Geometry of cuts and metrics:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | German |
| Published: |
Berlin [u.a.]
Springer
1997
|
| Series: | Algorithms and combinatorics
15 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 587 S. graph. Darst. |
| ISBN: | 354061611X |
Staff View
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| 245 | 1 | 0 | |a Geometry of cuts and metrics |c Michel Marie Deza ; Monique Laurent |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1997 | |
| 300 | |a XII, 587 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Contents
1. Outline of the Book 1
1.1 Outline of Part I. Measure Aspects: ^i Embeddability
and Probability 2
1.2 Outline of Part II. Hypermetric Spaces: an Approach
via Geometry of Numbers 4
1.3 Outline of Part III. Embeddings of Graphs 6
1.4 Outline of Part IV. Hypercube Embeddings and Designs .... 7
1.5 Outline of Part V. Facets of the Cut Cone and Polytope .... 8
2. Basic Definitions 11
2.1 Graphs 11
2.2 Polyhedra 14
2.3 Algorithms and Complexity 18
2.4 Matrices 19
Part I. Measure Aspects: ^ Embeddability and Probability 23
3. Preliminaries on Distances 27
3.1 Distance Spaces and lp Spaces 27
3.2 Measure Spaces and Lp Spaces 32
4. The Cut Cone and ^ Metrics 37
4.1 The Cut Cone and Polytope 37
4.2 £i Spaces 39
4.3 Realizations, Rigidity, Size and Scale 43
4.4 Complexity Questions 48
4.5 An Application to Statistical Physics 51
5. The Correlation Cone and {0,1} Covariances 53
5.1 The Correlation Cone and Polytope 53
5.2 The Covariance Mapping 55
5.3 Covariances 58
5.4 The Boole Problem 61
viii Contents
6. Conditions for Li Embeddability 67
6.1 Hypermetric and Negative Type Conditions 67
6.1.1 Hypermetric and Negative Type Inequalities 67
6.1.2 Hypermetric and Negative Type Distance Spaces 71
6.1.3 Analogues for Covariances 72
6.2 Characterization of L2 Embeddability 73
6.2.1 Schoenberg s Result and Cayley Menger Determinants . . 74
6.2.2 Menger s Result 78
6.2.3 Further Characterizations 80
6.3 A Chain of Implications 82
6.4 An Example: The Spherical Distance Space 86
6.5 An Example: Kalmanson Distances 91
7. Operations 93
7.1 The Gate Extension Operation 93
7.2 The Antipodal Extension Operation 94
7.3 The Spherical Extension Operation 97
7.4 An Example: The Cocktail Party Graph 98
7.5 The Direct Product and Tensor Product Operations 101
7.6 The 1 Sum Operation 103
8. Li Metrics from Lattices, Semigroups and Normed Spaces 105
8.1 la Metrics from Lattices 105
8.2 Lx Metrics from Semigroups 107
8.3 Lj Metrics from Normed Spaces 109
9. Metric Transforms of Li Spaces 113
9.1 The Schoenberg Transform 115
9.2 The Biotope Transform 118
9.3 The Power Transform 120
10. Lipschitz Embeddings 125
10.1 Embeddings with Distortion 125
10.2 The Negative Type Condition for Lipschitz Embeddings .... 130
10.3 An Application for Approximating Multicommodity Flows . . 132
11. Dimensionality Questions for ^ Embeddings 139
11.1 ^i Embeddings in Fixed Dimension 139
11.1.1 The Order of Congruence of the ^i Space 141
11.1.2 A Canonical Decomposition for Distances 145
11.1.3 Embedding Distances in the £rPlane 148
11.2 On the Minimum £p Dimension 156
Contents ix
12. Examples of the Use of the Lx Metric 161
12.1 The Li Metric in Probability Theory 161
12.2 The ^ Metric in Statistical Data Analysis 162
12.3 The ^ Metric in Computer Vision and Pattern Recognition . . 163
Part II. Hypermetric Spaces: an Approach via Geometry
of Numbers 167
13. Preliminaries on Lattices 175
13.1 Distance Spaces 175
13.2 Lattices and Delaunay Polytopes 177
13.2.1 Lattices 177
13.2.2 Delaunay Polytopes 179
13.2.3 Basic Facts on Delaunay Polytopes 182
13.2.4 Construction of Delaunay Polytopes 184
13.2.5 Additional Notes 187
13.3 Finiteness of the Number of Types of Delaunay Polytopes . . 189
14. Hypermetrics and Delaunay Polytopes 193
14.1 Connection between Hypermetrics and Delaunay Polytopes . . 193
14.2 Polyhedrality of the Hypermetric Cone 199
14.3 Delaunay Polytopes in Root Lattices 206
14.4 On the Radius of Delaunay Polytopes 211
15. Delaunay Polytopes: Rank and Hypermetric Faces 217
15.1 Rank of a Delaunay Polytope 217
15.2 Delaunay Polytopes Related to Faces 222
15.2.1 Hypermetric Faces 222
15.2.2 Hypermetric Facets 226
15.3 Bounds on the Rank of Basic Delaunay Polytopes 230
16. Extreme Delaunay Polytopes 235
16.1 Extreme Delaunay Polytopes and Equiangular Sets of Lines . . 236
16.2 The Schlafli and Gosset Polytopes are Extreme 239
16.3 Extreme Delaunay Polytopes in the Leech Lattice A24 244
16.4 Extreme Delaunay Polytopes in the Barnes Wall Lattice Ai6 . 245
16.5 Extreme Delaunay Polytopes and Perfect Lattices 248
17. Hypermetric Graphs 251
17.1 Characterizing Hypermetric and ^i Graphs 252
17.2 Hypermetric Regular Graphs 262
17.3 Extreme Hypermetric Graphs 269
x Contents
Part III. Isometric Embeddings of Graphs 275
18. Preliminaries on Graphs 279
19. Isometric Embeddings of Graphs into Hypercubes 283
19.1 Djokovic s Characterization 283
19.2 Further Characterizations 286
19.3 Additional Notes 293
20. Isometric Embeddings of Graphs into Cartesian Products 297
20.1 Canonical Metric Representation of a Graph 297
20.2 The Prime Factorization of a Graph 305
20.3 Metric Decomposition of Bipartite Graphs 306
20.4 Additional Notes 308
21. ^ Graphs 313
21.1 Results on £i Graphs 313
21.2 Construction of G via the Atom Graph 316
21.3 Proofs 322
21.4 More about ^i Graphs 325
Part IV. Hypercube Embeddings and Designs 331
22. Rigidity of the Equidistant Metric 335
23. Hypercube Embeddings of the Equidistant Metric 341
23.1 Preliminaries on Designs 341
23.1.1 (r, A, n) Designs and BIBD s 341
23.1.2 Intersecting Systems 344
23.2 Embeddings of 2 ln and Designs 345
23.3 The Minimum /i Size of 2tln 347
23.4 All Hypercube Embeddings of 2t n for Small n,t 350
24. Recognition of Hypercube Embeddable Metrics 353
24.1 Preliminary Results 354
24.2 Generalized Bipartite Metrics 357
24.3 Metrics with Few Values 362
24.3.1 Distances with Values 2a, b (b odd) 363
24.3.2 Distances with Values a, b,a + b {a,b odd) 365
24.3.3 Distances with Values b, 2a, b + 2a(b odd, 6 2a) .... 367
24.4 Truncated Distances of Graphs 370
24.4.1 The Class of Graphs Hk 372
24.5 Metrics with Restricted Extremal Graph 375
Contents xi
25. Cut Lattices, Quasi /i Distances and Hilbert Bases 381
25.1 Cut Lattices 381
25.2 Quasi /i Distances 385
25.3 Hilbert Bases of Cuts 391
Part V. Facets of the Cut Cone and Polytope 395
26. Operations on Valid Inequalities and Facets 401
26.1 Cut and Correlation Vectors 401
26.2 The Permutation Operation 403
26.3 The Switching Operation 403
26.3.1 Switching: A General Definition 403
26.3.2 Switching: Cut Polytope versus Cut Cone 405
