Advances in the Computational Complexity of Holant Problems

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ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (93 download)

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Book Synopsis Advances in the Computational Complexity of Holant Problems by :

Download or read book Advances in the Computational Complexity of Holant Problems written by and published by . This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the computational complexity of counting problems defined over graphs. Complexity dichotomies are proved for various sets of problems, which classify the complexity of each problem in the set as either computable in polynomial time or \#P-hard. These problems are expressible in the frameworks of counting graph homomorphisms, counting constraint satisfaction problems, or Holant problems. However, the proofs are always expressed within the framework of Holant problems, which contains the other two frameworks as special cases. Holographic transformations are naturally expressed using this framework. They represent proofs that two different-looking problems are actually the same. We use them to prove both hardness and tractability. Moreover, the tractable cases are often stated using a holographic transformation. The uniting theme in the proofs of every dichotomy is the technical advances achieved in order to prove the hardness. Specifically, polynomial interpolation appears prominently and is indispensable. We repeatedly strengthen and extend this technique and are rewarded with dichotomies for larger and larger classes of problems. We now have a thorough understanding of its power as well as its ultimate limitations. However, fundamental questions remain since polynomial interpolation is intimately connected with integer solutions of algebraic curves and determinations of Galois groups, subjects that remain active areas of research in pure mathematics. Our motivation for this work is to understand the limits of efficient computation. Without settling the P versus #P question, the best hope is to achieve such complexity classifications.

Advances in Computational Complexity Theory

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Publisher : American Mathematical Soc.
ISBN 13 : 9780821885758
Total Pages : 234 pages
Book Rating : 4.8/5 (857 download)

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Book Synopsis Advances in Computational Complexity Theory by : Jin-yi Cai

Download or read book Advances in Computational Complexity Theory written by Jin-yi Cai and published by American Mathematical Soc.. This book was released on 1993-01-01 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: * 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.

A Journey Through Some Computational Problems

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ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (142 download)

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Book Synopsis A Journey Through Some Computational Problems by : Yin Liu (Ph.D.)

Download or read book A Journey Through Some Computational Problems written by Yin Liu (Ph.D.) and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The research on Holant problems is an important topic in the field of Computational Complexity Theory, which is one of the most important subjects in computer science. This thesis discusses the beauty of several specific classes of Holant problems. We give a dichotomy of a bipartite Holant problem, then move on to higher domains, where we not only study the dichotomies, but also manage to give some new combinatorial insights over specific problems. First we prove a complexity dichotomy for a class of counting problems expressible as bipartite 3-regular Holant problems. For every problem of one specific form, we prove that it is either P-time computable or belongs to a class that is hard to compute in polynomial time under standard Complexity Theory hypothesis, and give an explicit criterion of the ternary integer valued function. Additionally, we discover a new phenomenon: there is a set of problems that their planar tractability has not seen before. We then move on to higher domains. We intend to study a class of problems on the domain of size 4 with a real-valued ternary function, but we have been blocked and we show the partial results in the thesis. However, we manage to prove a dichotomy when we restrict the function to take only 0 and 1 values. In addition, we are able to give a more concise dichotomy theorem on the domain of size 3 for real functions. To extend ``Fibonacci gates" on Boolean domain, we give a combinatorial view for Holant problems on the domain of size 3 and 4 respectively. Moreover, we derive the corresponding combinatorial algorithms. In part II of this thesis, I also describe some other work I carried out for more practical problems in computer science. We apply a novel method of optimizing the parameters in the neural network and discuss some adjustments to the traditional neural network which could improve the performance with the idea of Taylor approximation.

Theory of Computational Complexity

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Publisher : John Wiley & Sons
ISBN 13 : 1118594975
Total Pages : 512 pages
Book Rating : 4.1/5 (185 download)

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Book Synopsis Theory of Computational Complexity by : Ding-Zhu Du

Download or read book Theory of Computational Complexity written by Ding-Zhu Du and published by John Wiley & Sons. This book was released on 2014-07-18 with total page 512 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Complexity Dichotomies for Counting Problems

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Publisher : Cambridge University Press
ISBN 13 : 1107062373
Total Pages : 473 pages
Book Rating : 4.1/5 (7 download)

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Book Synopsis Complexity Dichotomies for Counting Problems by : Jin-Yi Cai

Download or read book Complexity Dichotomies for Counting Problems written by Jin-Yi Cai and published by Cambridge University Press. This book was released on 2017-11-16 with total page 473 pages. Available in PDF, EPUB and Kindle. Book excerpt: Volume 1. Boolean domain

Complexity Dichotomies for Counting Problems

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Publisher :
ISBN 13 : 9781107635609
Total Pages : pages
Book Rating : 4.6/5 (356 download)

