Submodular functions and optimization:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier
2005
|
| Edition: | Second edition |
| Series: | Annals of Discrete Mathematics
58 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 395 Seiten Diagramme |
| ISBN: | 0444520864 |
Staff View
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| 245 | 1 | 0 | |a Submodular functions and optimization |c Satoru Fujishige, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, japan |
| 250 | |a Second edition | ||
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| 650 | 4 | |a Optimisation combinatoire | |
| 650 | 4 | |a Combinatorial optimization | |
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Record in the Search Index
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| adam_text | ix Contents Preface ............................................................................................................ v Preface to the Second Edition...................................................................... vii PART I .......................................................................................................... 1 Chapter I. Introduction ......................................................................... 3 1. Introduction.............................................................................................3 1.1. Introduction .................................................................................. 3 1.2. Mathematical Preliminaries ........................................................ 4 (a) Sets.............................................................................................. 4 (b) Algebraic structures ............................................................... 5 (c) Graphs......................................................................................... 9 (d) Network flows ....................................................................... 13 (e) Elements of convex analysis and linear inequalities ......... 15 Chapter II. Submodular Systems and Base Polyhedra............ 21 2. From Matroids to Submodular Systems............................................ 21 2.1. Matroids .................................................................................... 21 2.2. Polymatroids ............................................................................. 25 2.3. Submodular Systems
............................................................... 33 3. Submodular Systems ......................................................................... 45 3.1. Fundamental Operations on Submodular Systems .............. 45 (a) Reductions and contractions by sets................................... 45 (b) Reductions and contractions by vectors ........................... 46 (c) Translations and sums ......................................................... 51 (d) Other operations .................................................................. 53
x 3.2. Greedy Algorithm ...................................................................... 55 (a) Distributive lattices and posets .......................................... 55 (b) Greedy algorithm .................................................................. 58 3.3. Structures of Base Polyhedra ................................................... 66 (a) Extreme points and rays ................... 66 (b) Elementary transformations of bases ................................. 70 (c) Tangent cones ........................................................................ 72 (d) Faces, dimensions and connected components .................. 75 3.4. Intersecting- and Crossing-Submodular Functions ................ 86 (a) Tree representations of cross-free families............................ 87 (b) Crossing-submodular functions.............................................. 91 (c) Intersecting-submodular functions..................................... 101 3.5. Related Polyhedra.................................................................... 102 (a) Generalized polymatroids ................................................... 102 (b) Polypseudomatroids ........................................................... 106 (c) Ternary semimodular polyhedra........................................ 112 3.6. Submodular Systems of Network Type................................. 122 Chapter III. Neoflows........................................................................... 127 4. The Intersection Problem ............................................................... 127 4.1. The
Intersection Theorem ...................................................... 127 (a) Preliminaries ........................................................................ 128 (b) An algorithm and the intersection theorem..................... 131 (c) A refinement of the algorithm............................................ 136 4.2. The Discrete Separation Theorem ........................................ 140 4.3. The Common Base Problem ................................................... 142
5. Neoflows ............................................................................................ 145 5.1. Neoflows .................................................................................... 145 (a) Submodular flows................................................................. 145 (b) Independent flows ............................................................... 146 (c) Polymatroidal flows ............................................................. 147 5.2. The Equivalence of the Neoflow Problems............................ 148 (a) From submodular flows to independent flows ................. 148 (b) Rom independent flows to polymatroidal flows ............ 149 (c) From polymatroidal flows to submodular flows .............. 150 5.3. Feasibility for Submodular Flows .......................................... 153 5.4. Optimality for Submodular Flows............................................ 155 5.5. Algorithms for Neoflows ......................................................... 167 (a) Maximum independent flows ............................................. 167 (b) Maximum submodular flows ............................................. 172 (c) Minimum-cost submodular flows ...................................... 175 5.6. Matroid Optimization ............................................................. 188 (a) Maximum independent matchings..................................... 188 (b) Optimal independent assignments ..................................... 194 Chapter IV. Submodular Analysis ................................................. 199 6.
Submodular Functions and Convexity .......................................... 199 6.1. Conjugate Functions and a Fenchel-Type Min-Max Theorem for Submodular and Supermodular Functions............ 199 (a) Conjugate functions ........................................................... 199 (b) A Fenchel-type min-max theorem ..................................... 201 6.2. Subgradients of Submodular Functions................................... 203 (a) Subgradients and subdifferentials ..................................... 203 (b) Structures of subdifferentials............................................... 209
xii 6.3. The Lovász Extensions of Submodular Functions .............. 211 7. Submodular Programs .................................................................... 216 7.1. Submodular Programs — Unconstrained Optimization .... 216 (a) Minimizing submodular functions ..................................... 217 (b) Minimizing modular functions .......... 223 7.2. Submodular Programs — Constrained Optimization ......... 228 (a) Lagrangian functions and optimality conditions ............. 229 (b) Related problems ................................................................. 234 (b.l) The principal partition ............................................. 234 (b.2) The principal structures of submodular systems .. 245 (b.3) The minimum-ratio problem ................................... 248 Chapter V. Nonlinear Optimization with Submodular Constraints............................... 253 8. Separable Convex Optimization ..................................................... 253 8.1. Optimality Conditions ............................................................. 253 8.2. A Decomposition Algorithm ................................................... 257 8.3. Discrete Optimization ............................................................. 260 9. The Lexicographically Optimal Base Problem ............................ 261 9.1. Nonlinear Weight Functions ................................................... 262 9.2. Linear Weight Functions.......................................................... 264 10. The Weighted Max-Min and Min-Max Problems ....................... 269 10.1.
