COMPLEXITY CLASSIFICATION of EXACT and APPROXIMATE COUNTING PROBLEMS by Heng Guo a Dissertation Submitted in Partial Fulfillment
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COMPLEXITY CLASSIFICATION OF EXACT AND APPROXIMATE COUNTING PROBLEMS by Heng Guo A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Computer Sciences) at the UNIVERSITY OF WISCONSIN–MADISON 2015 Date of final oral examination: 07/22/2015 The dissertation is approved by the following members of the Final Oral Committee: Jin-Yi Cai (Advisor), Professor Eric Bach, Professor Dieter van Melkebeek, Professor Shuchi Chawla, Associate Professor Sebastian Roch, Associate Professor © Copyright by Heng Guo 2015 All Rights Reserved i To my parents ii Acknowledgments First and foremost, I cannot express enough gratitude to Jin-Yi Cai, who has guided, advised, and tolerated me in the last five years. When I first came to Madison, I was an ambitious yet clueless young student. It is Jin-Yi who illustrated to me how to do research and solve problems. He also helped me to secure every possible opportunity: attending conferences, giving talks, visiting other institutions, and so on. His insights and knowledge helped me to develop my skill set, and his dedication and attitude inspired me to devote into academics. Truly my role model. There are also other people from whom I received wisdom and to whom I am in debt. Thanks to the hospitality of Leslie Ann Goldberg, I spent six amazing months in Oxford. It was a productive period full of wonderful events and stimulating discussions, and I enjoyed a lot the relaxed atmosphere there. Another big thanks goes to Pinyan Lu, whom I have been working with before and after starting my Ph. D. Pinyan’s cool tricks and observations are something I always enjoy. The area I am working in is largely due to the pioneer work of Leslie Valiant, and while working with Leslie his marvelous vision allows me to have a glimpse on how to be such an extraordinary scientist. I thank all my collaborators: Ivona Bezáková, Xi Chen, Andreas Galanis, Zhiguo Fu, Sangxia Huang, Mark Jerrum, Liang Li, Daniel Štefankoviˇc,Eric Vigoda, Mingji Xia, and Yitong Yin. It has been a great pleasure to be able to work and exchange ideas with you all. I thank the CS department of UW-Madison, which housed me in the past five years. In particular, I am lucky to be with the faculty and students of the theory group. I thank Eric Bach, Shuchi Chawla, Dieter van Melkebeek, and Sebastien Roch for the excellent classes they taught me, and for the burden they took to examine this dissertation. I would also like to thank Tyson Williams, a fellow student of Jin-Yi and who perhaps I have spent most time working with iii aside from Jin-Yi, for his help both in work and in life. In addition to brilliant ideas, Tyson’s tireless and meticulous work is essential to achieve what we have done. I am fortunate to have a group of friends who have helped me in all kinds of ways over the years. A surely incomplete list includes: Xixuan Feng, Andreas Goebel, Jiexing Li, Yinan Li, David Richerby, Wentao Wu, Xi Wu, Junming Xu, Chen Zeng, and Ce Zhang. Our “lunch group” meeting is usually the most relaxing time of the day, and it is always very enlightening to share thoughts with peers from outside of my area. In particular I would like to thank Ce Zhang, a brilliant guy who happened to be stuck under the same roof (a.k.a. roommate) with me for a few years. Without his careful explanation and patience for someone who knows nothing beyond theory, I would never be aware of the application side of some subjects I study. I could have achieved nothing without the selfless and unconditional support from my parents. Thank you, and your love is one of my main drives to keep me moving forward. Finally, I am very grateful to receive the Simons Awards in Theoretical Computer Science from the Simons Foundation. Their generous support opens a lot of opportunities for me and benefited my research in an unparalleled way. I also thank the financial support from NSF CCF-0914969 and NSF CCF-1217549. iv Contents Contents iv List of Tables vii List of Figures viii Abstract xi 1 Introduction 1 1.1 Counting Problems and Complexity Classification .................... 1 1.2 Approximate Counting .................................... 7 1.3 Problems and Definitions .................................. 11 1.4 Useful Reductions ....................................... 13 1.5 Tractable Signature Sets ................................... 17 1.6 Some Known Dichotomies .................................. 22 2 Parity Holant Problems 25 2.1 Tractable Z2-Functions .................................... 26 2.2 Hardness Results ....................................... 38 2.3 Parity Holantc Dichotomy .................................. 43 2.4 Parity Vanishing Signature Sets ............................... 