Computational Complexity

Computational Complexity

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

Author: Sanjeev Arora

Publisher: Cambridge University Press

ISBN: 9780521424264

Category: Computers

Page: 579

View: 823

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

Computational Complexity and Statistical Physics

Computational Complexity and Statistical Physics

This is leading to a new understanding of the structure of these problems, and of how algorithms perform on them.

Author: Allon Percus

Publisher: Oxford University Press on Demand

ISBN: 0195177371

Category: Computers

Page: 367

View: 982

Computer science and physics have been closely linked since the birth of modern computing. In recent years, an interdisciplinary area has blossomed at the junction of these fields, connecting insights from statistical physics with basic computational challenges. Researchers have successfully applied techniques from the study of phase transitions to analyze NP-complete problems such as satisfiability and graph coloring. This is leading to a new understanding of the structure of these problems, and of how algorithms perform on them. Computational Complexity and Statistical Physics will serve as a standard reference and pedagogical aid to statistical physics methods in computer science, with a particular focus on phase transitions in combinatorial problems. Addressed to a broad range of readers, the book includes substantial background material along with current research by leading computer scientists, mathematicians, and physicists. It will prepare students and researchers from all of these fields to contribute to this exciting area.
Categories: Computers

The Computational Complexity of Machine Learning

The Computational Complexity of Machine Learning

We also give algorithms for learning powerful concept classes under the uniform distribution, and give equivalences between natural models of efficient learnability.

Author: Harvard University Center for Research in Computing Technology

Publisher: MIT Press

ISBN: 0262111527

Category: Computers

Page: 165

View: 695

We also give algorithms for learning powerful concept classes under the uniform distribution, and give equivalences between natural models of efficient learnability. This thesis also includes detailed definitions and motivation for the distribution-free model, a chapter discussing past research in this model and related models, and a short list of important open problems."
Categories: Computers

Computational Complexity Theory

Computational Complexity Theory

At the core of the theory are some of the most alluring open problems in mathematics. This book presents three weeks of lectures from the IAS/Park City Mathematics Institute Summer School on computational complexity.

Author: Steven Rudich

Publisher: American Mathematical Soc.

ISBN: 9780821828724

Category: Mathematics

Page: 389

View: 241

Computational Complexity Theory is the study of how much of a given resource is required to perform the computations that interest us the most. Four decades of fruitful research have produced a rich and subtle theory of the relationship between different resource measures and problems. At the core of the theory are some of the most alluring open problems in mathematics. This book presents three weeks of lectures from the IAS/Park City Mathematics Institute Summer School on computational complexity. It is recommended for independent study by graduate students or researchers interested in computational complexity.
Categories: Mathematics

Computational Complexity of Counting and Sampling

Computational Complexity of Counting and Sampling

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.

Author: Istvan Miklos

Publisher: CRC Press

ISBN: 9781351971614

Category: Mathematics

Page: 390

View: 965

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
Categories: Mathematics

Computational Complexity Theory

Computational Complexity Theory

This volume contains the proceedings of the AMS Short Course on Computational Complexity Theory, held at the Joint Mathematics Meetings in Atlanta in January 1988.

Author: Juris Hartmanis

Publisher: American Mathematical Soc.

ISBN: 9780821801314

Category: Mathematics

Page: 128

View: 143

Computational complexity theory is the study of the quantitative laws that govern computing. During the last 25 years, this field has grown into a rich mathematical theory. Currently one of the most active research areas in computer science, complexity theory is of considerable interest to mathematicians as well, since some of the key open problems in this field raise basic questions about the nature of mathematics. Many experts in complexity theory believe that, in coming decades, the strongest influence on the development of mathematics will come from the extended use of computing and from concepts and problems arising in computer science.This volume contains the proceedings of the AMS Short Course on Computational Complexity Theory, held at the Joint Mathematics Meetings in Atlanta in January 1988. The purpose of the short course was to provide an overview of complexity theory and to describe some of the current developments in the field. The papers presented here represent contributions by some of the top experts in this burgeoning area of research.
Categories: Mathematics

Logic and Computational Complexity

Logic and Computational Complexity

Expressing Computational Complexity in Constructive Type Theory * Robert L.
Constable Cornell University Abstract It is notoriously hard to express
computational complexity properties of programs in programming logics based
on a ...

