMPONENTS
.
242
15.3
ARC
AND
EDGE-CONNECTIVITY
ALGORITHMICALLY
.
243
15.3A
COMPLEXITY
SURVEY
FOR
ARC
AND
EDGE-CONNECTIVITY
.
246
15.3B
FINDING
THE
2-EDGE-CONNECTED
COMPONENTS
.
247
15.4
GOMORY-HU
TREES
.
248
15.4A
MINIMUM-REQUIREMENT
SPANNING
TREE
.
251
15.5
FURTHER
RESULTS
AND
NOTES
.
252
15.5A
EAR-DECOMPOSITION
OF
UNDIRECTED
GRAPHS
.
252
15.5B
FURTHER
NOTES
.
253
XIV
TABLE
OF
CONTENTS
PART
II:
BIPARTITE
MATCHING
AND
COVERING
257
16
CARDINALITY
BIPARTITE
MATCHING
AND
VERTEX
COVER
.
259
16.1
AF-AUGMENTING
PATHS
.
259
16.2
FROBENIUS
'
AND
KONIG
'
S
THEOREMS
.
260
16.2A
FROBENIUS
'
PROOF
OF
HIS
THEOREM
.
262
16.2B
LINEAR-ALGEBRAIC
PROOF
OF
FROBENIUS
'
THEOREM
.
262
16.2C
RIZZI
'
S
PROOF
OF
KONIG
'
S
MATCHING
THEOREM
.
263
16.3
MAXIMUM-SIZE
BIPARTITE
MATCHING
ALGORITHM
.
263
16.4
AN
O(N^
2RRI)
ALGORITHM
.
264
16.5
FINDING
A
MINIMUM-SIZE
VERTEX
COVER
.
265
16.6
MATCHINGS
COVERING
GIVEN
VERTICES
.
265
16.7
FURTHER
RESULTS
AND
NOTES
.
267
16.7A
COMPLEXITY
SURVEY
FOR
CARDINALITY
BIPARTITE
MATCHING
.
267
16.7B
FINDING
PERFECT
MATCHINGS
IN
REGULAR
BIPARTITE
GRAPHS
.
267
16.7C
THE
EQUIVALENCE
OF
MONGER
'
S
THEOREM
AND
KONIG
'
S
THEOREM
.
275
16.
7D
EQUIVALENT
FORMULATIONS
IN
TERMS
OF
MATRICES
.
276
16.
7E
EQUIVALENT
FORMULATIONS
IN
TERMS
OF
PARTITIONS
.
276
16.
7F
ON
THE
COMPLEXITY
OF
BIPARTITE
MATCHING
AND
VERTEX
COVER
.
277
16.7G
FURTHER
NOTES
.
277
16.
7H
HISTORICAL
NOTES
ON
BIPARTITE
MATCHING
.
278
17
WEIGHTED
BIPARTITE
MATCHING
AND
THE
ASSIGNMENT
PROBLEM
.
285
17.1
WEIGHTED
BIPARTITE
MATCHING
.
285
17.2
THE
HUNGARIAN
METHOD
.
286
17.3
PERFECT
MATCHING
AND
ASSIGNMENT
PROBLEMS
.
288
17.4
FINDING
A
MINIMUM-SIZE
W-VERTEX
COVER
.
289
17.5
FURTHER
RESULTS
AND
NOTES
.
290
17.5A
COMPLEXITY
SURVEY
FOR
MAXIMUM-WEIGHT
BIPARTITE
MATCHING
.
290
17.5B
FURTHER
NOTES
.
290
17.5C
HISTORICAL
NOTES
ON
WEIGHTED
BIPARTITE
MATCHING
AND
OPTIMUM
ASSIGNMENT
.
292
18
LINEAR
PROGRAMMING
METHODS
AND
THE
BIPARTITE
MATCHING
POLYTOPE
.
301
18.1
THE
MATCHING
AND
THE
PERFECT
MATCHING
POLYTOPE
.
301
18.2
TOTALLY
UNIMODULAR
MATRICES
FROM
BIPARTITE
GRAPHS
.
303
18.3
CONSEQUENCES
OF
TOTAL
UNIMODULARITY
.
304
THBLE
OF
CONTENTS
XV
18.4
THE
VERTEX
COVER
POLYTOPE
.
305
18.5
FURTHER
RESULTS
AND
NOTES
.
305
18.5A
DERIVATION
OF
KONIG
'
S
MATCHING
THEOREM
FROM
THE
MATCHING
POLYTOPE
.
305
18.5B
DUAL,
PRIMAL-DUAL,
PRIMAL?
.
305
18.5C
ADJACENCY
AND
DIAMETER
OF
THE
MATCHING
POLYTOPE
.
307
18.5D
THE
PERFECT
MATCHING
SPACE
OF
A
BIPARTITE
GRAPH
.
308
18.
5E
UP
AND
DOWN
HULL
OF
THE
PERFECT
MATCHING
POLYTOPE
.
309
18.5F
MATCHINGS
OF
GIVEN
SIZE
.
310
18.5G
STABLE
MATCHINGS
.
311
18.
5H
FURTHER
NOTES
.
314
19
BIPARTITE
EDGE
COVER
AND
STABLE
SET
.
315
19.1
MATCHINGS,
EDGE
COVERS,
AND
GALLAI
'
S
THEOREM
.
315
19.2
THE
KONIG-RADO
EDGE
COVER
THEOREM
.
317
19.3
FINDING
A
MINIMUM-WEIGHT
EDGE
COVER
.
317
19.4
BIPARTITE
EDGE
COVERS
AND
TOTAL
UNIMODULARITY
.
318
19.5
THE
EDGE
COVER
AND
STABLE
SET
POLYTOPE
.
318
19.5A
SOME
HISTORICAL
NOTES
ON
BIPARTITE
EDGE
COVERS
.
319
20
BIPARTITE
EDGE-COLOURING
.
321
20.1
EDGE-COLOURINGS
OF
BIPARTITE
GRAPHS
.
321
20.1A
EDGE-COLOURING
REGULAR
BIPARTITE
GRAPHS
.
322
20.2
THE
CAPACITATED
CASE
.
322
20.3
EDGE-COLOURING
POLYHEDRALLY
.
323
20.4
PACKING
EDGE
COVERS
.
324
20.5
BALANCED
COLOURS
.
325
20.6
PACKING
PERFECT
MATCHINGS
.
326
20.6A
POLYHEDRAL
INTERPRETATION
.
327
20.6B
EXTENSIONS
.
328
20.7
COVERING
BY
PERFECT
MATCHINGS
.
329
20.7A
POLYHEDRAL
INTERPRETATION
.
330
20.8
THE
PERFECT
MATCHING
LATTICE
OF
A
BIPARTITE
GRAPH
.
331
20.9
FURTHER
RESULTS
AND
NOTES
.
333
20.9A
SOME
FURTHER
EDGE-COLOURING
ALGORITHMS
.
333
20.9B
COMPLEXITY
SURVEY
FOR
BIPARTITE
EDGE-COLOURING
.
334
20.9C
LIST-EDGE-COLOURING
.
335
20.
9D
FURTHER
NOTES
.
336
21
BIPARTITE
B-MATCHINGS
AND
TRANSPORTATION
.
337
21.1
B-MATCHINGS
AND
W
VERTEX
COVERS
.
337
21.2
THE
B-MATCHING
POLYTOPE
AND
THE
W
VERTEX
COVER
POLYHEDRON
.
338
21.3
SIMPLE
6-MATCHINGS
AND
B-FACTORS
.
339
21.4
CAPACITATED
B-MATCHINGS
.
341
XVI
TABLE
OF
CONTENTS
21.5
BIPARTITE
B-MATCHING
AND
W-VERTEX
COVER
ALGORITHMICALLY
.
342
21.6
TRANSPORTATION
.
343
21.6A
REDUCTION
OF
TRANSSHIPMENT
TO
TRANSPORTATION
.
345
21.6B
THE
TRANSPORTATION
POLYTOPE
.
346
21.7
B-EDGE
COVERS
AND
W-STABLE
SETS
.
347
21.8
THE
B-EDGE
COVER
AND
THE
W-STABLE
SET
POLYHEDRON
.
348
21.9
SIMPLE
B-EDGE
COVERS
.
349
21.10
CAPACITATED
B-EDGE
COVERS
.
350
21.11
RELATIONS
BETWEEN
B-MATCHINGS
AND
B-EDGE
COVERS
.
351
21.12
UPPER
AND
LOWER
BOUNDS
.
353
21.13
FURTHER
RESULTS
AND
NOTES
.
355
21.13A
COMPLEXITY
SURVEY
ON
WEIGHTED
BIPARTITE
B-MATCHING
AND
TRANSPORTATION
.
355
21.13B
THE
MATCHABLE
SET
POLYTOPE
.
359
21.13C
EXISTENCE
OF
MATRICES
.
359
21.13D
FURTHER
NOTES
.
361
21.13E
HISTORICAL
NOTES
ON
THE
TRANSPORTATION
AND
TRANSSHIPMENT
PROBLEMS
.
362
22
TRANSVERSALS
.
378
22.1
TRANSVERSALS
.
378
22.1A
ALTERNATIVE
PROOFS
OF
HALL
'
S
MARRIAGE
THEOREM
.
379
22.2
PARTIAL
TRANSVERSALS
.
380
22.3
WEIGHTED
TRANSVERSALS
.
382
22.4
MIN-MAX
RELATIONS
FOR
WEIGHTED
TRANSVERSALS
.
