Hardness Of Approximation Between P And Np

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Hardness of Approximation Between P and NP

Author : Aviad Rubinstein
Publisher : Morgan & Claypool
ISBN 13 : 1947487213
Total Pages : 319 pages
Book Rating : 4.9/5 (474 download)

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Book Synopsis Hardness of Approximation Between P and NP by : Aviad Rubinstein

Download or read book Hardness of Approximation Between P and NP written by Aviad Rubinstein and published by Morgan & Claypool. This book was released on 2019-06-07 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nash equilibrium is the central solution concept in Game Theory. Since Nash’s original paper in 1951, it has found countless applications in modeling strategic behavior of traders in markets, (human) drivers and (electronic) routers in congested networks, nations in nuclear disarmament negotiations, and more. A decade ago, the relevance of this solution concept was called into question by computer scientists, who proved (under appropriate complexity assumptions) that computing a Nash equilibrium is an intractable problem. And if centralized, specially designed algorithms cannot find Nash equilibria, why should we expect distributed, selfish agents to converge to one? The remaining hope was that at least approximate Nash equilibria can be efficiently computed. Understanding whether there is an efficient algorithm for approximate Nash equilibrium has been the central open problem in this field for the past decade. In this book, we provide strong evidence that even finding an approximate Nash equilibrium is intractable. We prove several intractability theorems for different settings (two-player games and many-player games) and models (computational complexity, query complexity, and communication complexity). In particular, our main result is that under a plausible and natural complexity assumption ("Exponential Time Hypothesis for PPAD"), there is no polynomial-time algorithm for finding an approximate Nash equilibrium in two-player games. The problem of approximate Nash equilibrium in a two-player game poses a unique technical challenge: it is a member of the class PPAD, which captures the complexity of several fundamental total problems, i.e., problems that always have a solution; and it also admits a quasipolynomial time algorithm. Either property alone is believed to place this problem far below NP-hard problems in the complexity hierarchy; having both simultaneously places it just above P, at what can be called the frontier of intractability. Indeed, the tools we develop in this book to advance on this frontier are useful for proving hardness of approximation of several other important problems whose complexity lies between P and NP: Brouwer’s fixed point, market equilibrium, CourseMatch (A-CEEI), densest k-subgraph, community detection, VC dimension and Littlestone dimension, and signaling in zero-sum games.

P, NP, and NP-Completeness

Author : Oded Goldreich
Publisher : Cambridge University Press
ISBN 13 : 1139490095
Total Pages : pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis P, NP, and NP-Completeness by : Oded Goldreich

Download or read book P, NP, and NP-Completeness written by Oded Goldreich and published by Cambridge University Press. This book was released on 2010-08-16 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The focus of this book is the P versus NP Question and the theory of NP-completeness. It also provides adequate preliminaries regarding computational problems and computational models. The P versus NP Question asks whether or not finding solutions is harder than checking the correctness of solutions. An alternative formulation asks whether or not discovering proofs is harder than verifying their correctness. It is widely believed that the answer to these equivalent formulations is positive, and this is captured by saying that P is different from NP. Although the P versus NP Question remains unresolved, the theory of NP-completeness offers evidence for the intractability of specific problems in NP by showing that they are universal for the entire class. Amazingly enough, NP-complete problems exist, and furthermore hundreds of natural computational problems arising in many different areas of mathematics and science are NP-complete.

Approximation Algorithms for NP-hard Problems

Author : Dorit S. Hochbaum
Publisher : Course Technology
ISBN 13 :
Total Pages : 632 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Approximation Algorithms for NP-hard Problems by : Dorit S. Hochbaum

Download or read book Approximation Algorithms for NP-hard Problems written by Dorit S. Hochbaum and published by Course Technology. This book was released on 1997 with total page 632 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book to fully address the study of approximation algorithms as a tool for coping with intractable problems. With chapters contributed by leading researchers in the field, this book introduces unifying techniques in the analysis of approximation algorithms. APPROXIMATION ALGORITHMS FOR NP-HARD PROBLEMS is intended for computer scientists and operations researchers interested in specific algorithm implementations, as well as design tools for algorithms. Among the techniques discussed: the use of linear programming, primal-dual techniques in worst-case analysis, semidefinite programming, computational geometry techniques, randomized algorithms, average-case analysis, probabilistically checkable proofs and inapproximability, and the Markov Chain Monte Carlo method. The text includes a variety of pedagogical features: definitions, exercises, open problems, glossary of problems, index, and notes on how best to use the book.