Author: International Workshop on Logic and Computational Complexity

Publisher: Springer Science & Business Media

ISBN: 3540601783

Category: Computers

Page: 514

View: 197

This book contains revised versions of papers invited for presentation at the International Workshop on Logic and Computational Complexity, LCC '94, held in Indianapolis, IN in October 1994. The synergy between logic and computational complexity has gained importance and vigor in recent years, cutting across many areas. The 25 revised full papers in this book contributed by internationally outstanding researchers document the state-of-the-art in this interdisciplinary field of growing interest; they are presented in sections on foundational issues, applicative and proof-theoretic complexity, complexity of proofs, computational complexity of functionals, complexity and model theory, and finite model theory.
Categories: Computers

Kolmogorov Complexity and Computational Complexity

Kolmogorov Complexity and Computational Complexity

In relation to computational complexity theory, resource-bounded Kolmogorov
complexities are frequently considered. For example, the t(n)-time bounded
Kolmogorov complexity of a string a is the size of the shortest string y from which
the ...

Author: Osamu Watanabe

Publisher: Springer Science & Business Media

ISBN: 9783642777356

Category: Computers

Page: 105

View: 858

The mathematical theory of computation has given rise to two important ap proaches to the informal notion of "complexity": Kolmogorov complexity, usu ally a complexity measure for a single object such as a string, a sequence etc., measures the amount of information necessary to describe the object. Compu tational complexity, usually a complexity measure for a set of objects, measures the compuational resources necessary to recognize or produce elements of the set. The relation between these two complexity measures has been considered for more than two decades, and may interesting and deep observations have been obtained. In March 1990, the Symposium on Theory and Application of Minimal Length Encoding was held at Stanford University as a part of the AAAI 1990 Spring Symposium Series. Some sessions of the symposium were dedicated to Kolmogorov complexity and its relations to the computational complexity the ory, and excellent expository talks were given there. Feeling that, due to the importance of the material, some way should be found to share these talks with researchers in the computer science community, I asked the speakers of those sessions to write survey papers based on their talks in the symposium. In response, five speakers from the sessions contributed the papers which appear in this book.
Categories: Computers

Theory of Computational Complexity

Theory of Computational Complexity

—Elizabeth Barrett Browning Informally, the complexity class NP contains the
decision problems inwhich an input instance is ... ofthese counting problems
andits relation to thecomplexity of nondeterministic and probabilistic computation.