382
22.5
THE
TRANSVERSAL
POLYTOPE
.
383
22.6
PACKING
AND
COVERING
OF
TRANSVERSALS
.
385
22.7
FURTHER
RESULTS
AND
NOTES
.
387
22.7A
THE
CAPACITATED
CASE
.
387
22.7B
A
THEOREM
OF
RADO
.
389
22.7C
FURTHER
NOTES
.
389
22.7D
HISTORICAL
NOTES
ON
TRANSVERSALS
.
390
23
COMMON
TRANSVERSALS
.
393
23.1
COMMON
TRANSVERSALS
.
393
23.2
WEIGHTED
COMMON
TRANSVERSALS
.
395
23.3
WEIGHTED
COMMON
PARTIAL
TRANSVERSALS
.
397
23.4
THE
COMMON
PARTIAL
TRANSVERSAL
POLYTOPE
.
399
23.5
THE
COMMON
TRANSVERSAL
POLYTOPE
.
401
23.6
PACKING
AND
COVERING
OF
COMMON
TRANSVERSALS
.
402
23.7
FURTHER
RESULTS
AND
NOTES
.
407
23.7A
CAPACITATED
COMMON
TRANSVERSALS
.
407
23.7B
EXCHANGE
PROPERTIES
.
407
23.7C
COMMON
TRANSVERSALS
OF
THREE
FAMILIES
.
408
23.7D
FURTHER
NOTES
.
409
TABLE
OF
CONTENTS
XVII
PART
III:
NONBIPARTITE
MATCHING
AND
COVERING
411
24
CARDINALITY
NONBIPARTITE
MATCHING
.
413
24.1
TUTTE
'
S
1-FACTOR
THEOREM
AND
THE
TUTTE-BERGE
FORMULA
.
413
24.1A
TUTTE
'
S
PROOF
OF
HIS
1-FACTOR
THEOREM
.
415
24.1B
PETERSEN
'
S
THEOREM
.
415
24.2
CARDINALITY
MATCHING
ALGORITHM
.
415
24.2A
AN
O(N
3
)
ALGORITHM
.
418
24.3
MATCHINGS
COVERING
GIVEN
VERTICES
.
421
24.4
FURTHER
RESULTS
AND
NOTES
.
422
24.4A
COMPLEXITY
SURVEY
FOR
CARDINALITY
NONBIPARTITE
MATCHING
.
422
24.4B
THE
EDMONDS-GALLAI
DECOMPOSITION
OF
A
GRAPH
.
423
24.4C
STRENGTHENING
OF
TUTTE
'
S
1-FACTOR
THEOREM
.
425
24.
4D
EAR-DECOMPOSITION
OF
FACTOR-CRITICAL
GRAPHS
.
425
24.4E
EAR-DECOMPOSITION
OF
MATCHING-COVERED
GRAPHS
.
426
24.4F
BARRIERS
IN
MATCHING-COVERED
GRAPHS
.
427
24.4G
TWO-PROCESSOR
SCHEDULING
.
428
24.
4H
THE
TUTTE
MATRIX
AND
AN
ALGEBRAIC
MATCHING
ALGORITHM
.
429
24.4I
FURTHER
NOTES
.
430
24.
4J
HISTORICAL
NOTES
ON
NONBIPARTITE
MATCHING
.
431
25
THE
MATCHING
POLYTOPE
.
438
25.1
THE
PERFECT
MATCHING
POLYTOPE
.
438
25.2
THE
MATCHING
POLYTOPE
.
439
25.3
TOTAL
DUAL
INTEGRALITY:
THE
CUNNINGHAM-MARSH
FORMULA
.
440
25.3A
DIRECT
PROOF
OF
THE
CUNNINGHAM-MARSH
FORMULA
.
442
25.4
ON
THE
TOTAL
DUAL
INTEGRALITY
OF
THE
PERFECT
MATCHING
CONSTRAINTS
.
443
25.5
FURTHER
RESULTS
AND
NOTES
.
444
25.5A
ADJACENCY
AND
DIAMETER
OF
THE
MATCHING
POLYTOPE
.
444
25.5B
FACETS
OF
THE
MATCHING
POLYTOPE
.
446
25.5C
POLYNOMIAL-TIME
SOLVABILITY
WITH
THE
ELLIPSOID
METHOD
.
448
25.5D
THE
MATCHABLE
SET
POLYTOPE
.
450
25.
5E
FURTHER
NOTES
.
452
26
WEIGHTED
NONBIPARTITE
MATCHING
ALGORITHMICALLY
.
453
26.1
INTRODUCTION
AND
PRELIMINARIES
.
453
26.2
WEIGHTED
MATCHING
ALGORITHM
.
454
26.2A
AN
O(N
3
)
ALGORITHM
.
456
26.3
FURTHER
RESULTS
AND
NOTES
.
458
XVIII
TABLE
OF
CONTENTS
26.3A
COMPLEXITY
SURVEY
FOR
WEIGHTED
NONBIPARTITE
MATCHING
.
458
26.3B
DERIVATION
OF
THE
MATCHING
POLYTOPE
CHARACTERIZATION
FROM
THE
ALGORITHM
.
459
26.3C
FURTHER
NOTES
.
459
27
NONBIPARTITE
EDGE
COVER
.
461
27.1
MINIMUM-SIZE
EDGE
COVER
.
461
27.2
THE
EDGE
COVER
POLYTOPE
AND
TOTED
DUAL
INTEGRALITY
.
462
27.3
FURTHER
NOTES
ON
EDGE
COVERS
.
464
27.3A
FURTHER
NOTES
.
464
27.3B
HISTORICAL
NOTES
ON
EDGE
COVERS
.
464
28
EDGE-COLOURING
.
465
28.1
VIZING
'
S
THEOREM
FOR
SIMPLE
GRAPHS
.
465
28.2
VIZING
'
S
THEOREM
FOR
GENERAL
GRAPHS
.467
28.3
NP-COMPLETENESS
OF
EDGE-COLOURING
.
468
28.4
NOWHERE-ZERO
FLOWS
AND
EDGE-COLOURING
.
470
28.5
FRACTIONAL
EDGE-COLOURING
.
474
28.6
CONJECTURES
.
475
28.7
EDGE-COLOURING
POLYHEDRALLY
.
477
28.8
PACKING
EDGE
COVERS
.
478
28.9
FURTHER
RESULTS
AND
NOTES
.
480
28.9A
SHANNON
'
S
THEOREM
.
480
28.9B
FURTHER
NOTES
.
480
28.9C
HISTORICAL
NOTES
ON
EDGE-COLOURING
.
482
29
T-JOINS,
UNDIRECTED
SHORTEST
PATHS,
AND
THE
CHINESE
POSTMAN
.
485
29.1
T-JOINS
.
485
29.2
THE
SHORTEST
PATH
PROBLEM
FOR
UNDIRECTED
GRAPHS
.
487
29.3
THE
CHINESE
POSTMAN
PROBLEM
.
487
29.4
T-JOINS
AND
T-CUTS
.
488
29.5
THE
UP
HULL
OF
THE
T-JOIN
POLYTOPE
.
.*
.
490
29.6
THE
T-JOIN
POLYTOPE
.
491
29.7
SUMS
OF
CIRCUITS
.
493
29.8
INTEGER
SUMS
OF
CIRCUITS
.
494
29.9
THE
T-CUT
POLYTOPE
.
498
29.10
FINDING
A
MINIMUM-CAPACITY
T-CUT
.
499
29.11
FURTHER
RESULTS
AND
NOTES
.
500
29.11A
MINIMUM-MEAN
LENGTH
CIRCUIT
.
500
29.11B
PACKING
T-CUTS
.
501
29.11C
PACKING
T-JOINS
.
507
29.
LID
MAXIMUM
JOINS
.
510
29.11E
ODD
PATHS
.
515
TABLE
OF
CONTENTS
XIX
29.
ILF
FURTHER
NOTES
.
517
29.11G
ON
THE
HISTORY
OF
THE
CHINESE
POSTMAN
PROBLEM
.
519
30
2-MATCHINGS,
2-COVERS,
AND
2-FACTORS
.
520
30.1
2-MATCHINGS
AND
2
VERTEX
COVERS
.
520
30.2
FRACTIONAL
MATCHINGS
AND
VERTEX
COVERS
.
521
30.3
THE
FRACTIONAL
MATCHING
POLYTOPE
.
522
30.4
THE
2
MATCHING
POLYTOPE
.
522
30.5
THE
WEIGHTED
2-MATCHING
PROBLEM
.
523
30.5A
MAXIMUM-SIZE
2-MATCHINGS
AND
MAXIMUM-SIZE
MATCHINGS
.
524
30.6
SIMPLE
2-MATCHINGS
AND
2-FACTORS
.
526
30.7
THE
SIMPLE
2-MATCHING
POLYTOPE
AND
THE
2-FACTOR
POLYTOPE
.
528
30.8
TOTAL
DUAL
INTEGRALITY
.
531
30.9
2-EDGE
COVERS
AND
2-STABLE
SETS
.
531
30.10
FRACTIONAL
EDGE
COVERS
AND
STABLE
SETS
.
532
30.11
THE
FRACTIONAL
EDGE
COVER
POLYHEDRON
.
533
30.12
THE
2-EDGE
COVER
POLYHEDRON
.
533
30.13
TOTAL
DUAL
INTEGRALITY
OF
THE
2-EDGE
COVER
CONSTRAINTS
.