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Book Synopsis Complexity Dichotomies for Counting Problems by : Jin-yi Cai

Download or read book Complexity Dichotomies for Counting Problems written by Jin-yi Cai and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Complexity Dichotomies for Counting Problems: Volume 1, Boolean Domain

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Publisher : Cambridge University Press
ISBN 13 : 1108508820
Total Pages : 473 pages
Book Rating : 4.1/5 (85 download)

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Book Synopsis Complexity Dichotomies for Counting Problems: Volume 1, Boolean Domain by : Jin-Yi Cai

Download or read book Complexity Dichotomies for Counting Problems: Volume 1, Boolean Domain written by Jin-Yi Cai and published by Cambridge University Press. This book was released on 2017-11-16 with total page 473 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Complexity In Numerical Optimization

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Publisher : World Scientific
ISBN 13 : 9814504084
Total Pages : 538 pages
Book Rating : 4.8/5 (145 download)

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Book Synopsis Complexity In Numerical Optimization by : Panos M Pardalos

Download or read book Complexity In Numerical Optimization written by Panos M Pardalos and published by World Scientific. This book was released on 1993-07-31 with total page 538 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational complexity, originated from the interactions between computer science and numerical optimization, is one of the major theories that have revolutionized the approach to solving optimization problems and to analyzing their intrinsic difficulty.The main focus of complexity is the study of whether existing algorithms are efficient for the solution of problems, and which problems are likely to be tractable.The quest for developing efficient algorithms leads also to elegant general approaches for solving optimization problems, and reveals surprising connections among problems and their solutions.This book is a collection of articles on recent complexity developments in numerical optimization. The topics covered include complexity of approximation algorithms, new polynomial time algorithms for convex quadratic minimization, interior point algorithms, complexity issues regarding test generation of NP-hard problems, complexity of scheduling problems, min-max, fractional combinatorial optimization, fixed point computations and network flow problems.The collection of articles provide a broad spectrum of the direction in which research is going and help to elucidate the nature of computational complexity in optimization. The book will be a valuable source of information to faculty, students and researchers in numerical optimization and related areas.

Complexity in Numerical Optimization

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Publisher : World Scientific
ISBN 13 : 9789810214159
Total Pages : 536 pages
Book Rating : 4.2/5 (141 download)

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Book Synopsis Complexity in Numerical Optimization by : Panos M. Pardalos

Download or read book Complexity in Numerical Optimization written by Panos M. Pardalos and published by World Scientific. This book was released on 1993 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational complexity, originated from the interactions between computer science and numerical optimization, is one of the major theories that have revolutionized the approach to solving optimization problems and to analyzing their intrinsic difficulty.The main focus of complexity is the study of whether existing algorithms are efficient for the solution of problems, and which problems are likely to be tractable.The quest for developing efficient algorithms leads also to elegant general approaches for solving optimization problems, and reveals surprising connections among problems and their solutions.This book is a collection of articles on recent complexity developments in numerical optimization. The topics covered include complexity of approximation algorithms, new polynomial time algorithms for convex quadratic minimization, interior point algorithms, complexity issues regarding test generation of NP-hard problems, complexity of scheduling problems, min-max, fractional combinatorial optimization, fixed point computations and network flow problems.The collection of articles provide a broad spectrum of the direction in which research is going and help to elucidate the nature of computational complexity in optimization. The book will be a valuable source of information to faculty, students and researchers in numerical optimization and related areas.

Complexity and Approximation

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Publisher : Springer Nature
ISBN 13 : 3030416720
Total Pages : 298 pages
Book Rating : 4.0/5 (34 download)

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Book Synopsis Complexity and Approximation by : Ding-Zhu Du

Download or read book Complexity and Approximation written by Ding-Zhu Du and published by Springer Nature. This book was released on 2020-02-20 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Festschrift is in honor of Ker-I Ko, Professor in the Stony Brook University, USA. Ker-I Ko was one of the founding fathers of computational complexity over real numbers and analysis. He and Harvey Friedman devised a theoretical model for real number computations by extending the computation of Turing machines. He contributed significantly to advancing the theory of structural complexity, especially on polynomial-time isomorphism, instance complexity, and relativization of polynomial-time hierarchy. Ker-I also made many contributions to approximation algorithm theory of combinatorial optimization problems. This volume contains 17 contributions in the area of complexity and approximation. Those articles are authored by researchers over the world, including North America, Europe and Asia. Most of them are co-authors, colleagues, friends, and students of Ker-I Ko.