46 2.5 Dichotomy for Parity Holant ................................ 56 3 Holant Problems on 4-regular graphs 62 3.1 Complex Vanishing Signatures ............................... 62 v 3.2 Redundant 4-by-4 Matrices ................................. 75 3.3 Counting Eulerian Orientations in 4-Regular Planar Graphs . 79 3.4 Redundant Signatures with Non-Singular Compressed Matrices . 84 3.5 A Unary Interpolation Lemma ............................... 94 3.6 Pl-Holant Dichotomy for a Symmetric Arity 4 Signature . 96 3.7 Vanishing Signatures Revisited ...............................103 4 Planar Counting CSP 120 4.1 Domain Pairing ........................................122 4.2 Mixing of Tractable Signatures ...............................127 4.3 Pinning for Planar #CSP ...................................132 4.4 Planar #CSP Dichotomy ...................................141 5 A Closer Look at Tractable Signatures 145 5.1 Characterization of A-transformable Signatures . 146 5.2 Characterization of P-transformable Signatures . 152 5.3 Characterization of M-transformable Signatures . 154 5.4 Related Hardness Results ..................................160 5.5 Hardness results for the Inductive Step . 171 5.6 Single Signature Dichotomy .................................179 6 Dichotomy for Holant Problems 183 6.1 Another Planar Tractable Case ...............................185 6.2 Complementing Hardness Results .............................196 6.3 Mixing M4 and P2 with Other Signatures . 204 6.4 #PM in Planar Hypergraphs .................................210 6.5 The Full Dichotomy ......................................211 7 Anti-Ferromagnetic 2-Spin Systems 217 7.1 Definitions and Backgrounds ................................217 7.2 The Self-Avoiding Walk Tree ................................222 7.3 The Potential Method .....................................227 vi 7.4 A Degree Dependent Potential Function . 237 7.5 An Improved Potential Function ..............................241 7.6 Monotonicity of the Uniqueness ...............................249 7.7 Uniqueness Thresholds ....................................256 7.8 Correlation Decay at the Threshold ............................266 7.9 Concluding and Bibliographic Remarks . 269 8 Phase Transitions and Computational Hardness 272 8.1 Phase Transitions in Anti-Ferromagnetic Systems . 272 8.2 Spin Systems in Bipartite Graphs ..............................276 8.3 Balancing Nearly-Independent Phase-Correlated Spins . 279 8.4 Symmetry Breaking ......................................283 8.5 The Reduction from #BIS ..................................285 9 Ferromagnetic 2-Spin Systems 293 9.1 The Uniqueness Condition on Regular Trees . 294 9.2 The Potential Method for Ferromagnetic Systems . 297 9.3 Bounded Degree Graphs ...................................300 9.4 General Graphs ........................................301 9.5 Correlation Decay Beyond λc ................................304 9.6 Complementary Hardness Results .............................307 10 Complex Weighted Ising Models 309 10.1 Approximating Complex Numbers .............................309 10.2 Problem Definitions and Results ..............................310 10.3 The Tutte polynomial .....................................317 10.4 Series and Parallel Compositions ..............................318 10.5 Hardness Results for the Ising Model ............................321 10.6 Quantum Circuits and Counting Complexity . 334 10.7 Complex Ising with External Fields .............................346 Bibliography 358 vii List of Tables 1.1 List of all non-degenerate affine signatures ......................... 19 viii List of Figures 1.1 An F-gate with 5 dangling edges ................................ 16 2.1 A triangular gadget ........................................ 37 2.2 The gadget for [0, 1, 0, 0, 0] and [1, 1, 0, 0, 0] .......................... 37 2.3 The gadget to construct [1, 1, 1, 0] ............................... 41 2.4 An asymmetric gadget for domain pairing .......................... 41 2.5 The gadget to construct [1, 0, 0, 0, 1] from [0, 0, 1, 0, 0] ................... 43 2.6 The gadget to construct [1, 0, 0, 0, 1] from [0, 0, 1, 0, 1] ................... 43 2.7 The signature inner product .................................. 50 2.8 Simulating [1, 0] using [1, 0, 0] or [1, 1, 0] ........................... 57 2.9 A tetrahedron gadget ...................................... 58 2.10 The gadget for [1, 0, 0, 0, 1, 0] .................................. 58 3.1 Degenerate vanishing gadget .................................. 69 3.2 Entry movement for rotations of arity 4 signatures ..................... 78 3.3 Medial graphs ........................................... 80 3.4 Planar tetrahedron gadget .................................... 82 3.5 Recursive construction ....................................