534
30.14
SIMPLE
2-EDGE
COVERS
.
535
30.15
GRAPHS
WITH
P(G)
=
R(G)
AND
A(G)
=
P(G)
.
536
30.16
EXCLUDING
TRIANGLES
.
539
30.16A
EXCLUDING
HIGHER
POLYGONS
.
544
30.16B
PACKING
EDGES
AND
FACTOR-CRITICAL
SUBGRAPHS
.
544
30.16C
2-FACTORS
WITHOUT
SHORT
CIRCUITS
.
545
31
B-MATCHINGS
.
546
31.1
B-MATCHINGS
.
546
31.2
THE
B-MATCHING
POLYTOPE
.
547
31.3
TOTAL
DUAL
INTEGRALITY
.
550
31.4
THE
WEIGHTED
B-MATCHING
PROBLEM
.
554
31.5
IF
B
IS
EVEN
.
556
31.6
IF
B
IS
CONSTANT
.
558
31.7
FURTHER
RESULTS
AND
NOTES
.
559
31.7A
COMPLEXITY
SURVEY
FOR
THE
B-MATCHING
PROBLEM
.
559
31.7B
FACETS
AND
MINIMAL
SYSTEMS
FOR
THE
B-MATCHING
POLYTOPE
.
559
31.7C
REGULARIZABLE
GRAPHS
.
560
31.
7D
FURTHER
NOTES
.
561
32
CAPACITATED
B-MATCHINGS
.
562
32.1
CAPACITATED
B-MATCHINGS
.
562
32.2
THE
CAPACITATED
B-MATCHING
POLYTOPE
.
564
32.3
TOTAL
DUAL
INTEGRALITY
.
566
32.4
THE
WEIGHTED
CAPACITATED
B-MATCHING
PROBLEM
.
567
XX
TABLE
OF
CONTENTS
32.4A
FURTHER
NOTES
.
567
33
SIMPLE
B-MATCHINGS
AND
B-FACTORS
.
569
33.1
SIMPLE
B-MATCHINGS
AND
B-FACTORS
.
569
33.2
THE
SIMPLE
B-MATCHING
POLYTOPE
AND
THE
B-FACTOR
POLYTOPE
.
570
33.3
TOTAL
DUAL
INTEGRALITY
.
570
33.4
THE
WEIGHTED
SIMPLE
B-MATCHING
AND
B-FACTOR
PROBLEM
.
571
33.5
IF
B
IS
CONSTANT
.
572
33.6
FURTHER
RESULTS
AND
NOTES
.
573
33.6A
COMPLEXITY
RESULTS
.
573
33.6B
DEGREE-SEQUENCES
.
573
33.6C
FURTHER
NOTES
.
574
34
B-EDGE
COVERS
.
575
34.1
B-EDGE
COVERS
.
575
34.2
THE
B-EDGE
COVER
POLYHEDRON
.
576
34.3
TOTAL
DUAL
INTEGRALITY
.
576
34.4
THE
WEIGHTED
B-EDGE
COVER
PROBLEM
.
577
34.5
IF
B
IS
EVEN
.
578
34.6
IF
B
IS
CONSTANT
.
578
34.7
CAPACITATED
B-EDGE
COVERS
.
579
34.8
SIMPLE
B-EDGE
COVERS
.
581
34.8A
SIMPLE
B-EDGE
COVERS
AND
B-MATCHINGS
.
582
34.8B
CAPACITATED
B-EDGE
COVERS
AND
B-MATCHINGS
.
583
35
UPPER
AND
LOWER
BOUNDS
.
584
35.1
UPPER
AND
LOWER
BOUNDS
.
584
35.2
CONVEX
HULL
.
586
35.3
TOTAL
DUAL
INTEGRALITY
.
589
35.4
FURTHER
RESULTS
AND
NOTES
.
591
35.4A
FURTHER
RESULTS
ON
SUBGRAPHS
WITH
PRESCRIBED
DEGREES
.
591
35.4B
ODD
WALKS
.
593
36
BIDIRECTED
GRAPHS
.
594
36.1
BIDIRECTED
GRAPHS
.
594
36.2
CONVEX
HULL
.
597
36.3
TOTAL
DUAL
INTEGRALITY
.
598
36.4
INCLUDING
PARITY
CONDITIONS
.
600
36.5
CONVEX
HULL
.
604
36.5A
CONVEX
HULL
OF
VERTEX-DISJOINT
CIRCUITS
.
605
36.6
TOTAL
DUAL
INTEGRALITY
.
605
36.7
FURTHER
RESULTS
AND
NOTES
.
607
36.7A
THE
CHVATAL
RANK
.
607
36.7B
FURTHER
NOTES
.
608
TBBLE
OF
CONTENTS
XXI
37
THE
DIMENSION
OF
THE
PERFECT
MATCHING
POLYTOPE
.
609
37.1
THE
DIMENSION
OF
THE
PERFECT
MATCHING
POLYTOPE
.
609
37.2
THE
PERFECT
MATCHING
SPACE
.
611
37.3
THE
BRICK
DECOMPOSITION
.
612
37.4
THE
BRICK
DECOMPOSITION
OF
A
BIPARTITE
GRAPH
.
613
37.5
BRACES
.
614
37.6
BRICKS
.
614
37.7
MATCHING-COVERED
GRAPHS
WITHOUT
NONTRIVIAL
TIGHT
CUTS
.
617
38
THE
PERFECT
MATCHING
LATTICE
.
619
38.1
THE
PERFECT
MATCHING
LATTICE
.
619
38.2
THE
PERFECT
MATCHING
LATTICE
OF
THE
PETERSEN
GRAPH
.
620
38.3
A
FURTHER
FACT
ON
THE
PETERSEN
GRAPH
.
621
38.4
VARIOUS
USEFUL
OBSERVATIONS
.
622
38.5
SIMPLE
BARRIERS
.
624
38.6
THE
PERFECT
MATCHING
LATTICE
OF
A
BRICK
.
630
38.7
SYNTHESIS
AND
FURTHER
CONSEQUENCES
OF
THE
PREVIOUS
RESULTS
.
643
38.8
WHAT
FURTHER
MIGHT
(NOT)
BE
TRUE
.
644
38.9
FURTHER
RESULTS
AND
NOTES
.
646
38.9A
THE
PERFECT
2-MATCHING
SPACE
AND
LATTICE
.
646
38.9B
FURTHER
NOTES
.
647
XXII
TABLE
OF
CONTENTS
VOLUME
B
PART
IV:
MATROIDS
AND
SUBMODULAR
FUNCTIONS
649
39
MATROIDS
.
651
39.1
MATROIDS
.
651
39.2
THE
DUAL
MATROID
.
652
39.3
DELETION,
CONTRACTION,
AND
TRUNCATION
.
653
39.4
EXAMPLES
OF
MATROIDS
.
654
39.4A
RELATIONS
BETWEEN
TRANSVERSAL
MATROIDS
AND
GAMMOIDS
.
659
39.5
CHARACTERIZING
MATROIDS
BY
BASES
.
662
39.6
CHARACTERIZING
MATROIDS
BY
CIRCUITS
.
662
39.6A
A
CHARACTERIZATION
OF
LEHMAN
.
663
39.7
CHARACTERIZING
MATROIDS
BY
RANK
FUNCTIONS
.
664
39.8
THE
SPAN
FUNCTION
AND
FLATS
.
666
39.8A
CHARACTERIZING
MATROIDS
BY
SPAN
FUNCTIONS
.
666
39.8B
CHARACTERIZING
MATROIDS
BY
FLATS
.
667
39.8C
CHARACTERIZING
MATROIDS
IN
TERMS
OF
LATTICES
.
668
39.9
FURTHER
EXCHANGE
PROPERTIES
.
669
39.9A
FURTHER
PROPERTIES
OF
BASES
.
671
39.10
FURTHER
RESULTS
AND
NOTES
.
671
39.10A
FURTHER
NOTES
.
671
39.10B
HISTORICAL
NOTES
ON
MATROIDS
.
672
40
THE
GREEDY
ALGORITHM
AND
THE
INDEPENDENT
SET
POLYTOPE
.
688
40.1
THE
GREEDY
ALGORITHM
.
688
40.2
THE
INDEPENDENT
SET
POLYTOPE
.
690
40.3
THE
MOST
VIOLATED
INEQUALITY
.
693
40.3A
FACETS
AND
ADJACENCY
ON
THE
INDEPENDENT
SET
POLYTOPE
.
698
40.3B
FURTHER
NOTES
.
699
41
MATROID
INTERSECTION
.
700
41.1
MATROID
INTERSECTION
THEOREM
.
700
41.1A
APPLICATIONS
OF
THE
MATROID
INTERSECTION
THEOREM
.
702
41.1B
WOODALL
'
S
PROOF
OF
THE
MATROID
INTERSECTION
THEOREM.
.
704
41.2
CARDINALITY
MATROID
INTERSECTION
ALGORITHM
.
705
41.3
WEIGHTED
MATROID
INTERSECTION
ALGORITHM
.
707
41.3A
SPEEDING
UP
THE
WEIGHTED
MATROID
INTERSECTION
ALGORITHM
.
710
41.4
INTERSECTION
OF
THE
INDEPENDENT
SET
POLYTOPES
.
712
41.4A
FACETS
OF
THE
COMMON
INDEPENDENT
SET
POLYTOPE
.
717
41.4B
UP
AND
DOWN
HULL
OF
THE
COMMON
BASE
POLYTOPE
.