Perspectives in Computational Complexity

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Publisher : Springer
ISBN 13 : 3319054465
Total Pages : 206 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Perspectives in Computational Complexity by : Manindra Agrawal

Download or read book Perspectives in Computational Complexity written by Manindra Agrawal and published by Springer. This book was released on 2014-07-16 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Complexity and Information

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Publisher : Cambridge University Press
ISBN 13 : 9780521485067
Total Pages : 152 pages
Book Rating : 4.4/5 (85 download)

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Book Synopsis Complexity and Information by : J. F. Traub

Download or read book Complexity and Information written by J. F. Traub and published by Cambridge University Press. This book was released on 1998-12-10 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: The twin themes of computational complexity and information pervade this 1998 book. It starts with an introduction to the computational complexity of continuous mathematical models, that is, information-based complexity. This is then used to illustrate a variety of topics, including breaking the curse of dimensionality, complexity of path integration, solvability of ill-posed problems, the value of information in computation, assigning values to mathematical hypotheses, and new, improved methods for mathematical finance. The style is informal, and the goals are exposition, insight and motivation. A comprehensive bibliography is provided, to which readers are referred for precise statements of results and their proofs. As the first introductory book on the subject it will be invaluable as a guide to the area for the many students and researchers whose disciplines, ranging from physics to finance, are influenced by the computational complexity of continuous problems.

Complexity Classification of Counting Problems on Boolean Variables

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Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (124 download)

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Book Synopsis Complexity Classification of Counting Problems on Boolean Variables by : Shuai Shao

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 0 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?

Frontiers in Algorithmics and Algorithmic Aspects in Information and Management

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Publisher : Springer
ISBN 13 : 3642212042
Total Pages : 400 pages
Book Rating : 4.6/5 (422 download)

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Book Synopsis Frontiers in Algorithmics and Algorithmic Aspects in Information and Management by : Mikhail Atallah

Download or read book Frontiers in Algorithmics and Algorithmic Aspects in Information and Management written by Mikhail Atallah and published by Springer. This book was released on 2011-05-30 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the joint refereed proceedings of the 5th International Frontiers of Algorithmics Workshop, FAW 2011, and the 7th International Conference on Algorithmic Aspects in Information and Management, AAIM 2011, jointly held in Jinhua, China, in May 2011. The 35 revised full papers presented together with 4 invited talks were carefully reviewed and selected from 92 submissions. The papers cover a wide range of topics in the areas of algorithmics, combinatorial optimization and their applications presenting current trends of research.

On Some Central Problems in Computational Complexity

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Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (897 download)

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Book Synopsis On Some Central Problems in Computational Complexity by : Jacques Simon

Download or read book On Some Central Problems in Computational Complexity written by Jacques Simon and published by . This book was released on 1975 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Language and Automata Theory and Applications

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Publisher : Springer
ISBN 13 : 3642370640
Total Pages : 592 pages
Book Rating : 4.6/5 (423 download)

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Book Synopsis Language and Automata Theory and Applications by : Adrian-Horia Dediu

Download or read book Language and Automata Theory and Applications written by Adrian-Horia Dediu and published by Springer. This book was released on 2013-03-15 with total page 592 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 7th International Conference on Language and Automata Theory and Applications, LATA 2013, held in Bilbao, Spain in April 2013. The 45 revised full papers presented together with 5 invited talks were carefully reviewed and selected from 97 initial submissions. The volume features contributions from both classical theory fields and application areas (bioinformatics, systems biology, language technology, artificial intelligence, etc.). Among the topics covered are algebraic language theory; algorithms for semi-structured data mining; algorithms on automata and words; automata and logic; automata for system analysis and program verification; automata, concurrency and Petri nets; automatic structures; cellular automata; combinatorics on words; computability; computational complexity; computational linguistics; data and image compression; decidability questions on words and languages; descriptional complexity; DNA and other models of bio-inspired computing; document engineering; foundations of finite state technology; foundations of XML; fuzzy and rough languages; grammars (Chomsky hierarchy, contextual, multidimensional, unification, categorial, etc.); grammars and automata architectures; grammatical inference and algorithmic learning; graphs and graph transformation; language varieties and semigroups; language-based cryptography; language-theoretic foundations of artificial intelligence and artificial life; parallel and regulated rewriting; parsing; pattern recognition; patterns and codes; power series; quantum, chemical and optical computing; semantics; string and combinatorial issues in computational biology and bioinformatics; string processing algorithms; symbolic dynamics; symbolic neural networks; term rewriting; transducers; trees, tree languages and tree automata; weighted automata.

The Computational Complexity of Algebraic and Numeric Problems

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Publisher : Elsevier Publishing Company
ISBN 13 :
Total Pages : 198 pages
Book Rating : 4.4/5 (91 download)

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Book Synopsis The Computational Complexity of Algebraic and Numeric Problems by : Allan Borodin

Download or read book The Computational Complexity of Algebraic and Numeric Problems written by Allan Borodin and published by Elsevier Publishing Company. This book was released on 1975 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: