Computational Complexity Of Counting And Sampling

Computational Complexity of Counting and Sampling provides readers with comprehensive and detailed coverage of the subject of computational complexity. It is primarily geared toward researchers in enumerative combinatorics, discrete mathematics, and theoretical computer science. The book covers the following topics: Counting and sampling problems that are solvable in polynomial running time, including holographic algorithms; #P-complete counting problems; and approximation algorithms for counting and sampling. First, it opens with the basics, such as the theoretical computer science background and dynamic programming algorithms. Later, the book expands its scope to focus on advanced topics, like stochastic approximations of counting discrete mathematical objects and holographic algorithms. After finishing the book, readers will agree that the subject is well covered, as the book starts with the basics and gradually explores the more complex aspects of the topic. Features: Each chapter includes exercises and solutions Ideally written for researchers and scientists Covers all aspects of the topic, beginning with a solid introduction, before shifting to computational complexity’s more advanced features, with a focus on counting and sampling

The subject of these notes is counting and related topics, viewed from a computational perspective. A major theme of the book is the idea of accumulating information about a set of combinatorial structures by performing a random walk on those structures. These notes will be of value not only to teachers of postgraduate courses on these topics, but also to established researchers. For the first time this body of knowledge has been brought together in a single volume.

This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory. Requiring essentially no background apart from mathematical maturity, the book can be used as a reference for self-study for anyone interested in complexity, including physicists, mathematicians, and other scientists, as well as a textbook for a variety of courses and seminars. More than 300 exercises are included with a selected hint set. The book starts with a broad introduction to the field and progresses to advanced results. Contents include: definition of Turing machines and basic time and space complexity classes, probabilistic algorithms, interactive proofs, cryptography, quantum computation, lower bounds for concrete computational models (decision trees, communication complexity, constant depth, algebraic and monotone circuits, proof complexity), average-case complexity and hardness amplification, derandomization and pseudorandom constructions, and the PCP theorem.

New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.

This book constitutes the refereed proceedings of the 6th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2002, held in Cambridge, MA, USA in September 2002. The 21 revised full papers presented were carefully reviewed and selected from 48 submissions. Among the topics addressed are coding, geometric computations, graph colorings, random hypergraphs, graph computations, lattice computations, proof systems, probabilistic algorithms, derandomization, constraint satisfaction, and web graphs analysis.

This book constitutes the refereed proceedings of the 33rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2008, held in Torun, Poland, in August 2008. The 45 revised full papers presented together with 5 invited lectures were carefully reviewed and selected from 119 submissions. All current aspects in theoretical computer science and its mathematical foundations are addressed, ranging from algorithmic game theory, algorithms and data structures, artificial intelligence, automata and formal languages, bioinformatics, complexity, concurrency and petrinets, cryptography and security, logic and formal specifications, models of computations, parallel and distributed computing, semantics and verification.

The six- and eight-vertex models originate in statistical mechanics for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935 to account for the residual entropy of water ice. The family of models not only are among the most extensively studied topics in physics, but also have fascinated chemists, mathematicians, theoretical computer scientists, and others, with thousands of papers studying their properties and connections to other fields. In this dissertation, we study the computational complexity of approximately counting and sampling in the six- and eight-vertex models on various classes of underlying graphs. First, we study the approximability of the partition function on general 4-regular graphs, classified according to the parameters of the models. Our complexity results conform to the phase transition phenomenon from physics due to the change in temperature. We introduce a quantum decomposition of the six- and eight-vertex models and prove a set of closure properties in various regions of the parameter space. These regions of the parameter space are concordant with the phase transition threshold. Using these closure properties, we derive polynomial time approximation algorithms via Markov chain Monte Carlo in some parameter space in the high temperature regime. In some other parameter space in the high temperature regime, we prove that the problem is (at least) as hard as approximately counting perfect matchings, a central open problem in this field. We also show that the six- and eight-vertex models are NP-hard to approximate in the whole low temperature regime on general 4-regular graphs. We then study the six- and eight-vertex models on more restricted classes of 4-regular graphs, including planar graphs and bipartite graphs. We give the first polynomial time approximation algorithm for the partition function in the low temperature regime on planar and on bipartite graphs. Our results show that the six- and eight-vertex models are the first problems with the provable property that while NP-hard to approximate on general graphs (even #P-hard for planar graphs in exact complexity), they possess efficient approximation schemes on both bipartite graphs and planar graphs in substantial regions of the parameter space. Finally, we study the square lattice six- and eight-vertex models. We prove that natural Markov chains for these models are mixing torpidly in the low temperature regime. Moreover, we give the first efficient approximate counting and sampling algorithms for the six- and the eight-vertex models on the square lattice at sufficiently low temperatures.

Here are the refereed proceedings of the Second International Workshop on Parameterized and Exact Computation, IWPEC 2006, held in the context of the combined conference ALGO 2006. The book presents 23 revised full papers together with 2 invited lectures. Coverage includes research in all aspects of parameterized and exact computation and complexity, including new techniques for the design and analysis of parameterized and exact algorithms, parameterized complexity theory, and more.