The Golden Ticket

Author : Lance Fortnow
Publisher : Princeton University Press
ISBN 13 : 0691175780
Total Pages : 188 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis The Golden Ticket by : Lance Fortnow

Download or read book The Golden Ticket written by Lance Fortnow and published by Princeton University Press. This book was released on 2017-02-28 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: The P-NP problem is the most important open problem in computer science, if not all of mathematics. Simply stated, it asks whether every problem whose solution can be quickly checked by computer can also be quickly solved by computer. The Golden Ticket provides a nontechnical introduction to P-NP, its rich history, and its algorithmic implications for everything we do with computers and beyond. Lance Fortnow traces the history and development of P-NP, giving examples from a variety of disciplines, including economics, physics, and biology. He explores problems that capture the full difficulty of the P-NP dilemma, from discovering the shortest route through all the rides at Disney World to finding large groups of friends on Facebook. The Golden Ticket explores what we truly can and cannot achieve computationally, describing the benefits and unexpected challenges of this compelling problem.

Kernelization

Author : Fedor V. Fomin
Publisher : Cambridge University Press
ISBN 13 : 1107057760
Total Pages : 531 pages
Book Rating : 4.1/5 (7 download)

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Book Synopsis Kernelization by : Fedor V. Fomin

Download or read book Kernelization written by Fedor V. Fomin and published by Cambridge University Press. This book was released on 2019-01-10 with total page 531 pages. Available in PDF, EPUB and Kindle. Book excerpt: A complete introduction to recent advances in preprocessing analysis, or kernelization, with extensive examples using a single data set.

Computational Complexity

Author : Sanjeev Arora
Publisher : Cambridge University Press
ISBN 13 : 0521424267
Total Pages : 609 pages
Book Rating : 4.5/5 (214 download)

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Book Synopsis Computational Complexity by : Sanjeev Arora

Download or read book Computational Complexity written by Sanjeev Arora and published by Cambridge University Press. This book was released on 2009-04-20 with total page 609 pages. Available in PDF, EPUB and Kindle. Book excerpt: New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.

Classification Theory

Author : S. Shelah
Publisher : Elsevier
ISBN 13 : 9780080880242
Total Pages : 740 pages
Book Rating : 4.8/5 (82 download)

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Book Synopsis Classification Theory by : S. Shelah

Download or read book Classification Theory written by S. Shelah and published by Elsevier. This book was released on 1990-12-06 with total page 740 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this research monograph, the author's work on classification and related topics are presented. This revised edition brings the book up to date with the addition of four new chapters as well as various corrections to the 1978 text. The additional chapters X - XIII present the solution to countable first order T of what the author sees as the main test of the theory. In Chapter X the Dimensional Order Property is introduced and it is shown to be a meaningful dividing line for superstable theories. In Chapter XI there is a proof of the decomposition theorems. Chapter XII is the crux of the matter: there is proof that the negation of the assumption used in Chapter XI implies that in models of T a relation can be defined which orders a large subset of m|M|. This theorem is also the subject of Chapter XIII.

A Survey of Lower Bounds for Satisfiability and Related Problems

Author : Dieter van Melkebeek
Publisher : Now Publishers Inc
ISBN 13 : 1601980841
Total Pages : 124 pages
Book Rating : 4.6/5 (19 download)

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Book Synopsis A Survey of Lower Bounds for Satisfiability and Related Problems by : Dieter van Melkebeek

Download or read book A Survey of Lower Bounds for Satisfiability and Related Problems written by Dieter van Melkebeek and published by Now Publishers Inc. This book was released on 2007 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: Surveys the recently discovered lower bounds for the time and space complexity of satisfiability and closely related problems. It overviews the state-of-the-art results on general deterministic, randomized, and quantum models of computation, and presents the underlying arguments in a unified framework.

The Design of Approximation Algorithms

Author : David P. Williamson
Publisher : Cambridge University Press
ISBN 13 : 1139498177
Total Pages : 517 pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis The Design of Approximation Algorithms by : David P. Williamson

Download or read book The Design of Approximation Algorithms written by David P. Williamson and published by Cambridge University Press. This book was released on 2011-04-26 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discrete optimization problems are everywhere, from traditional operations research planning (scheduling, facility location and network design); to computer science databases; to advertising issues in viral marketing. Yet most such problems are NP-hard; unless P = NP, there are no efficient algorithms to find optimal solutions. This book shows how to design approximation algorithms: efficient algorithms that find provably near-optimal solutions. The book is organized around central algorithmic techniques for designing approximation algorithms, including greedy and local search algorithms, dynamic programming, linear and semidefinite programming, and randomization. Each chapter in the first section is devoted to a single algorithmic technique applied to several different problems, with more sophisticated treatment in the second section. The book also covers methods for proving that optimization problems are hard to approximate. Designed as a textbook for graduate-level algorithm courses, it will also serve as a reference for researchers interested in the heuristic solution of discrete optimization problems.