26.3.3 The Symmetry Group of the Cut Polytope 409
26.4 The Collapsing Operation 410
26.5 The Lifting Operation 413
26.6 Facets by Projection 416
27. Triangle Inequalities 421
27.1 Triangle Inequalities for the Correlation Polytope 423
27.2 Rooted Triangle Inequalities 424
27.2.1 An Integer Programming Formulation for Max Cut . . . 425
27.2.2 Volume of the Rooted Semimetric Polytope 426
27.2.3 Additional Notes 428
27.3 Projecting the Triangle Inequalities 430
27.3.1 The Semimetric Polytope of a Graph 431
27.3.2 The Cut Polytope for Graphs with no K5 Minor 434
27.4 An Excursion to Cycle Polytopes of Binary Matroids 435
27A.I Preliminaries on Binary Matroids 436
27.4.2 The Cycle Cone and the Cycle Polytope 438
27.4.3 More about Cycle Spaces 441
28. Hypermetric Inequalities 445
28.1 Hypermetric Inequalities: Validity 445
28.2 Hypermetric Facets 447
28.3 Separation of Hypermetric Inequalities 453
28.4 Gap Inequalities 457
28.4.1 A Positive Semidefinite Relaxation for Max Cut 459
28.5 Additional Notes 463
29. Clique Web Inequalities 467
29.1 Pure Clique Web Inequalities 467
29.2 General Clique Web Inequalities 470
29.3 Clique Web Inequalities: Validity and Roots 472
29.4 Clique Web Facets 477
xii Contents
29.5 Separation of Clique Web Inequalities 481
29.6 An Example of Proof for Clique Web Facets 483
30. Other Valid Inequalities and Facets 487
30.1 Suspended Tree Inequalities 487
30.2 Path Block Cycle Inequalities 492
30.3 Circulant Inequalities 496
30.4 The Parachute Inequality 497
30.4.1 Roots and Fibonacci Numbers 498
30.4.2 Generalizing the Parachute Inequality 500
30.5 Some Sporadic Examples 502
30.6 Complete Description of CUTn and CUT° for n 7 503
30.7 Additional Notes 506
31. Geometric Properties 511
31.1 Disproval of a Conjecture of Borsuk Using Cuts 512
31.2 Inequalities for Angles of Vectors 514
31.3 The Positive Semidefinite Completion Problem 515
31.3.1 Results 516
31.3.2 Characterizing Graphs with Excluded Induced Wheels . 522
31.4 The Euclidean Distance Matrix Completion Problem 527
31.4.1 Results 529
31.4.2 Links Between the Two Completion Problems 531
31.5 Geometry of the Elliptope 534
31.6 Adjacency Properties 539
31.6.1 Low Dimension Faces 539
31.6.2 Small Polytopes 542
31.7 Distance of Facets to the Barycentrum 546
31.8 Simplex Facets 549
Bibliography 551
Notation Index 575
Subject Index 579
|
| any_adam_object | 1 |
| author | Deza, Michel 1939- Laurent, Monique |
| author_GND | (DE-588)118064495 |
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| callnumber-first | Q - Science |
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| callnumber-raw | QA166 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 170 SK 890 |
| classification_tum | MAT 515f |
| ctrlnum | (OCoLC)246637297 (DE-599)BVBBV011341744 |
| dewey-full | 511.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.6 |
| dewey-search | 511.6 |
| dewey-sort | 3511.6 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV011341744 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:08:07Z |
| institution | BVB |
| isbn | 354061611X |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007620797 |
| oclc_num | 246637297 |
| open_access_boolean | |
| owner | DE-20 DE-91G DE-BY-TUM DE-384 DE-824 DE-703 DE-739 DE-355 DE-BY-UBR DE-634 DE-83 DE-11 DE-29T DE-188 |