Continuous Variables ............................................................. 269 10.2. Discrete Variables .................................................................. 272 11. The Fair Resource Allocation Problem ........................................ 273 11.1. Continuous Variables ............................................................. 273 11.2. Discrete Variables .................................................................. 274
xiii 12. The Neoflow Problem with a Separable Convex Cost Function ...................................................................................... 280 PART II ................................................................................................... 285 Chapter VI. Submodular Function Minimization ..................... 287 13. Symmetric Submodular Function Minimization: Queyranne’s Algorithm...................................................................................... 287 14. Submodular Function Minimization ............................................. 290 14.1. The Iwata-Fleischer-Fujishige Algorithm ............................ 293 (a) A weakly polynomial algorithm ........................................ 293 (b) A strongly polynomial algorithm........................................ 300 (c) Modification with multiple exchanges .............................. 303 (d) Submodular functions on distributive lattices ................ 305 14.2. Schrijver’s Algorithm ............................................................. 308 14.3. Further Progress in Submodular Function Minimization ................................................................... 313 Chapter VII. Discrete Convex Analysis ...................................... 315 15. Locally Polyhedral Convex Functions and Conjugacy ................ 315 16. L- and L’-convex Functions ........................................................... 319 16.1. L- and L^-convex Sets ........................................................... 319 16.2. L- and L^-convex
Functions.....................................................322 16.3. Domain-integral L- and L^-convex Functions ..................... 326 17. Μ- and M^-convex Functions .......................................................... 331 18. Conjugacy between L /L^-convex Functions and М-ДР-сопѵех Functions .................................................................................... 338 19. The Discrete Fenchel-Duality Theorem ........................................ 341
xiv 20. Algorithmic and Structural Properties of Discrete Convex Function ............................................................................................ 344 20.1. L- and LAconvex Functions..................................................... 344 20.2. Μ- and M^-convex Functions ............................................... 345 20.3. Proximity Theorems................................................................. 351 21. Other Related Topics ...................................................................... 356 21.1. The M-convex Submodular Flow Problem ......................... 356 21.2. A Two-sided Discrete-Concave Market Model ................... 357 22. Historical Notes ................................................................................ 360 References ................................................................................................... 365 Index............................................. 389
|
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| edition | Second edition |
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| id | DE-604.BV020825741 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T20:20:07Z |
| institution | BVB |
| isbn | 0444520864 |
| language | English |
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| physical | XIV, 395 Seiten Diagramme |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Elsevier |
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| series | Annals of Discrete Mathematics |
| series2 | Annals of Discrete Mathematics |
| spelling | Fujishige, Satoru 1947- Verfasser (DE-588)171825837 aut Submodular functions and optimization Satoru Fujishige, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, japan Second edition Amsterdam [u.a.] Elsevier 2005 XIV, 395 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Annals of Discrete Mathematics 58 Fonctions sous modulaires Optimisation combinatoire Combinatorial optimization Submodular functions Kombinatorische Optimierung (DE-588)4031826-6 gnd rswk-swf Submodulare Funktion (DE-588)4323153-6 gnd rswk-swf Submodulare Funktion (DE-588)4323153-6 s Kombinatorische Optimierung (DE-588)4031826-6 s DE-604 Annals of Discrete Mathematics 58 (DE-604)BV004511910 58 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013831028&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fujishige, Satoru 1947- Submodular functions and optimization Annals of Discrete Mathematics Fonctions sous modulaires Optimisation combinatoire Combinatorial optimization Submodular functions Kombinatorische Optimierung (DE-588)4031826-6 gnd Submodulare Funktion (DE-588)4323153-6 gnd |
| subject_GND | (DE-588)4031826-6 (DE-588)4323153-6 |
| title | Submodular functions and optimization |
| title_auth | Submodular functions and optimization |
| title_exact_search | Submodular functions and optimization |
| title_full | Submodular functions and optimization Satoru Fujishige, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, japan |
| title_fullStr | Submodular functions and optimization Satoru Fujishige, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, japan |
| title_full_unstemmed | Submodular functions and optimization Satoru Fujishige, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, japan |
| title_short | Submodular functions and optimization |
| title_sort | submodular functions and optimization |
| topic | Fonctions sous modulaires Optimisation combinatoire Combinatorial optimization Submodular functions Kombinatorische Optimierung (DE-588)4031826-6 gnd Submodulare Funktion (DE-588)4323153-6 gnd |
| topic_facet | Fonctions sous modulaires Optimisation combinatoire Combinatorial optimization Submodular functions Kombinatorische Optimierung Submodulare Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013831028&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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| work_keys_str_mv | AT fujishigesatoru submodularfunctionsandoptimization |