719
TABLE
OF
CONTENTS
XXIII
41.5
FURTHER
RESULTS
AND
NOTES
.
720
41.5A
MENGER
'
S
THEOREM
FOR
MATROIDS
.
720
41.5B
EXCHANGE
PROPERTIES
.
721
41.5C
JUMP
SYSTEMS
.
722
41.
5D
FURTHER
NOTES
.
723
42
MATROID
UNION
.
725
42.1
MATROID
UNION
THEOREM
.
725
42.1A
APPLICATIONS
OF
THE
MATROID
UNION
THEOREM
.
727
42.1B
HORN
'
S
PROOF
.
729
42.2
POLYHEDRAL
APPLICATIONS
.
730
42.3
MATROID
UNION
ALGORITHM
.
731
42.4
THE
CAPACITATED
CASE:
FRACTIONAL
PACKING
AND
COVERING
OF
BASES
.
732
42.5
THE
CAPACITATED
CASE:
INTEGER
PACKING
AND
COVERING
OF
BASES.
.
734
42.6
FURTHER
RESULTS
AND
NOTES
.
736
42.6A
INDUCTION
OF
MATROIDS
.
736
42.6B
LIST-COLOURING
.
737
42.6C
STRONGLY
BASE
ORDERABLE
MATROIDS
.
738
42.
6D
BLOCKING
AND
ANTIBLOCKING
POLYHEDRA
.
741
42.6E
FURTHER
NOTES
.
743
42.6F
HISTORICAL
NOTES
ON
MATROID
UNION
.
743
43
MATROID
MATCHING
.
745
43.1
INFINITE
MATROIDS
.
745
43.2
MATROID
MATCHINGS
.
746
43.3
CIRCUITS
.
747
43.4 A
SPECIAL
CLASS
OF
MATROIDS
.
747
43.5
A
MIN-MAX
FORMULA
FOR
MAXIMUM-SIZE
MATROID
MATCHING.
.
751
43.6
APPLICATIONS
OF
THE
MATROID
MATCHING
THEOREM
.
753
43.7
A
GALLAI
THEOREM
FOR
MATROID
MATCHING
AND
COVERING
.
756
43.8
LINEAR
MATROID
MATCHING
ALGORITHM
.
757
43.9
MATROID
MATCHING
IS
NOT
POLYNOMIAL-TIME
SOLVABLE
IN
GENERAL
.
762
43.10
FURTHER
RESULTS
AND
NOTES
.
763
43.10A
OPTIMAL
PATH-MATCHING
.
763
43.10B
FURTHER
NOTES
.
764
44
SUBMODULAR
FUNCTIONS
AND
POLYMATROIDS
.
766
44.1
SUBMODULAR
FUNCTIONS
AND
POLYMATROIDS
.
766
44.1A
EXAMPLES
.
768
44.2
OPTIMIZATION
OVER
POLYMATROIDS
BY
THE
GREEDY
METHOD
.
771
44.3
TOTAL
DUAL
INTEGRALITY
.
773
44.4
F
IS
DETERMINED
BY
EPF
.
773
44.5
SUPERMODULAR
FUNCTIONS
AND
CONTRAPOLYMATROIDS
.
774
XXIV
TABLE
OF
CONTENTS
44.6
FURTHER
RESULTS
AND
NOTES
.
775
44.6A
SUBMODULAR
FUNCTIONS
AND
MATROIDS
.
775
44.6B
REDUCING
INTEGER
POLYMATROIDS
TO
MATROIDS
.
776
44.6C
THE
STRUCTURE
OF
POLYMATROIDS
.
776
44.6D
CHARACTERIZATION
OF
POLYMATROIDS
.
779
44.
6E
OPERATIONS
ON
SUBMODULAR
FUNCTIONS
AND
POLYMATROIDS
.
.781
44.6F
DUALS
OF
POLYMATROIDS
.
782
44.6G
INDUCTION
OF
POLYMATROIDS
.
782
44.
6H
LOVASZ
'
S
GENERALIZATION
OF
KONIG
'
S
MATCHING
THEOREM
.
783
44.
6I
FURTHER
NOTES
.
784
45
SUBMODULAR
FUNCTION
MINIMIZATION
.
786
45.1
SUBMODULAR
FUNCTION
MINIMIZATION
.
786
45.2
ORDERS
AND
BASE
VECTORS
.
787
45.3
A
SUBROUTINE
.
787
45.4
MINIMIZING
A
SUBMODULAR
FUNCTION
.
789
45.5
RUNNING
TIME
OF
THE
ALGORITHM
.
790
45.6
MINIMIZING
A
SYMMETRIC
SUBMODULAR
FUNCTION
.
792
45.7
MINIMIZING
A
SUBMODULAR
FUNCTION
OVER
THE
ODD
SETS
.
793
46
POLYMATROID
INTERSECTION
.
795
46.1
BOX-TOTAL
DUAL
INTEGRALITY
OF
POLYMATROID
INTERSECTION
.
795
46.2
CONSEQUENCES
.
796
46.3
CONTRAPOLYMATROID
INTERSECTION
.
797
46.4
INTERSECTING
A
POLYMATROID
AND
A
CONTRAPOLYMATROID
.
798
46.5
FRANK
'
S
DISCRETE
SANDWICH
THEOREM
.
799
46.6
INTEGER
DECOMPOSITION
.
;
.
800
46.7
FURTHER
RESULTS
AND
NOTES
.
801
46.7A
UP
AND
DOWN
HULL
OF
THE
COMMON
BASE
VECTORS
.
801
46.7B
FURTHER
NOTES
.
804
47
POLYMATROID
INTERSECTION
ALGORITHMICALLY
.
805
47.1
A
MAXIMUM-SIZE
COMMON
VECTOR
IN
TWO
POLYMATROIDS
.
805
47.2
MAXIMIZING
A
COORDINATE
OF
A
COMMON
BASE
VECTOR
.
807
47.3
WEIGHTED
POLYMATROID
INTERSECTION
IN
POLYNOMIAL
TIME
.
809
47.4
WEIGHTED
POLYMATROID
INTERSECTION
IN
STRONGLY
POLYNOMIAL
TIME
.
811
47.5
CONTRAPOLYMATROIDS
.
818
47.6
INTERSECTING
A
POLYMATROID
AND
A
CONTRAPOLYMATROID
.
818
47.6A
FURTHER
NOTES
.
819
TABLE
OF
CONTENTS
XXV
48
DILWORTH
TRUNCATION
.
820
48.1
IF
/(0)
0
.
820
48.2
DILWORTH
TRUNCATION
.
821
48.2A
APPLICATIONS
AND
INTERPRETATIONS
.
823
48.3
INTERSECTION
.
825
49
SUBMODULARITY
MORE
GENERALLY
.
826
49.1
SUBMODULAR
FUNCTIONS
ON
A
LATTICE
FAMILY
.
826
49.2
INTERSECTION
.
828
49.3
COMPLEXITY
.
829
49.4
SUBMODULAR
FUNCTIONS
ON
AN
INTERSECTING
FAMILY
.
832
49.5
INTERSECTION
.
833
49.6
FROM
AN
INTERSECTING
FAMILY
TO
A
LATTICE
FAMILY
.
834
49.7
COMPLEXITY
.
835
49.8
INTERSECTING
A
POLYMATROID
AND
A
CONTRAPOLYMATROID
.
837
49.9
SUBMODULAR
FUNCTIONS
ON
A
CROSSING
FAMILY
.
838
49.10
COMPLEXITY
.
840
49.10A
NONEMPTINESS
OF
THE
BASE
POLYHEDRON
.
841
49.11
FURTHER
RESULTS
AND
NOTES
.
842
49.11A
MINIMIZING
A
SUBMODULAR
FUNCTION
OVER
A
SUBCOLLECTION
OF
A
LATTICE
FAMILY
.
842
49.11B
GENERALIZED
POLYMATROIDS
.
845
49.11C
SUPERMODULAR
COLOURINGS
.
849
49.
LID
FURTHER
NOTES
.
851
PART
V:
TREES,
BRANCHINGS,
AND
CONNECTORS
853
50
SHORTEST
SPANNING
TREES
.
855
50.1
SHORTEST
SPANNING
TREES
.
855
50.2
IMPLEMENTING
PRIM
'
S
METHOD
.
857
50.3
IMPLEMENTING
KRUSKAL
'
S
METHOD
.
858
50.3A
PARALLEL
FOREST-MERGING
.
859
50.3B
A
DUAL
GREEDY
ALGORITHM
.
859
50.4
THE
LONGEST
FOREST
AND
THE
FOREST
POLYTOPE
.
860
50.5
THE
SHORTEST
CONNECTOR
AND
THE
CONNECTOR
POLYTOPE
.
862
50.6
FURTHER
RESULTS
AND
NOTES
.
864
50.6A
COMPLEXITY
SURVEY
FOR
SHORTEST
SPANNING
TREE
.
864
50.6B
CHARACTERIZATION
OF
SHORTEST
SPANNING
TREES
.
865
50.6C
THE
MAXIMUM
RELIABILITY
PROBLEM
.
866
50.6D
EXCHANGE
PROPERTIES
OF
FORESTS
.
867
50.
6E
UNIQUENESS
OF
SHORTEST
SPANNING
TREE
.
868
50.
6F
FOREST
COVERS
.
869
50.6G
FURTHER
NOTES
.