Author: Ding-Zhu Du

Publisher: John Wiley & Sons

ISBN: 9781118594971

Category: Mathematics

Page: 512

View: 819

Praise for the First Edition "...complete, up-to-date coverage of computational complexitytheory...the book promises to become the standard reference oncomputational complexity." -Zentralblatt MATH A thorough revision based on advances in the field ofcomputational complexity and readers’ feedback, the SecondEdition of Theory of Computational Complexity presentsupdates to the principles and applications essential tounderstanding modern computational complexity theory. The newedition continues to serve as a comprehensive resource on the useof software and computational approaches for solving algorithmicproblems and the related difficulties that can be encountered. Maintaining extensive and detailed coverage, Theory ofComputational Complexity, Second Edition, examines the theoryand methods behind complexity theory, such as computational models,decision tree complexity, circuit complexity, and probabilisticcomplexity. The Second Edition also features recentdevelopments on areas such as NP-completeness theory, as wellas: A new combinatorial proof of the PCP theorem based on thenotion of expander graphs, a research area in the field of computerscience Additional exercises at varying levels of difficulty to furthertest comprehension of the presented material End-of-chapter literature reviews that summarize each topic andoffer additional sources for further study Theory of Computational Complexity, Second Edition, is anexcellent textbook for courses on computational theory andcomplexity at the graduate level. The book is also a usefulreference for practitioners in the fields of computer science,engineering, and mathematics who utilize state-of-the-art softwareand computational methods to conduct research. Athorough revision based on advances in the field of computationalcomplexity and readers’feedback,the Second Edition of Theory of Computational Complexity presentsupdates to theprinciplesand applications essential to understanding modern computationalcomplexitytheory.The new edition continues to serve as a comprehensive resource onthe use of softwareandcomputational approaches for solving algorithmic problems and therelated difficulties thatcanbe encountered.Maintainingextensive and detailed coverage, Theory of ComputationalComplexity, SecondEdition,examines the theory and methods behind complexity theory, such ascomputationalmodels,decision tree complexity, circuit complexity, and probabilisticcomplexity. The SecondEditionalso features recent developments on areas such as NP-completenesstheory, as well as:•A new combinatorial proof of the PCP theorem based on the notion ofexpandergraphs,a research area in the field of computer science•Additional exercises at varying levels of difficulty to furthertest comprehension ofthepresented material•End-of-chapter literature reviews that summarize each topic andoffer additionalsourcesfor further studyTheoryof Computational Complexity, Second Edition, is an excellenttextbook for courses oncomputationaltheory and complexity at the graduate level. The book is also auseful referenceforpractitioners in the fields of computer science, engineering, andmathematics who utilizestate-of-the-artsoftware and computational methods to conduct research.
Categories: Mathematics

Computational Complexity and Feasibility of Data Processing and Interval Computations

Computational Complexity and Feasibility of Data Processing and Interval Computations

The book is written with all explanations and definitions added, so that it can be used as a graduate level textbook. What this book .is about Data processing.

Author: V. Kreinovich

Publisher: Springer Science & Business Media

ISBN: 9781475727937

Category: Mathematics

Page: 459

View: 500

Targeted audience • Specialists in numerical computations, especially in numerical optimiza tion, who are interested in designing algorithms with automatie result ver ification, and who would therefore be interested in knowing how general their algorithms caIi in principle be. • Mathematicians and computer scientists who are interested in the theory 0/ computing and computational complexity, especially computational com plexity of numerical computations. • Students in applied mathematics and computer science who are interested in computational complexity of different numerical methods and in learning general techniques for estimating this computational complexity. The book is written with all explanations and definitions added, so that it can be used as a graduate level textbook. What this book .is about Data processing. In many real-life situations, we are interested in the value of a physical quantity y that is diflicult (or even impossible) to measure directly. For example, it is impossible to directly measure the amount of oil in an oil field or a distance to a star. Since we cannot measure such quantities directly, we measure them indirectly, by measuring some other quantities Xi and using the known relation between y and Xi'S to reconstruct y. The algorithm that transforms the results Xi of measuring Xi into an estimate fj for y is called data processing.
Categories: Mathematics

Perspectives in Computational Complexity

Perspectives in Computational Complexity

53(2), 267–282 (1996) [Agr01] M. Agrawal, Towards uniform AC0–isomorphisms.
in Proceedings IEEE Conference on Computational Complexity (2001). pp. 13–
20 [Agr02] M. Agrawal, Pseudo-random generators and structure of complete ...

Author: Manindra Agrawal

Publisher: Springer

ISBN: 9783319054469

Category: Mathematics

Page: 202

View: 908

This book brings together contributions by leading researchers in computational complexity theory written in honor of Somenath Biswas on the occasion of his sixtieth birthday. They discuss current trends and exciting developments in this flourishing area of research and offer fresh perspectives on various aspects of complexity theory. The topics covered include arithmetic circuit complexity, lower bounds and polynomial identity testing, the isomorphism conjecture, space-bounded computation, graph isomorphism, resolution and proof complexity, entropy and randomness. Several chapters have a tutorial flavor. The aim is to make recent research in these topics accessible to graduate students and senior undergraduates in computer science and mathematics. It can also be useful as a resource for teaching advanced level courses in computational complexity.
Categories: Mathematics

Computational Complexity A Quantitative Perspective

Computational Complexity  A Quantitative Perspective

The main purpose of computational complexity is to measure the amount of time,
or of space, or of Some other resource, that is necessary to solve a computational
problem. Thus, by its very nature, computational complexity is a quantitative ...

