Global Methods for Combinatorial Isoperimetric Problems
Certain constrained combinatorial optimization problems have a natural analogue in the continuous setting of the classical isoperimetric problem. The study of so called combinatorial isoperimetric problems exploits similarities between these two, seemingly disparate, settings. This text focuses on g...
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| Hlavní autor: | |
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
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Cambridge
Cambridge University Press
09.02.2004
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| Vydání: | 1 |
| Edice: | Cambridge Studies in Advanced Mathematics |
| Témata: | |
| ISBN: | 0521183839, 9780521832687, 9780521183833, 0521832683 |
| On-line přístup: | Získat plný text |
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- 7 Isoperimetric problems on infinite graphs -- 7.1 Euclidean tessellations -- 7.1.1 Cubical -- 7.1.1.1 d = 1 -- 7.1.1.2 d = 2 -- 7.1.1.3 d > -- 2 -- 7.1.2 Triangular -- 7.1.2.1 What are the solutions? -- 7.1.2.2 The stability order of V -- 7.1.2.3 Solutions for all k -- 7.1.3 The hexagonal tessellation -- 7.2 Comments -- 7.2.1 The last Euclidean tessellations -- 7.2.2 Powers of T and H -- 7.2.3 Tessellations of hyperbolic space -- 7.2.4 The VIP on Z -- 8 Isoperimetric problems on complexes -- 8.1 Minimum shadow problems -- 8.1.1 Combinatorial complexes -- 8.1.1.1 The face lattice of the d-cube -- 8.1.1.2 The face lattice of the d-simplex -- 8.1.2 Shadows -- 8.1.3 Duality -- 8.1.4 The Kruskal-Katona theorem -- 8.1.5 Macaulay posets -- 8.2 Steiner operations for MSP -- 8.2.1 Stabilization -- 8.2.2 Compression -- 8.2.3 Lindström's theorem -- 8.2.4 Extensions of Lindström's theorem -- 8.2.4.1 The theorems of Leeb and Bezrukov-Elsässer -- 8.2.4.2 The theorems of Vasta and Leck -- 8.3 Scheduling problems -- 8.3.1 Profile scheduling -- 8.3.2 Bandwidth scheduling and wirelength scheduling -- 8.4 Comments -- 8.4.1 More MSPs -- 8.4.2 Other complexes -- 8.4.3 Bezrukov's equivalence principle -- 8.4.4 Combinatorics since the 1960s -- 9 Morphisms for MWI problems -- 9.1 MWI-morphisms -- 9.1.1 Quotients -- 9.1.2 The main definitions -- 9.1.2.1 Strong MWI-morphisms -- 9.1.2.2 Weak MWI-morphisms -- 9.2 Examples -- 9.2.1 EIP on the dodecahedron -- 9.2.2 EIP on BS -- 9.2.3 EIP on the 24-cell -- 9.2.4 Z × Z -- 9.3 Application I: The pairwise product of Petersen graphs -- 9.3.1 The product of Petersen graphs -- 9.4 Application II: The EIP on the 600-vertex -- 9.4.1 How to repair broken inequalities -- 9.5 The calculation for V -- 9.5.1 Reducing the EIP -- 9.5.1.1 Representation -- 9.5.1.2 Fricke-Klein point -- 9.5.1.3 Reflective symmetries -- 9.5.1.4 Basic reflections
- Cover -- Half-title -- Series-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- 1 The edge-isoperimetric problem -- 1.1 Basic definitions -- 1.2 Examples -- 1.2.1 K, the complete graph on n vertices -- 1.2.2 Z, the n-cycle -- 1.2.3 The d-cube, Q -- 1.3 Application to layout problems -- 1.3.1 The wirelength problem -- 1.3.1.1 Example -- 1.3.1.2 Another representation of wl -- 1.3.2 The deBruijn graph of order 4 -- 1.3.3 Partitioning problems -- 1.3.3.1 Examples -- 1.4 Comments -- 2 The minimum path problem -- 2.1 Introduction -- 2.1.1 Basic definitions -- 2.1.2 Example -- 2.2 Algorithms -- 2.2.1 The acyclic case -- 2.2.2 Positive weights: Dijkstra's algorithm -- 2.2.3 No negative cycles: the Bellman-Ford algorithm -- 2.2.4 The general case -- 2.2.5 An observation by Klee -- 2.3 Reduction of wirelength to minpath -- 2.4 Pathmorphisms -- 2.4.1 Definitions -- 2.4.2 Examples -- 2.4.3 Steiner operations -- 2.5 Comments -- 3 Stabilization and compression -- 3.1 Introduction -- 3.2 Stabilization -- 3.2.1 Diagrams -- 3.2.1.1 The n-gon, Z -- 3.2.1.2 The