ISBN: 3-540-66752-0
TITLE: Completeness and Reduction in Algebraic Complexity Theory
AUTHOR: Brgisser, Peter
TOC:

1 Introduction 1
1.1 Classical Model 1
1.2 Blum-Shub-Smale Model 3
1.3 Valiant's Model 4
1.4 Overview of Main Results 6
2 Valiant's Algebraic Model of NP-Completeness 11
2.1 The Complexity Classes VP and VNP 11
2.2 Completeness of the Permanent Family 16
2.2.1 p-Definability and Formula Size 16
2.2.2 Universality of the Permanent 19
2.2.3 Completeness of the Permanent 21
2.3 Closure Properties 25
2.4 Parallel Complexity 30
2.5 Completeness of the Determinant Family 34
3 Some Complete Families of Polynomials 37
3.1 Generating Functions of Graph Properties 37
3.2 p-Computable Families 39
3.3 VNP-Complete Families 41
3.3.1 Matchings 42
3.3.2 Cliques 43
3.3.3 Cycle Format Polynomials 44
3.3.4 Graph Factors 48
3.3.5 Hamilton Cycles of Planar Graphs 50
3.3.6 Self Avoiding Walks 53
3.3.7 Connectivity 55
4 Cook's versus Valiant's Hypothesis 61
4.1 Dependence on the Field 61
4.2 Statement of Main Results 64
4.3 Review of Discrete Complexity Classes 66
4.4 Relating NP to Counting Classes 68
4.5 A Bound on the Heights 70
4.6 Roots of Univariate Polynomials Modulo a Prime 75
4.7 Proof of Thm. 4.5 76
5 The Structure of Valiant's Complexity Classes 81
5.1 Outline and Comparison with Previous Work 81
5.2 An Abstract Diagonalization Theorem 82
5.3 An Abstract Embedding Theorem 85
5.4 Structure of Valiant's Complexity Classes 89
5.5 A Specific Family Neither Complete Nor p-Computable 92
5.6 Relativized Complexity Classes 96
6 Fast Evaluation of Representations of General Linear Groups 105
6.1 Description of the Problem 105
6.2 Preliminaries on Representations of GLm 107
6.3 Auxiliary Fast Linear Algebra Algorithms 109
6.4 An Algorithm for Evaluating Representations 112
6.5 A Lower Bound 114
6.6 Fast Evaluation of Legendre Functions 115
7 The Complexity of Immanants 117
7.1 Motivation and Outline of Chapter 117
7.2 Fast Evaluation of Immanants 118
7.3 Completeness Results for Immanants 120
7.4 Character Formulas for the Symmetric Group 121
7.5 p-Definability of Immanants 126
7.6 Completeness Proofs 128
8 Separation Results and Future Directions 135
8.1 Specific Families Which Are Not p-Definable 135
8.2 Separations for the Complexity Class VQP 139
8.3 Possible Connections to Univariate Polynomials 141
8.4 Connections to the BSS-Model 143
References 149
List of Notation 159
Index 163
END