| owner_facet | DE-20 DE-91G DE-BY-TUM DE-384 DE-824 DE-703 DE-739 DE-355 DE-BY-UBR DE-634 DE-83 DE-11 DE-29T DE-188 |
| physical | XII, 587 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Springer |
| record_format | marc |
| series | Algorithms and combinatorics |
| series2 | Algorithms and combinatorics |
| spelling | Deza, Michel 1939- Verfasser (DE-588)118064495 aut Geometry of cuts and metrics Michel Marie Deza ; Monique Laurent Berlin [u.a.] Springer 1997 XII, 587 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algorithms and combinatorics 15 Embeddings (Mathematics) Graph theory Metric spaces Diskrete Geometrie (DE-588)4130271-0 gnd rswk-swf Kombinatorische Optimierung (DE-588)4031826-6 gnd rswk-swf Graphentheorie (DE-588)4113782-6 gnd rswk-swf Metrik Mathematik (DE-588)4193458-1 gnd rswk-swf Geometrische Kombinatorik (DE-588)4156713-4 gnd rswk-swf Geometrie der Zahlen (DE-588)4227477-1 gnd rswk-swf Schnitt Mathematik (DE-588)4458889-6 gnd rswk-swf Graphentheorie (DE-588)4113782-6 s DE-604 Schnitt Mathematik (DE-588)4458889-6 s Metrik Mathematik (DE-588)4193458-1 s Diskrete Geometrie (DE-588)4130271-0 s Geometrie der Zahlen (DE-588)4227477-1 s Geometrische Kombinatorik (DE-588)4156713-4 s Kombinatorische Optimierung (DE-588)4031826-6 s Laurent, Monique Verfasser aut Algorithms and combinatorics 15 (DE-604)BV000617357 15 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007620797&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Deza, Michel 1939- Laurent, Monique Geometry of cuts and metrics Algorithms and combinatorics Embeddings (Mathematics) Graph theory Metric spaces Diskrete Geometrie (DE-588)4130271-0 gnd Kombinatorische Optimierung (DE-588)4031826-6 gnd Graphentheorie (DE-588)4113782-6 gnd Metrik Mathematik (DE-588)4193458-1 gnd Geometrische Kombinatorik (DE-588)4156713-4 gnd Geometrie der Zahlen (DE-588)4227477-1 gnd Schnitt Mathematik (DE-588)4458889-6 gnd |
| subject_GND | (DE-588)4130271-0 (DE-588)4031826-6 (DE-588)4113782-6 (DE-588)4193458-1 (DE-588)4156713-4 (DE-588)4227477-1 (DE-588)4458889-6 |
| title | Geometry of cuts and metrics |
| title_auth | Geometry of cuts and metrics |
| title_exact_search | Geometry of cuts and metrics |
| title_full | Geometry of cuts and metrics Michel Marie Deza ; Monique Laurent |
| title_fullStr | Geometry of cuts and metrics Michel Marie Deza ; Monique Laurent |
| title_full_unstemmed | Geometry of cuts and metrics Michel Marie Deza ; Monique Laurent |
| title_short | Geometry of cuts and metrics |
| title_sort | geometry of cuts and metrics |
| topic | Embeddings (Mathematics) Graph theory Metric spaces Diskrete Geometrie (DE-588)4130271-0 gnd Kombinatorische Optimierung (DE-588)4031826-6 gnd Graphentheorie (DE-588)4113782-6 gnd Metrik Mathematik (DE-588)4193458-1 gnd Geometrische Kombinatorik (DE-588)4156713-4 gnd Geometrie der Zahlen (DE-588)4227477-1 gnd Schnitt Mathematik (DE-588)4458889-6 gnd |
| topic_facet | Embeddings (Mathematics) Graph theory Metric spaces Diskrete Geometrie Kombinatorische Optimierung Graphentheorie Metrik Mathematik Geometrische Kombinatorik Geometrie der Zahlen Schnitt Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007620797&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000617357 |
| work_keys_str_mv | AT dezamichel geometryofcutsandmetrics AT laurentmonique geometryofcutsandmetrics |