870
50.6H
HISTORICAL
NOTES
ON
SHORTEST
SPANNING
TREES
.
871
XXVI
TABLE
OF
CONTENTS
51
PACKING
AND
COVERING
OF
TREES
.
877
51.1
UNIONS
OF
FORESTS
.
877
51.2
DISJOINT
SPANNING
TREES
.
877
51.3
COVERING
BY
FORESTS
.
878
51.4
COMPLEXITY
.
879
51.5
FURTHER
RESULTS
AND
NOTES
.
889
51.5A
COMPLEXITY
SURVEY
FOR
TREE
PACKING
AND
COVERING
.
889
51.5B
FURTHER
NOTES
.
892
52
LONGEST
BRANCHINGS
AND
SHORTEST
ARBORESCENCES
.
893
52.1
FINDING
A
SHORTEST
R-ARBORESCENCE
.
893
52.1A
R-ARBORESCENCES
AS
COMMON
BASES
OF
TWO
MATROIDS.
.
895
52.2
RELATED
PROBLEMS
.
895
52.3
A
MIN-MAX
RELATION
FOR
SHORTEST
R-ARBORESCENCES
.
896
52.4
THE
R-ARBORESCENCE
POLYTOPE
.897
52.4A
UNCROSSING
CUTS
.
899
52.5
A
MIN-MAX
RELATION
FOR
LONGEST
BRANCHINGS
.
900
52.6
THE
BRANCHING
POLYTOPE
.
901
52.7
THE
ARBORESCENCE
POLYTOPE
.
901
52.8
FURTHER
RESULTS
AND
NOTES
.
902
52.8A
COMPLEXITY
SURVEY
FOR
SHORTEST
R-ARBORESCENCE
.
902
52.8B
CONCISE
LP-FORMULATION
FOR
SHORTEST
R-ARBORESCENCE
.
902
52.8C
FURTHER
NOTES
.
903
53
PACKING
AND
COVERING
OF
BRANCHINGS
AND
ARBORESCENCES
.
904
53.1
DISJOINT
BRANCHINGS
.
904
53.2
DISJOINT
R-ARBORESCENCES
.
905
53.3
THE
CAPACITATED
CASE
.
907
53.4
DISJOINT
ARBORESCENCES
.
908
53.5
COVERING
BY
BRANCHINGS
.
908
53.6
AN
EXCHANGE
PROPERTY
OF
BRANCHINGS
.
909
53.7
COVERING
BY
R-ARBORESCENCES
.
911
53.8
MINIMUM-LENGTH
UNIONS
OF
K
R-ARBORESCENCES
.
913
53.9
THE
COMPLEXITY
OF
FINDING
DISJOINT
ARBORESCENCES
.
918
53.10
FURTHER
RESULTS
AND
NOTES
.
921
53.10A
COMPLEXITY
SURVEY
FOR
DISJOINT
ARBORESCENCES
.
921
53.10B
ARBORESCENCES
WITH
ROOTS
IN
GIVEN
SUBSETS
.
923
53.10C
DISCLAIMERS
.
925
53.10D
FURTHER
NOTES
.
926
54
BICONNECTORS
AND
BIBRANCHINGS
.
928
54.1
SHORTEST
R
-
S
BICONNECTORS
.
928
54.2
LONGEST
R
-
S
BIFORESTS
.
930
54.3
DISJOINT
R
-
S
BICONNECTORS
.
931
54.4
COVERING
BY
R
-
S
BIFORESTS
.
934
TABLE
OF
CONTENTS
XXVII
54.5
MINIMUM-SIZE
BIBRANCHINGS
.
934
54.6
SHORTEST
BIBRANCHINGS
.
935
54.6A
LONGEST
BIFURCATIONS
.
937
54.7
DISJOINT
BIBRANCHINGS
.
940
54.7A
PROOF
USING
SUPERMODULAR
COLOURINGS
.
943
54.7B
COVERING
BY
BIFURCATIONS
.
943
54.7C
DISJOINT
R
-
S
BICOJMECTORS
AND
R
-
S
BIBRANCHINGS
.
944
54.
7D
COVERING
BY
R
-
S
BIFORESTS
AND
BY
R
-
S
BIFURCATIONS
.
944
55
MINIMUM
DIRECTED
CUT
COVERS
AND
PACKING
DIRECTED
CUTS
.
946
55.1
MINIMUM
DIRECTED
CUT
COVERS
AND
PACKING
DIRECTED
CUTS
.
946
55.2
THE
LUCCHESI-YOUNGER
THEOREM
.
947
55.3
DIRECTED
CUT
FC-COVERS.
949
55.4
FEEDBACK
ARC
SETS
.
951
55.5
COMPLEXITY
.
953
55.5A
FINDING
A
DUAL
SOLUTION
.
954
55.6
FURTHER
RESULTS
AND
NOTES
.
956
55.6A
COMPLEXITY
SURVEY
FOR
MINIMUM-SIZE
DIRECTED
CUT
COVER
.
956
55.6B
FEEDBACK
ARC
SETS
IN
LINKLESSLY
EMBEDDABLE
DIGRAPHS
.
956
55.6C
FEEDBACK
VERTEX
SETS
.
958
55.
6D
THE
BIPARTITE
CASE
.
959
55.
6E
FURTHER
NOTES
.
960
56
MINIMUM
DIRECTED
CUTS
AND
PACKING
DIRECTED
CUT
COVERS
.
962
56.1
MINIMUM
DIRECTED
CUTS
AND
PACKING
DIRECTED
CUT
COVERS
.
962
56.2
SOURCE-SINK
CONNECTED
DIGRAPHS
.
964
56.3
OTHER
CASES
WHERE
WOODALL
'
S
CONJECTURE
IS
TRUE
.
967
56.3A
FURTHER
NOTES
.
968
57
STRONG
CONNECTORS
.
969
57.1
MAKING
A
DIRECTED
GRAPH
STRONGLY
CONNECTED
.
969
57.2
SHORTEST
STRONG
CONNECTORS
.
970
57.3
POLYHEDRALLY
.
973
57.4
DISJOINT
STRONG
CONNECTORS
.
973
57.5
COMPLEXITY
.
975
57.5A
CROSSING
FAMILIES
.
976
58
THE
TRAVELING
SALESMAN
PROBLEM
.
981
58.1
THE
TRAVELING
SALESMAN
PROBLEM
.
981
58.2
NP-COMPLETENESS
OF
THE
TSP
.
982
58.3
BRANCH-AND-BOUND
TECHNIQUES
.
982
58.4
THE
SYMMETRIC
TRAVELING
SALESMAN
POLYTOPE
.
983
58.5
THE
SUBTOUR
ELIMINATION
CONSTRAINTS
.
984
XXVIII
THBLE
OF
CONTENTS
58.6
1-TREES
AND
LAGRANGEAN
RELAXATION
.
985
58.7
THE
2-FACTOR
CONSTRAINTS
.
986
58.8
THE
CLIQUE
TREE
INEQUALITIES
.
987
58.8A
CHRISTOFIDES
'
HEURISTIC
FOR
THE
TSP
.
989
58.8B
FURTHER
NOTES
ON
THE
SYMMETRIC
TRAVELING
SALESMAN
PROBLEM
.
990
58.9
THE
ASYMMETRIC
TRAVELING
SALESMAN
PROBLEM
.
992
58.10
DIRECTED
1-TREES
.
993
58.10A
AN
INTEGER
PROGRAMMING
FORMULATION
.
993
58.10B
FURTHER
NOTES
ON
THE
ASYMMETRIC
TRAVELING
SALESMAN
PROBLEM
.
994
58.11
FURTHER
NOTES
ON
THE
TRAVELING
SALESMAN
PROBLEM
.
995
58.11A
FURTHER
NOTES
.
995
58.11B
HISTORICAL
NOTES
ON
THE
TRAVELING
SALESMAN
PROBLEM
.
996
59
MATCHING
FORESTS
.
1005
59.1
INTRODUCTION
.
1005
59.2
THE
MAXIMUM
SIZE
OF
A
MATCHING
FOREST
.
1006
59.3
PERFECT
MATCHING
FORESTS
.
1007
59.4
AN
EXCHANGE
PROPERTY
OF
MATCHING
FORESTS
.
1008
59.5
THE
MATCHING
FOREST
POLYTOPE
.
1011
59.6
FURTHER
RESULTS
AND
NOTES
.
1015
59.6A
MATCHING
FORESTS
IN
PARTITIONABLE
MIXED
GRAPHS
.
1015
59.6B
FURTHER
NOTES
.
1017
60
SUBMODULAR
FUNCTIONS
ON
DIRECTED
GRAPHS
.
1018
60.1
THE
EDMONDS-GILES
THEOREM
.
1018
60.1A
APPLICATIONS
.
1020
60.1B
GENERALIZED
POLYMATROIDS
AND
THE
EDMONDS-GILES
THEOREM
.
1020
60.2
A
VARIANT
.
1021
60.2A
APPLICATIONS
.
1023
60.3
FURTHER
RESULTS
AND
NOTES
.
1025
60.3A
LATTICE
POLYHEDRA
.
1025
60.3B
POLYMATROIDAL
NETWORK
FLOWS
.
1028
60.3C
A
GENERAL
MODEL
.
1029
60.3D
PACKING
CUTS
AND
GYORI
'
S
THEOREM
.
1030
60.
3E
FURTHER
NOTES
.
1034
61
GRAPH
ORIENTATION
.