d-cube, Q -- 3.2.1.3 The d-crosspolytope, square -- 3.2.2 Symmetries -- 3.2.3 Examples -- 3.2.3.1 The dihedral group, D -- 3.2.3.2 The cuboctahedral group -- 3.2.4 Definition -- 3.2.5 Basic properties of stabilization -- 3.2.6 Multiple stabilizations -- 3.2.7 Stability order -- 3.2.7.1 Definition -- 3.2.7.2 Examples -- 3.2.8 Ideals -- 3.2.9 The derived network -- 3.2.9.1 Definition -- 3.2.9.2 Examples -- 3.2.10 Summary -- 3.3 Compression -- 3.3.1 Introduction -- 3.3.2 Definition -- 3.3.3 Basic properties of compression -- 3.3.4 The compressibility order -- 3.3.4.1 Definition -- 3.3.4.2 Example -- 3.3.5 Another solution of the wirelength problem for Q -- 3.4 Comments -- 4 The vertex-isoperimetric problem -- 4.1 Definitions and examples -- 4.1.1 The VIP -- 4.1.1.1 Exercise
- 4.2 Stabilization and VIP -- 4.3 Compression for VIP -- 4.3.1 Compressibility order -- 4.4 Optimality of Hales numbering -- 4.5 Applications to layout problems -- 4.5.1 The bandwidth problem -- 4.5.1.1 Example -- 4.5.2 Reducing bandwidth to minimum path -- 4.5.3 Partitioning to minimize pins -- 4.6 Comments -- 5 Stronger stabilization -- 5.1 Graphs of regular solids -- 5.1.1 The dodecahedron and icosahedron -- 5.2 A summary of Coxeter theory -- 5.2.1 Basic facts -- 5.2.1.1 Chambers -- 5.2.1.2 Generators and relations -- 5.2.1.3 Classification -- 5.2.1.4 Length -- 5.2.1.5 Bruhat order -- 5.2.1.6 Example -- 5.2.1.7 Parabolic subgroups -- 5.3 The structure of stability orders -- 5.3.1 The components of stability orders -- 5.3.1.1 Definition -- 5.3.2 Duality -- 5.3.2.1 Definition -- 5.4 Calculating stability orders -- 5.4.1 The algorithm -- 5.4.2 The deBruijn graph revisited -- 5.5 Into the fourth dimension -- 5.5.1 The 24-vertex (24-cell) -- 5.5.1.1 Generating the stability order -- 5.5.1.2 Generating the derived network -- 5.5.2 The 120-vertex -- 5.5.3 Cayley graphs of Coxeter groups -- 5.6 Extended stabilization -- 5.6.1 The k-pather of G -- 5.6.1.1 Example -- 5.6.2 Applications -- 5.6.2.1 Z the Harary graph -- 5.6.2.2 Q, the k-pather of the d-cube -- 5.7 Comments -- 5.7.1 Stabilization compared to isomorph rejection -- 5.7.2 More on deBruijn graphs -- 6 Higher compression -- 6.1 Additivity -- 6.2 The MWI problem -- 6.2.1 Definitions -- 6.3 The Ahlswede-Cai theorem -- 6.3.1 Applications -- 6.3.1.1 Products of complete graphs -- 6.3.1.2 Products of complete bipartite graphs -- 6.3.1.3 Products of crosspolytopes -- 6.4 The Bezrukov-Das-Elsässer theorem -- 6.4.1 The Petersen graph and its products -- 6.4.2 The solution for d = 1 -- 6.4.3 The solution for d = 2 -- 6.4.4 Solution for all d > -- 2 -- 6.5 Comments
- 9.5.1.5 The stability order -- 9.5.1.6 The weight, Delta -- 9.5.2 Calculating the quotient, Q -- 9.5.2.1 Finding phi -- 9.5.2.2 The MinShadow function -- 9.5.2.3 The local MWI-function -- 9.5.2.4 Checking inequalities -- 9.5.2.5 Repairing broken inequalities -- 9.5.3 Solving the MWI problem on Q -- 9.5.3.1 Generating ideals -- 9.5.3.2 Generating ideals of Q -- 9.6 Comments -- 10 Passage to the limit -- 10.1 The Bollobas-Leader theorem -- 10.2 The Kleitman-West problem -- 10.2.1 Simplifying Kleitman-West by stabilization -- 10.3 VIP on the Hamming graph -- 10.4 Sapozhenko's problem -- 10.4.1 The crooked neighborhood of v ∈ L (n, m) -- 10.4.2 Dido's principle -- 10.4.2.1 The Didonean embedding of L (n) into [0, 1] -- 10.5 Comments -- 10.5.1 Applications -- 10.5.1.1 Modeling the brain -- 10.5.1.2 Multicasting to minimize noise -- 10.5.1.3 Dedekind's problem again -- 10.5.1.4 The doctor's waiting room problem -- 10.5.2 The Hwang-Lagarias theorem -- 10.5.3 Discontinuous Steiner operations -- 10.5.4 Kleitman-West again -- Afterword -- Appendix The classical isoperimetric problem -- A.5 Steiner symmetrization -- A.6 The legend of the founding of Carthage -- A.6.1 Dido's principle -- References -- Index