Author: Marius Zimand

Publisher: Elsevier

ISBN: 008047666X

Category: Computers

Page: 352

View: 690

There has been a common perception that computational complexity is a theory of "bad news" because its most typical results assert that various real-world and innocent-looking tasks are infeasible. In fact, "bad news" is a relative term, and, indeed, in some situations (e.g., in cryptography), we want an adversary to not be able to perform a certain task. However, a "bad news" result does not automatically become useful in such a scenario. For this to happen, its hardness features have to be quantitatively evaluated and shown to manifest extensively. The book undertakes a quantitative analysis of some of the major results in complexity that regard either classes of problems or individual concrete problems. The size of some important classes are studied using resource-bounded topological and measure-theoretical tools. In the case of individual problems, the book studies relevant quantitative attributes such as approximation properties or the number of hard inputs at each length. One chapter is dedicated to abstract complexity theory, an older field which, however, deserves attention because it lays out the foundations of complexity. The other chapters, on the other hand, focus on recent and important developments in complexity. The book presents in a fairly detailed manner concepts that have been at the centre of the main research lines in complexity in the last decade or so, such as: average-complexity, quantum computation, hardness amplification, resource-bounded measure, the relation between one-way functions and pseudo-random generators, the relation between hard predicates and pseudo-random generators, extractors, derandomization of bounded-error probabilistic algorithms, probabilistically checkable proofs, non-approximability of optimization problems, and others. The book should appeal to graduate computer science students, and to researchers who have an interest in computer science theory and need a good understanding of computational complexity, e.g., researchers in algorithms, AI, logic, and other disciplines. · Emphasis is on relevant quantitative attributes of important results in complexity. · Coverage is self-contained and accessible to a wide audience. · Large range of important topics including: derandomization techniques, non-approximability of optimization problems, average-case complexity, quantum computation, one-way functions and pseudo-random generators, resource-bounded measure and topology.
Categories: Computers

Theories of Computational Complexity

Theories of Computational Complexity

The computational complexity studies fall into a variety of directions, some of
which are now available in monographs ... for the complexity of specific functions,
BORODIN and MUNRO [1975), for algebraic complexity, GAREY and JOHNSON
 ...

Author: C. Calude

Publisher: Elsevier

ISBN: 0080867758

Category: Mathematics

Page: 486

View: 533

This volume presents four machine-independent theories of computational complexity, which have been chosen for their intrinsic importance and practical relevance. The book includes a wealth of results - classical, recent, and others which have not been published before. In developing the mathematics underlying the size, dynamic and structural complexity measures, various connections with mathematical logic, constructive topology, probability and programming theories are established. The facts are presented in detail. Extensive examples are provided, to help clarify notions and constructions. The lists of exercises and problems include routine exercises, interesting results, as well as some open problems.
Categories: Mathematics

Logical Foundations of Mathematics and Computational Complexity

Logical Foundations of Mathematics and Computational Complexity

Complexity. of. Computations. Hiding in the alternating patterns of digits, deep
inside the transcendental number, was a ... Originally the motivation for studying
computational complexity was to understand which algorithms can be used in ...

Author: Pavel Pudlák

Publisher: Springer Science & Business Media

ISBN: 9783319001197

Category: Mathematics

Page: 695

View: 597

The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability. Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs. Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.
Categories: Mathematics

The Computational Complexity of Logical Theories

The Computational Complexity of Logical Theories

This book asks not only how the study of white-collar crime can enrich our understanding of crime and justice more generally, but also how criminological advances over the last few decades can enhance our understanding of white-collar ...

Author: J. Ferrante

Publisher: Springer

ISBN: 9783540351979

Category: Mathematics

Page: 244

View: 182

Categories: Mathematics

Analytic Computational Complexity

Analytic Computational Complexity

... circumstances we give some surprising work estimates, which first appeared in
Eisenstat, Schultz, and Sherman [74a], and whose complete derivation is given in
Sherman [ 75 | . These estimates indicate that the study of the complexity of ...