1035
61.1
ORIENTATIONS
WITH
BOUNDS
ON
IN
AND
OUTDEGREES
.
1035
61.2
2-EDGE-CONNECTIVITY
AND
STRONGLY
CONNECTED
ORIENTATIONS
.
1037
61.2A
STRONGLY
CONNECTED
ORIENTATIONS
WITH
BOUNDS
ON
DEGREES
.
1038
61.3
NASH-WILLIAMS
'
ORIENTATION
THEOREM
.
1040
TABLE
OF
CONTENTS
XXIX
61.4
FC-ARC-CONNECTED
ORIENTATIONS
OF
2FC-EDGE-CONNECTED
GRAPHS
.
1044
61.4A
COMPLEXITY
.
1045
61.4B
FC-ARC-CONNECTED
ORIENTATIONS
WITH
BOUNDS
ON
DEGREES
.
1045
61.4C
ORIENTATIONS
OF
GRAPHS
WITH
LOWER
BOUNDS
ON
INDEGREES
OF
SETS
.
1046
61.
4D
FURTHER
NOTES
.
1047
62
NETWORK
SYNTHESIS
.
1049
62.1
MINIMAL
FC-(
EDGE-)
CONNECTED
GRAPHS
.
1049
62.2
THE
NETWORK
SYNTHESIS
PROBLEM
.
1051
62.3
MINIMUM-CAPACITY
NETWORK
DESIGN
.
1052
62.4
INTEGER
REALIZATIONS
AND
R-EDGE-CONNECTED
GRAPHS
.
1055
63
CONNECTIVITY
AUGMENTATION
.
1058
63.1
MAKING
A
DIRECTED
GRAPH
FC-ARC-CONNECTED
.
1058
63.1A
FC-ARC-CONNECTORS
WITH
BOUNDS
ON
DEGREES
.
1061
63.2
MAKING
AN
UNDIRECTED
GRAPH
2-EDGE-CONNECTED
.
1062
63.3
MAKING
AN
UNDIRECTED
GRAPH
FC-EDGE-CONNECTED
.
1063
63.3A
FC-EDGE-CONNECTORS
WITH
BOUNDS
ON
DEGREES
.
1066
63.4
R-EDGE-CONNECTIVITY
AND
R-EDGE-CONNECTORS
.
1067
63.5
MAKING
A
DIRECTED
GRAPH
FC-VERTEX-CONNECTED
.
1074
63.6
MAKING
AN
UNDIRECTED
GRAPH
FC-VERTEX-CONNECTED
.
1077
63.6A
FURTHER
NOTES
.
1078
PART
VI:
CLIQUES,
STABLE
SETS,
AND
COLOURING
1081
64
CLIQUES,
STABLE
SETS,
AND
COLOURING
.
1083
64.1
TERMINOLOGY
AND
NOTATION
.
1083
64.2
NP-COMPLETENESS
.
1084
64.3
BOUNDS
ON
THE
COLOURING
NUMBER
.
1085
64.3A
BROOKS
'
UPPER
BOUND
ON
THE
COLOURING
NUMBER
.
1086
64.3B
HADWIGER
'
S
CONJECTURE
.
1086
64.4
THE
STABLE
SET,
CLIQUE,
AND
VERTEX
COVER
POLYTOPE
.
1088
64.4A
FACETS
AND
ADJACENCY
ON
THE
STABLE
SET
POLYTOPE
.
1088
64.5
FRACTIONAL
STABLE
SETS
.
1090
64.5A
FURTHER
ON
THE
FRACTIONAL
STABLE
SET
POLYTOPE
.
1091
64.6
FRACTIONAL
VERTEX
COVERS
.
1093
64.6A
A
BOUND
OF
LORENTZEN
.
1095
64.7
THE
CLIQUE
INEQUALITIES
.
1095
64.8
FRACTIONAL
AND
WEIGHTED
COLOURING
NUMBERS
.
1096
64.8A
THE
RATIO
OF
X(G)
AND
X*(G)
.
1098
64.8B
THE
CHVATAL
RANK
.
1098
64.9
FURTHER
RESULTS
AND
NOTES
.
1099
XXX
TABLE
OF
CONTENTS
64.9A
GRAPHS
WITH
POLYNOMIAL-TIME
STABLE
SET
ALGORITHM
.
1099
64.9B
COLOURINGS
AND
ORIENTATIONS
.
1101
64.9C
ALGEBRAIC
METHODS
.
1102
64.9D
APPROXIMATION
ALGORITHMS
.
1103
64.9E
FURTHER
NOTES
.
1104
65
PERFECT
GRAPHS:
GENERAL
THEORY
.
1106
65.1
INTRODUCTION
TO
PERFECT
GRAPHS
.
1106
65.2
THE
PERFECT
GRAPH
THEOREM
.
1108
65.3
REPLICATION
.
1109
65.4
PERFECT
GRAPHS
AND
POLYHEDRA
.
1110
65.4A
LOVASZ
'
S
PROOF
OF
THE
REPLICATION
LEMMA
.
1111
65.5
DECOMPOSITION
OF BERGE
GRAPHS
.
1112
65.5A
0
AND
1-JOINS
.
1112
65.5B
THE
2-JOIN
.
1113
65.6
PRE-PROOF
WORK
ON
THE
STRONG
PERFECT
GRAPH
CONJECTURE
.
1115
65.6A
PARTITIONABLE
GRAPHS
.
1116
65.6B
MORE
CHARACTERIZATIONS
OF
PERFECT
GRAPHS
.
1118
65.6C
THE
STABLE
SET
POLYTOPE
OF
MINIMALLY
IMPERFECT
GRAPHS
.
1118
65.
6D
GRAPH
CLASSES
.
1120
65.
6E
THE
^-STRUCTURE
OF
A
GRAPH
AND
A
SEMI-STRONG
PERFECT
GRAPH
THEOREM
.
1122
65.
6F
FURTHER
NOTES
ON
THE
STRONG
PERFECT
GRAPH
CONJECTURE
.
1123
65.7
FURTHER
RESULTS
AND
NOTES
.
1125
65.7A
PERZ
AND
ROLEWICZ
'
S
PROOF
OF
THE
PERFECT
GRAPH
THEOREM
.
1125
65.7B
KERNEL
SOLVABILITY
.
1126
65.7C
THE
AMALGAM
.
1130
65.
7D
DIPERFECT
GRAPHS
.
1131
65.
7E
FURTHER
NOTES
.
1133
66
CLASSES
OF
PERFECT
GRAPHS
.
1135
66.1
BIPARTITE
GRAPHS
AND
THEIR
LINE
GRAPHS
.
1135
66.2
COMPARABILITY
GRAPHS
.
1137
66.3
CHORDAL
GRAPHS
.
1138
66.3A
CHORDAL
GRAPHS
AS
INTERSECTION
GRAPHS
OF
SUBTREES
OF
A
TREE
.
1142
66.4
MEYNIEL
GRAPHS
.
1143
66.5
FURTHER
RESULTS
AND
NOTES
.
1145
66.5A
STRONGLY
PERFECT
GRAPHS
.
1145
66.5B
PERFECTLY
ORDERABLE
GRAPHS
.
1146
66.5C
UNIMODULAR
GRAPHS
.
1147
66.5D
FURTHER
CLASSES
OF
PERFECT
GRAPHS
.
1148
TABLE
OF
CONTENTS
XXXI
66.
5E
FURTHER
NOTES
.
1149
67
PERFECT
GRAPHS:
POLYNOMIAL-TIME
SOLVABILITY
.
1152
67.1
OPTIMUM
CLIQUE
AND
COLOURING
IN
PERFECT
GRAPHS
ALGORITHMICALLY
.
1152
67.2
WEIGHTED
CLIQUE
AND
COLOURING
ALGORITHMICALLY
.
1155
67.3
STRONG
POLYNOMIAL-TIME
SOLVABILITY
.
1159
67.4
FURTHER
RESULTS
AND
NOTES
.
1159
67.4A
FURTHER
ON
I?(G)
.
1159
67.4B
THE
SHANNON
CAPACITY
0(G)
.
1167
67.4C
CLIQUE
COVER
NUMBERS
OF
PRODUCTS
OF
GRAPHS
.
1172
67.4D
A
SHARPER
UPPER
BOUND
$'(&)
ON
A(G)
.
1173
67.4E
AN
OPERATOR
STRENGTHENING
CONVEX
BODIES
.
1173
67.4F
FURTHER
NOTES
.
1175
67.4G
HISTORICAL
NOTES
ON
PERFECT
GRAPHS
.
1176
68
T-PERFECT
GRAPHS
.
1186
68.1
T-PERFECT
GRAPHS
.
1186
68.2
STRONGLY
T-PERFECT
GRAPHS
.
1187
68.3
STRONG
T-PERFECTION
OF
ODD-JCI-FREE
GRAPHS
.
1188
68.4
ON
CHARACTERIZING
T-PERFECTION
.
1194
68.5
A
COMBINATORIAL
MIN-MAX
RELATION
.
1196
68.6
FURTHER
RESULTS
AND
NOTES
.
1200
68.6A
THE
W-STABLE
SET
POLYHEDRON
.
1200
68.6B
BIDIRECTED
GRAPHS
.
1201
68.6C
CHARACTERIZING
ODD-JQ-FREE
GRAPHS
BY
MIXING
STABLE
SETS
AND
VERTEX
COVERS
.
1203
68.
6D
ORIENTATIONS
OF DISCREPANCY
1
.
