Download or read book Complexity Classification of Counting Problems on Boolean Variables written by Shuai Shao and published by . This book was released on 2020 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation furthers a systematic study of the complexity classification of counting problems. A central goal of this study is to prove complexity classification theorems which state that every problem in some large class is either polynomial-time computable (tractable) or #P-hard. Such classification results are important as they tend to give a unified explanation for the tractability of certain counting problems and a reasonable basis for the conjecture that the remaining problems are inherently intractable. In this dissertation, we focus on the framework of Holant problems on Boolean variables, as well as other frameworks that are expressible as Holant problems, such as counting constraint satisfaction problems and counting Eulerian orientation problems. First, we prove a complexity dichotomy for Holant problems on the Boolean domain with arbitrary sets of real-valued constraint functions. It is proved that for every set F of real-valued constraint functions, Holant(F) is either tractable or #P-hard. The classification has an explicit criterion. This is a culmination of much research on this decade-long study, and it uses many previous results and techniques. On the other hand, to achieve the present result, many new tools were developed, and a novel connection with quantum information theory was built. In particular, two functions exhibiting intriguing and extraordinary closure properties are related to Bell states in quantum information theory. Dealing with these functions plays an important role in the proof. Then, we consider the complexity of Holant problems with respect to planar graphs, where physicists had discovered some remarkable algorithms, such as the FKT algorithm for counting planar perfecting matchings in polynomial time. For a basic case of Holant problems, called six-vertex models, we discover a new tractable class over planar graphs beyond the reach of the FKT algorithm. After carving out this new planar tractable class which had not been discovered for six-vertex models in the past six decades, we prove that everything else is #P-hard, even for the planar case. This leads to a complete complexity classification for planar six-vertex models. This result is the first substantive advance towards a planar Holant classification with asymmetric constraints. We hope this work can help us better understand a fundamental question in theoretical computer science: What does it mean for a computational counting problem to be easy or to be hard?