Complexity Dichotomies for Counting Problems: Volume 1, Boolean Domain

Complexity theory aims to understand and classify computational problems, especially decision problems, according to their inherent complexity. This book uses new techniques to expand the theory for use with counting problems. The authors present dichotomy classifications for broad classes of counting problems in the realm of P and NP. Classifications are proved for partition functions of spin systems, graph homomorphisms, constraint satisfaction problems, and Holant problems. The book assumes minimal prior knowledge of computational complexity theory, developing proof techniques as needed and gradually increasing the generality and abstraction of the theory. This volume presents the theory on the Boolean domain, and includes a thorough presentation of holographic algorithms, culminating in classifications of computational problems studied in exactly solvable models from statistical mechanics.

Table of Contents:
1. Counting problems; 2. Fibonacci gates and Holant problems; 3. Boolean #CSP; 4. Matchgates and holographic algorithms; 5. 2-spin systems on regular graphs; 6. Holant problems and #CSP; 7. Holant dichotomy for symmetric constraints; 8. Planar #CSP for symmetric constraints; 9. Planar Holant for symmetric constraints; 10. Dichotomies for asymmetric constraints.

About the Author :
Jin-Yi Cai is Professor of Computer Science and the Steenbock Professor of Mathematical Sciences at the University of Wisconsin, Madison. He studied at Fudan University, Shanghai (class of 77) and at Cornell University, New York, receiving his Ph.D. in 1986. He held faculty positions at Yale University, Connecticut (1986–1989), Princeton University, New Jersey (1989–1993), and State University of New York, Buffalo (1993–2000), where he rose from Assistant Professor to Full Professor in 1996. He received a Presidential Young Investigator Award (1990), an Alfred P. Sloan Fellowship (1994), and a John Simon Guggenheim Fellowship (1998). He is a Fellow of the Association for Computing Machinery (ACM) and the American Association for the Advancement of Science (AAAS). Xi Chen is Associate Professor of Computer Science at Columbia University, New York. He studied at Tsinghua University and received his Ph.D. in 2007. His research focuses on complexity theory and algorithmic game theory. He is the recipient of a NSF CAREER Award, an Alfred P. Sloan Fellowship (2012), and an European Association for Theoretical Computer Science (EATCS) Presburger Award (2015).

Review :
'This remarkable volume presents persuasive evidence that computer applications obey beautiful unities: within the significant classes considered, the problems that are not known to be polynomial time computable are all reducible to each other by a small number of elegant techniques. The treatment is original, comprehensive and thought provoking.' Leslie Valiant, Harvard University, Massachusetts 'This book provides a thorough study of the complexity of counting. These basic problems arise in statistical physics, optimization, algebraic combinatorics and computational complexity. The past fifteen years of research have led to a (surprisingly clean) complete characterization of their complexity in the form of a 'dichotomy theorem', whose proof is the main goal of this volume. Along the way, the authors provide detailed explanations of basic methods for studying such problems, including holographic algorithms and reductions. The book is very well written and organized, and should be useful to researchers and graduate students in the fields above.' Avi Wigderson, Institute for Advanced Study, Princeton, New Jersey 'This book would make an excellent introduction to the field of counting complexity for a mathematically literate reader with little or no background in computational complexity … Many of the results included in the book are recent, and have not appeared in book form before. As such, this thorough, self-contained treatment of the subject makes a very valuable contribution to the literature.' Kitty Meeks, MathSciNet 'There's no doubt in my mind that anyone interested in computational complexity would benefit greatly by reading it. It is a remarkable and indispensable book about a deep and important topic.' Frederic Green, SIGACT News