1204
68.6E
COLOURINGS
AND
ODD
^-SUBDIVISIONS
.
1206
68.
6F
HOMOMORPHISMS
.
1207
68.6G
FURTHER
NOTES
.
1207
69
CLAW-FREE
GRAPHS
.
1208
69.1
INTRODUCTION
.
1208
69.2
MAXIMUM-SIZE
STABLE
SET
IN
A
CLAW-FREE
GRAPH
.
1208
69.3
MAXIMUM-WEIGHT
STABLE
SET
IN
A
CLAW-FREE
GRAPH
.
1213
69.4
FURTHER
RESULTS
AND
NOTES
.
1216
69.4A
ON
THE
STABLE
SET
POLYTOPE
OF
A
CLAW-FREE
GRAPH
.
1216
69.4B
FURTHER
NOTES
.
1217
XXXII
TABLE
OF
CONTENTS
VOLUME
C
PART
VII:
MULTIFLOWS
AND
DISJOINT
PATHS
1219
70
MULTIFLOWS
AND
DISJOINT
PATHS
.
1221
70.1
DIRECTED
MULTIFLOW
PROBLEMS
.
1221
70.2
UNDIRECTED
MULTIFLOW
PROBLEMS
.
1222
70.3
DISJOINT
PATHS
PROBLEMS
.
1223
70.4
REDUCTIONS
.
1223
70.5
COMPLEXITY
OF
THE
DISJOINT
PATHS
PROBLEM
.
1224
70.6
COMPLEXITY
OF
THE
FRACTIONAL
MULTIFLOW
PROBLEM
.
1225
70.7
THE
CUT
CONDITION
FOR
DIRECTED
GRAPHS
.
1227
70.8
THE
CUT
CONDITION
FOR
UNDIRECTED
GRAPHS
.
1228
70.9
RELATIONS
BETWEEN
FRACTIONAL,
HALF-INTEGER,
AND
INTEGER
SOLUTIONS
.
1230
70.10
THE
EULER
CONDITION
.
1233
70.11
SURVEY
OF
CASES
WHERE
A
GOOD
CHARACTERIZATION
HAS
BEEN
FOUND
.
1234
70.12
RELATION
BETWEEN
THE
CUT
CONDITION
AND
FRACTIONAL
CUT
PACKING
.
1236
70.12A
SUFFICIENCY
OF
THE
CUT
CONDITION
SOMETIMES
IMPLIES
AN
INTEGER
MULTIFLOW
.
1238
70.12B
THE
CUT
CONDITION
AND
INTEGER
MULTIFLOWS
IN
DIRECTED
GRAPHS
.
1241
70.13
FURTHER
RESULTS
AND
NOTES
.
1242
70.13A
FIXING
THE
NUMBER
OF
COMMODITIES
IN
UNDIRECTED
GRAPHS
.
1242
70.13B
FIXING
THE
NUMBER
OF
COMMODITIES
IN
DIRECTED
GRAPHS
.
1243
70.13C
DISJOINT
PATHS
IN
ACYCLIC
DIGRAPHS
.
1244
70.13D
A
COLUMN
GENERATION
TECHNIQUE
FOR
MULTIFLOWS
.
1245
70.13E
APPROXIMATE
MAX-FLOW
MIN-CUT
THEOREMS
FOR
MULTIFLOWS
.
1247
70.13F
FURTHER
NOTES
.
1248
70.13G
HISTORICAL
NOTES
ON
MULTICOMMODITY
FLOWS
.
1249
71
TWO
COMMODITIES
.
1251
71.1
THE
ROTHSCHILD-WHINSTON
THEOREM
AND
HU
'
S
2-COMMODITY
FLOW
THEOREM
.
1251
71.1A
NASH-WILLIAMS
'
PROOF
OF
THE
ROTHSCHILD-WHINSTON
THEOREM
.
1254
71.2
CONSEQUENCES
.
1255
71.3
2-COMMODITY
CUT
PACKING
.
1257
71.4
FURTHER
RESULTS
AND
NOTES
.
1261
TABLE
OF
CONTENTS
XXXIII
71.4A
TWO
DISJOINT
PATHS
IN
UNDIRECTED
GRAPHS
.
1261
71.4B
A
DIRECTED
2-COMMODITY
FLOW
THEOREM
.
1262
71.4C
KLEITMAN,
MARTIN-LOF,
ROTHSCHILD,
AND
WHINSTON
'
S
THEOREM
.
1263
71.
4D
FURTHER
NOTES
.
1265
72
THREE
OR
MORE
COMMODITIES
.
1266
72.1
DEMAND
GRAPHS
FOR
WHICH
THE
CUT
CONDITION
IS
SUFFICIENT
.
1266
72.2
THREE
COMMODITIES
.
1271
72.2A
THE
A
2I
3-METRIC
CONDITION
.
1273
72.2B
SIX
TERMINALS
.
1275
72.3
CUT
PACKING
.
1276
73
T-PATHS
.
1279
73.1
DISJOINT
T-PATHS
.
1279
73.1A
DISJOINT
T-PATHS
WITH
THE
MATROID
MATCHING
ALGORITHM
.
1283
73.1B
POLYNOMIAL-TIME
FINDABILITY
OF
EDGE-DISJOINT
T-PATHS
.
1285
73.1C
A
FEASIBILITY
CHARACTERIZATION
FOR
INTEGER
JG-FLOWS
.
1286
73.2
FRACTIONAL
PACKING
OF
T-PATHS
.
1287
73.2A
DIRECT
PROOF
OF
COROLLARY
73.2D
.
1288
73.3
FURTHER
RESULTS
AND
NOTES
.
1289
73.3A
FURTHER
NOTES
ON
MADER
'
S
THEOREM
.
1289
73.3B
A
GENERALIZATION
OF
FRACTIONALLY
PACKING
T-PATHS
.
1290
73.3C
LOCKABLE
COLLECTIONS
.
1291
73.3D
MADER
MATROIDS
.
1292
73.
3E
MINIMUM-COST
MAXIMUM-VALUE
MULTIFLOWS
.
1294
73.
3F
FURTHER
NOTES
.
1295
74
PLANAR
GRAPHS
.
1296
74.1
ALL
NETS
SPANNED
BY
ONE
FACE:
THE
OKAMURA-SEYMOUR
THEOREM
.
1296
74.1A
COMPLEXITY
SURVEY
.
1299
74.1B
GRAPHS
ON
THE
PROJECTIVE
PLANE
.
1299
74.1C
IF
ONLY
INNER
VERTICES
SATISFY
THE
EULER
CONDITION
.
1302
74.
ID
DISTANCES
AND
CUT
PACKING
.
1304
74.
IE
LINEAR
ALGEBRA
AND
DISTANCE
REALIZABILITY
.
1305
74.
IF
DIRECTED
PLANAR
GRAPHS
WITH
ALL
TERMINALS
ON
THE
OUTER
BOUNDARY
.
1307
74.2
G
+
H
PLANAR
.
1307
74.2A
DISTANCES
AND
CUT
PACKING
.
1308
74.2B
DELETING
THE
EULER
CONDITION
IF
G
+
H
IS
PLANAR
.
1309
74.3
OKAMURA
'
S
THEOREM
.
1311
74.3A
DISTANCES
AND
CUT
PACKING
.
1313
XXXIV
TABLE
OF
CONTENTS
74.3B
THE
KLEIN
BOTTLE
.
1314
74.3C
COMMODITIES
SPANNED
BY
THREE
OR
MORE
FACES
.
1316
74.4
FURTHER
RESULTS
AND
NOTES
.
1318
74.4A
ANOTHER
THEOREM
OF
OKAMURA
.
1318
74.4B
SOME
OTHER
PLANAR
CASES
WHERE
THE
CUT
CONDITION
IS
SUFFICIENT
.
1320
74.4C
VERTEX-DISJOINT
PATHS
IN
PLANAR
GRAPHS
.
1320
74.4D
GRID
GRAPHS
.
1323
74.4E
FURTHER
NOTES
.
1325
75
CUTS,
ODD
CIRCUITS,
AND
MULTIFLOWS
.
1326
75.1
WEAKLY
AND
STRONGLY
BIPARTITE
GRAPHS
.
1326
75.1A
NP-COMPLETENESS
OF
MAXIMUM
CUT
.
1328
75.1B
PLANAR
GRAPHS
.
1328
75.2
SIGNED
GRAPHS
.
1329
75.3
WEAKLY,
EVENLY,
AND
STRONGLY
BIPARTITE
SIGNED
GRAPHS
.
1330
75.4
CHARACTERIZING
STRONGLY
BIPARTITE
SIGNED
GRAPHS
.
1331
75.5
CHARACTERIZING
WEAKLY
AND
EVENLY
BIPARTITE
SIGNED
GRAPHS
.
1334
75.6
APPLICATIONS
TO
MULTIFLOWS
.
1341
75.7
THE
CUT
CONE
AND
THE
CUT
POLYTOPE
.
1342
75.8
THE
MAXIMUM
CUT
PROBLEM
AND
SEMIDEFINITE
PROGRAMMING.
.
1345
75.9
FURTHER
RESULTS
AND
NOTES
.
1348
75.9A
CUTS
AND
STABLE
SETS
.
1348
75.9B
FURTHER
NOTES
.
1350
76
HOMOTOPY
AND
GRAPHS
ON
SURFACES
.
1352
76.1
GRAPHS,
CURVES,
AND
THEIR
INTERSECTIONS:
TERMINOLOGY
AND
NOTATION
.
