The Complexity of Partition Functions DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Informatik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakultät II Humboldt-Universität zu Berlin

von Herr Dipl.-Inf. Marc Thurley geboren am 28.03.1981 in Staaken Präsident der Humboldt-Universität zu Berlin: Prof. Dr. Dr. h.c. Christoph Markschies Dekan der Mathematisch-Naturwissenschaftlichen Fakultät II: Prof. Dr. Peter Frensch Gutachter: 1. Prof. Dr. Martin Grohe 2. Prof. Dr. Heribert Vollmer 3. Prof. Andrei Bulatov eingereicht am: Tag der mündlichen Prüfung:

07. Juli 2009 18. September 2009

Contents 1 Introduction 1.1 Contributions of this Thesis 1.2 Basic Notation

9 14 16

2 Origins of Partition Functions 2.1 Statistical Physics 2.1.1 Statistical Mechanics Models on Graphs 2.1.2 Solving Models and Computing Partition Functions 2.2 Combinatorics 2.2.1 The Tutte Polynomial 2.2.2 Partition Functions

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3 A Combinatorial Property of Partition Functions 3.1 Determining Matrices by Partition Functions 3.1.1 The Twin Reduction Lemma 3.1.2 The Reconstruction Lemma 3.1.3 The Proof of Theorem 3.1.1

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4 Partition Functions and Their Complexity 4.1 Counting Complexity 4.2 Counting Complexity and Partition Functions 4.3 The Arithmetical Structure of Partition Functions 4.3.1 Fundamental Technical Tools 4.3.2 The Equivalence of EVALpin(A,£>) and COUNT pin (A, £>) 4.3.3 Looking at the Structure: The Proof of Lemma 4.3.1 4.4 Further Preparatory Considerations 4.4.1 General Principles 4.4.2 Edge Products 4.4.3 Rank 1 Matrices and Tractability 4.5 Pinning Vertices 4.6 Bounded Degrees and Eliminating Vertex Weights 4.6.1 Some Technical Tools 4.6.2 The proof of Lemma 4.6.2

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4.6.3 5

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The proof of Lemma 4.6.1

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Non-Negative Matrices 5.1 Preliminaries 5.2 An Itinerary 5.3 The General Conditioning Lemma 5.3.1 From General Matrices to Positive Matrices 5.3.2 From Positive Matrices to X-matrices 5.3.3 From X-matrices to the General Conditioning Lemma 5.4 The Two 1-Cell Lemma 5.5 The Single 1-Cell Lemma 5.5.1 Proof of the Single-1-Cell Technical Core Lemma 5.5.4 Partition Functions on Hermitian Matrices 6.1 Congruential Partition Functions 6.2 An Itinerary 6.2.1 The General Case 6.2.2 Hadamard Components 6.2.3 The Polynomial Time Case 6.2.4 The Proof of Theorem 6.1 6.3 Technical Preliminaries 6.3.1 General Principles for Congruential Partition Functions 6.3.2 Edge Products 6.3.3 Basic Complexity Results for Congruential Partition Functions . . . .

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79 80 81 82 82 82 84 85 89 94 99 100 100 101 102 107 107 109 109 110 110

7 Connected Hermitian Matrices 111 7.1 Some Technical Preliminaries 112 7.2 The Non-Bipartite Case . 114 7.2.1 Satisfying Shape Conditions ( C I ) and ( D l ) 118 7.2.2 The Remaining Conditions (C2), (D2) and (D3) 124 7.2.3 Finishing the Proof of Lemma 7.1 131 7.3 The Proof of the Bipartite Case . 132 7.3.1 Satisfying the Bipartite Shape Conditions ( B - C l ) and ( B - D l ) . . . 139 7.3.2 The Remaining Conditions ( B - C 2 ) , ( B - D 2 ) and ( B - D 3 ) 146 7.3.3 Finishing the Proof of Lemma 7.2 153 8 Hadamard Components 8.1 Bounding The Maximum Degree 8.2 Non-Bipartite Hadamard Components 8.2.1 The Group Condition ( G C ) 8.2.2 The Representation Conditions ( R l ) through (R5) 8.2.3 The Affinity Condition (AF) 8.2.4 The Proof of the Non-Bipartite Case Lemma 8.1 vi

155 156 158 159 161 164 173

8.3

Bipartite Hadamard Components 8.3.1 The Group Condition (GC) 8.3.2 The Representation According to ( B - R l ) through ( B - R 5 ) 8.3.3 The Affinity Condition ( B - A F ) 8.3.4 The Proof of the Bipartite Case Lemma 8.2

173 173 176 180 189

9 Polynomial Time Computable Partition Functions 9.1 A Polynomial Time Computable Problem 9.1.1 Solving EVAL(g) for q a Power of an Odd Prime 9.1.2 Solving EVAL(q) for q a Power of 2 9.2 Computing Partition Functions — The Non-Bipartite Case 9.2.1 The Structure of the Mappings pc 9.2.2 The Final Reduction 9.3 Computing Partition Functions — The Bipartite Case 9.3.1 The Structure of the Mappings pCjrow and pCiCOi 9.3.2 The Proof of Lemma 9.2

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10 Epilogue 10.1 Implications on Polynomial Time Computability 10.2 Open Questions

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A Some Mathematical Background A.l Vandermonde Determinants A.2 Important Facts from Group Theory A.2.1 Fourier Analysis of Abelian Groups

225 225 225 226

Bibliography

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