Submodular Functions and Electrical Networks

Peter L. Hammer (Eds.)978-0-444-82523-0, 0444825231

There is a strong case for electrical network topologists and submodular function theorists being aware of each other’s fields.
Presenting a topological approach to electrical network theory, this book demonstrates the strong links that exist between submodular functions and electrical networks.
The book contains:
• a detailed discussion of graphs, matroids, vector spaces and the algebra of generalized minors, relevant to network analysis (particularly to the construction of efficient circuit simulators)
• a detailed discussion of submodular function theory in its own right; topics covered include, various operations, dualization, convolution and Dilworth truncation as well as the related notions of prinicpal partition and principal lattice of partitions.
In order to make the book useful to a wide audience, the material on electrical networks and that on submodular functions is presented independently of each other.
The hybrid rank problem, the bridge between (topological) electrical network theory and submodular functions, is covered in the final chapter.
The emphasis in the book is on low complexity algorithms, particularly based on bipartite graphs.
The book is intended for self-study and is recommended to designers of VLSI algorithms. More than 300 problems, almost all of them with solutions, are included at the end of each chapter.

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Table of contents :
Content:
Advisory Editors
Page ii

Edited by
Page iii

Copyright page
Page iv

Dedication
Page v

Preface
Pages vii-ix

Note to the Reader
Pages xi-xiii

List of Commonly Used Symbols
Pages xiv-xxi

Chaper 1 Introduction
Pages 1-13
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Chapter 2 Mathematical Preliminaries
Pages 15-30
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Chapter 3 Graphs
Pages 31-101
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Chapter 4 Matroids
Pages 103-129
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Chapter 5 Electrical Networks
Pages 131-171
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Chapter 6 Topological Hybrid Analysis
Pages 173-211
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Chapter 7 The Implicit Duality Theorem and Its Applications
Pages 213-268
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Chapter 8 Multiport Decomposition
Pages 269-323
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Chapter 9 Submodular Functions
Pages 325-378
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Chapter 10 Convolution of Submodular Functions
Pages 379-451
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Chapter 11 Matroid Union
Pages 453-480
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Chapter 12 Dilworth Truncation of Submodular Functions
Pages 481-532
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Chapter 13 Algorithms for the PLP of a Submodular Function
Pages 533-570
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Chapter 14 The Hybrid Rank Problem
Pages 571-627
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Bibliography
Pages 629-643

Index
Pages 644-650