Author: J.F. Traub

Publisher: Academic Press

ISBN: 9781483257891

Category: Mathematics

Page: 250

View: 128

Analytic Computational Complexity contains the proceedings of the Symposium on Analytic Computational Complexity held by the Computer Science Department, Carnegie-Mellon University, Pittsburgh, Pennsylvania, on April 7-8, 1975. The symposium provided a forum for assessing progress made in analytic computational complexity and covered topics ranging from strict lower and upper bounds on iterative computational complexity to numerical stability of iterations for solution of nonlinear equations and large linear systems. Comprised of 14 chapters, this book begins with an introduction to analytic computational complexity before turning to proof techniques used in analytic complexity. Subsequent chapters focus on the complexity of obtaining starting points for solving operator equations by Newton's method; maximal order of multipoint iterations using n evaluations; the use of integrals in the solution of nonlinear equations in N dimensions; and the complexity of differential equations. Algebraic constructions in an analytic setting are also discussed, along with the computational complexity of approximation operators. This monograph will be of interest to students and practitioners in the fields of applied mathematics and computer science.
Categories: Mathematics

Randomness and Completeness in Computational Complexity

Randomness and Completeness in Computational Complexity

Computational. Complexity. 1.1.1 The Power of Randomness Randomness
provides a too lthatwe can often use to our advantage. King Arthur,o n his questfo
r the Holy Grail,already knew that. One day,Merlin claimed to have found the
sword ...

Author: Dieter van Melkebeek

Publisher: Springer

ISBN: 9783540445456

Category: Computers

Page: 198

View: 104

This book contains a revised version of the dissertation the author wrote at the Department of Computer Science of the University of Chicago. The thesis was submitted to the Faculty of Physical Sciences in conformity with the requirements for the PhD degree in June 1999. It was honored with the 1999 ACM Doctoral Dissertation Award in May 2000. Summary Computational complexity is the study of the inherent di culty of compu- tional problems and the power of the tools we may use to solve them. It aims to describe how many resources we need to compute the solution as a function of the problem size. Typical resources include time on sequential and parallel architectures and memory space. As we want to abstract away from details of input representation and speci cs of the computer model, we end up with classes of problems that we can solve within certain robust resource bounds such as polynomial time, parallel logarithmic time, and logarithmic space. Research in complexity theory boils down to determining the relationships between these classes { inclusions and separations. In this dissertation, we focus on the role of randomness and look at various properties of hard problems in order to obtain separations. We also investigate the power of nondeterminism and alternation, as well as space versus time issues. Randomness provides a resource that seems to help in various situations.
Categories: Computers

Noisy Information and Computational Complexity

Noisy Information and Computational Complexity

Leszek Plaskota. COMPUTATIONAL COMPLEXITY COMPUTATIONAL
COMPLEXITY Leszek Plaskota Institute of Applied Mathematics and Mechanics
NOISY INFORMATION AND Half-title.

Author: Leszek Plaskota

Publisher: Cambridge University Press

ISBN: 9780521553681

Category: Computers

Page: 308

View: 551

In this volume, which was originally published in 1996, noisy information is studied in the context of computational complexity; in other words the text deals with the computational complexity of mathematical problems for which information is partial, noisy and priced.
Categories: Computers

Advances in Computational Complexity Theory

Advances in Computational Complexity Theory

L would not be considered sufficiently "complex", and thus two stronger notions of
complexity have often been considered: randomness (or Church- randomness)
and almost-everywhere complexity. A set L is said to be Church-random with ...

Author: Jin-yi Cai

Publisher: American Mathematical Soc.

ISBN: 0821885758

Category: Mathematics

Page: 209

View: 118

* Recent papers on computational complexity theory * Contributions by some of the leading experts in the field This book will prove to be of lasting value in this fast-moving field as it provides expositions not found elsewhere. The book touches on some of the major topics in complexity theory and thus sheds light on this burgeoning area of research.
Categories: Mathematics