1352
76.2
MAKING
CURVES
MINIMALLY
CROSSING
BY
REIDEMEISTER
MOVES
.
1353
76.3
DECOMPOSING
THE
EDGES
OF
AN
EULERIAN
GRAPH
ON
A
SURFACE
.
1354
76.4
A
COROLLARY
ON
LENGTHS
OF
CLOSED
CURVES
.
1356
76.5
A
HOMOTOPIC
CIRCULATION
THEOREM
.
1357
76.6
HOMOTOPIC
PATHS
IN
PLANAR
GRAPHS
WITH
HOLES
.
1361
76.7
VERTEX-DISJOINT
PATHS
AND
CIRCUITS
OF
PRESCRIBED
HOMOTOPIES.
.
1367
76.7A
VERTEX-DISJOINT
CIRCUITS
OF
PRESCRIBED
HOMOTOPIES
.
1367
76.7B
VERTEX-DISJOINT
HOMOTOPIC
PATHS
IN
PLANAR
GRAPHS
WITH
HOLES
.
1368
76.7C
DISJOINT
TREES
.
1371
TABLE
OF
CONTENTS
XXXV
PART
VIII:
HYPERGRAPHS
1373
77
PACKING
AND
BLOCKING
IN
HYPERGRAPHS:
ELEMENTARY
NOTIONS
.
1375
77.1
ELEMENTARY
HYPERGRAPH
TERMINOLOGY
AND
NOTATION
.
1375
77.2
DELETION,
RESTRICTION,
AND
CONTRACTION
.
1376
77.3
DUPLICATION
AND
PARALLELIZATION
.
1376
77.4
CLUTTERS
.
1376
77.5
PACKING
AND
BLOCKING
.
1377
77.6
THE
BLOCKER
.
1377
77.7
FRACTIONAL
MATCHINGS
AND
VERTEX
COVERS
.
1378
77.8
K-MATCHINGS
AND
FC-VERTEX
COVERS
.
1378
77.9
FURTHER
RESULTS
AND
NOTES
.
1379
77.9A
BOTTLENECK
EXTREMA
.
1379
77.9B
THE
RATIO
OF
R
AND
T
*
.
1380
77.9C
FURTHER
NOTES
.
1381
78
IDEAL
HYPERGRAPHS
.
1383
78.1
IDEAL
HYPERGRAPHS
.
1383
78.2
CHARACTERIZATIONS
OF
IDEAL
HYPERGRAPHS
.
1384
78.3
MINIMALLY
NONIDEAL
HYPERGRAPHS
.
1386
78.4
PROPERTIES
OF
MINIMALLY
NONIDEAL
HYPERGRAPHS:
LEHMAN
'
S
THEOREM
.
1387
78.4A
APPLICATION
OF
LEHMAN
'
S
THEOREM:
GUENIN
'
S
THEOREM
.
1392
78.4B
IDEALITY
IS
IN
CO-NP
.
1394
78.5
FURTHER
RESULTS
AND
NOTES
.
1395
78.5A
COMPOSITION
OF
CLUTTERS
.
1395
78.5B
FURTHER
NOTES
.
1395
79
MENGERIAN
HYPERGRAPHS
.
1397
79.1
MENGERIAN
HYPERGRAPHS
.
1397
79.1A
EXAMPLES
OF MENGERIAN
HYPERGRAPHS
.
1399
79.2
MINIMALLY
NON-MENGERIAN
HYPERGRAPHS
.
1400
79.3
FURTHER
RESULTS
AND
NOTES
.
1401
79.3A
PACKING
HYPERGRAPHS
.
1401
79.3B
RESTRICTIONS
INSTEAD
OF
PARALLELIZATIONS
.
1402
79.3C
EQUIVALENCES
FOR
FC-MATCHINGS
AND
FC-VERTEX
COVERS
.
1402
79.3D
A
GENERAL
TECHNIQUE
.
1403
79.
3E
FURTHER
NOTES
.
1404
XXXVI
TABLE
OF
CONTENTS
80
BINARY
HYPERGRAPHS
.
1406
80.1
BINARY
HYPERGRAPHS
.
1406
80.2
BINARY
HYPERGRAPHS
AND
BINARY
MATROIDS
.
1406
80.3
THE
BLOCKER
OF
A
BINARY
HYPERGRAPH
.
1407
80.3A
FURTHER
CHARACTERIZATIONS
OF
BINARY
CLUTTERS
.
1408
80.4
ON
CHARACTERIZING
BINARY
IDEAL
HYPERGRAPHS
.
1408
80.5
SEYMOUR
'
S
CHARACTERIZATION
OF
BINARY
MENGERIAN
HYPERGRAPHS
.
1409
80.5A
APPLICATIONS
OF
SEYMOUR
'
S
THEOREM
.
1413
80.6
MENGERIAN
MATROIDS
.
1415
80.6A
ORIENTED
MATROIDS
.
1415
80.7
FURTHER
RESULTS
AND
NOTES
.
1416
80.7A
T
2
(H)
=
2
T
(H)
FOR
BINARY
HYPERGRAPHS
H
.
1416
80.7B
APPLICATION:
T-JOINS
AND
T-CUTS
.
1417
80.7C
BOX-INTEGRALITY
OF
K
YY
P
H
.
1418
81
MATROIDS
AND
MULTIFLOWS
.
1419
81.1
MULTIFLOWS
IN
MATROIDS
.
1419
81.2
INTEGER
FC-FLOWING
.
1420
81.3
1-FLOWING
AND
1-CYCLING
.
1421
81.4
2-FLOWING
AND
2-CYCLING
.
1421
81.5
3-FLOWING
AND
3-CYCLING
.
1422
81.6
4-FLOWING,
4-CYCLING,
OO-FLOWING,
AND
OO-CYCLING
.
1423
81.7
THE
CIRCUIT
CONE
AND
CYCLE
POLYTOPE
OF
A
MATROID
.
1424
81.8
THE
CIRCUIT
SPACE
AND
CIRCUIT
LATTICE
OF
A
MATROID
.
1425
81.9
NONNEGATIVE
INTEGER
SUMS
OF
CIRCUITS
.
1425
81.10
NOWHERE-ZERO
FLOWS
AND
CIRCUIT
DOUBLE
COVERS
IN
MATROIDS
.
1426
82
COVERING
AND
ANTIBLOCKING
IN
HYPERGRAPHS
.
1428
82.1
ELEMENTARY
CONCEPTS
.
1428
82.2
FRACTIONAL
EDGE
COVERS
AND
STABLE
SETS
.
1429
82.3
A:-EDGE
COVERS
AND
FC-STABLE
SETS
.
1429
82.4
THE
ANTIBLOCKER
AND
CONFORMALITY
.
1430
82.4A
GILMORE
'
S
CHARACTERIZATION
OF
CONFORMALITY
.
1431
82.5
PERFECT
HYPERGRAPHS
.
1431
82.6
FURTHER
NOTES
.
1434
82.6A
SOME
EQUIVALENCES
FOR
THE
FC-PARAMETERS
.
1434
82.6B
FURTHER
NOTES
.
1437
83
BALANCED
AND
UNIMODULAR
HYPERGRAPHS
.
1439
83.1
BALANCED
HYPERGRAPHS
.
1439
83.2
CHARACTERIZATIONS
OF
BALANCED
HYPERGRAPHS
.
1440
83.2A
TOTALLY
BALANCED
MATRICES
.
1444
83.2B
EXAMPLES
OF
BALANCED
HYPERGRAPHS
.
1447
83.2C
BALANCED
0,
1
MATRICES
.
1447
TABLE
OF
CONTENTS
XXXVII
83.3
UNIMODULAR
HYPERGRAPHS
.
1448
83.3A
FURTHER
NOTES
.
1450
SURVEY
OF
PROBLEMS,
QUESTIONS,
AND
CONJECTURES
.
1453
REFERENCES
.
1463
NAME
INDEX
.
1767
SUBJECT
INDEX
.
1807
GREEK
GRAPH
AND
HYPERGRAPH
FUNCTIONS
.
1880 |
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| spelling | Schrijver, Alexander 1948- Verfasser (DE-588)13576355X aut Combinatorial optimization polyhedra and efficiency Vol. A. Path, flows, matchings : chapters 1 - 38 Alexander Schrijver Berlin [u.a.] Springer 2003 XXXVII, 647 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algorithms and combinatorics 24 (DE-604)BV016881642 1 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016949353&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schrijver, Alexander 1948- Combinatorial optimization polyhedra and efficiency |
| title | Combinatorial optimization polyhedra and efficiency |
| title_auth | Combinatorial optimization polyhedra and efficiency |
| title_exact_search | Combinatorial optimization polyhedra and efficiency |
| title_exact_search_txtP | Combinatorial optimization polyhedra and efficiency |
| title_full | Combinatorial optimization polyhedra and efficiency Vol. A. Path, flows, matchings : chapters 1 - 38 Alexander Schrijver |
| title_fullStr | Combinatorial optimization polyhedra and efficiency Vol. A. Path, flows, matchings : chapters 1 - 38 Alexander Schrijver |
| title_full_unstemmed | Combinatorial optimization polyhedra and efficiency Vol. A. Path, flows, matchings : chapters 1 - 38 Alexander Schrijver |
| title_short | Combinatorial optimization |
| title_sort | combinatorial optimization polyhedra and efficiency path flows matchings chapters 1 38 |
| title_sub | polyhedra and efficiency |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016949353&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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