Approximation Algorithms

Author : Vijay V. Vazirani
Publisher : Springer Science & Business Media
ISBN 13 : 3662045656
Total Pages : 380 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Approximation Algorithms by : Vijay V. Vazirani

Download or read book Approximation Algorithms written by Vijay V. Vazirani and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of work in the field. He gives clear, lucid explanations of key results and ideas, with intuitive proofs, and provides critical examples and numerous illustrations to help elucidate the algorithms. Many of the results presented have been simplified and new insights provided. Of interest to theoretical computer scientists, operations researchers, and discrete mathematicians.

WALCOM: Algorithms and Computation

Author : NAOKI KATOH
Publisher : Springer Science & Business Media
ISBN 13 : 3642190936
Total Pages : 243 pages
Book Rating : 4.6/5 (421 download)

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Book Synopsis WALCOM: Algorithms and Computation by : NAOKI KATOH

Download or read book WALCOM: Algorithms and Computation written by NAOKI KATOH and published by Springer Science & Business Media. This book was released on 2011-02-11 with total page 243 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of the 5th International Workshop on Algorithms and Computation, WALCOM 2011, held in New Delhi, India, in February 2011. The 20 papers presented in this volume were carefully reviewed and selected from 57 submissions. The papers are grouped in topical sections on approximation algorithms, hardness, algorithm engineering, computational geometry, string algorithms, and graph algorithms.

Algorithms in Bioinformatics

Author : Dan Brown
Publisher : Springer
ISBN 13 : 3662447533
Total Pages : 379 pages
Book Rating : 4.6/5 (624 download)

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Book Synopsis Algorithms in Bioinformatics by : Dan Brown

Download or read book Algorithms in Bioinformatics written by Dan Brown and published by Springer. This book was released on 2014-08-15 with total page 379 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 14th International Workshop on Algorithms in Bioinformatics, WABI 2014, held in Wroclaw, Poland, in September 2014. WABI 2014 was one of seven conferences that were organized as part of ALGO 2014. WABI is an annual conference series on all aspects of algorithms and data structure in molecular biology, genomics and phylogeny data analysis. The 26 full papers presented together with a short abstract were carefully reviewed and selected from 61 submissions. The selected papers cover a wide range of topics from sequence and genome analysis through phylogeny reconstruction and networks to mass spectrometry data analysis.

Complexity of Lattice Problems

Author : Daniele Micciancio
Publisher : Springer Science & Business Media
ISBN 13 : 1461508975
Total Pages : 229 pages
Book Rating : 4.4/5 (615 download)

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Book Synopsis Complexity of Lattice Problems by : Daniele Micciancio

Download or read book Complexity of Lattice Problems written by Daniele Micciancio and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 229 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lattices are geometric objects that can be pictorially described as the set of intersection points of an infinite, regular n-dimensional grid. De spite their apparent simplicity, lattices hide a rich combinatorial struc ture, which has attracted the attention of great mathematicians over the last two centuries. Not surprisingly, lattices have found numerous ap plications in mathematics and computer science, ranging from number theory and Diophantine approximation, to combinatorial optimization and cryptography. The study of lattices, specifically from a computational point of view, was marked by two major breakthroughs: the development of the LLL lattice reduction algorithm by Lenstra, Lenstra and Lovasz in the early 80's, and Ajtai's discovery of a connection between the worst-case and average-case hardness of certain lattice problems in the late 90's. The LLL algorithm, despite the relatively poor quality of the solution it gives in the worst case, allowed to devise polynomial time solutions to many classical problems in computer science. These include, solving integer programs in a fixed number of variables, factoring polynomials over the rationals, breaking knapsack based cryptosystems, and finding solutions to many other Diophantine and cryptanalysis problems.

Randomization and Approximation Techniques in Computer Science

Author : Jose Rolim
Publisher : Springer Science & Business Media
ISBN 13 : 9783540632481
Total Pages : 240 pages
Book Rating : 4.6/5 (324 download)

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Book Synopsis Randomization and Approximation Techniques in Computer Science by : Jose Rolim

Download or read book Randomization and Approximation Techniques in Computer Science written by Jose Rolim and published by Springer Science & Business Media. This book was released on 1997-06-25 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: Astronomy is the oldest and most fundamental of the natural sciences. From the early beginnings of civilization astronomers have attempted to explain not only what the Universe is and how it works, but also how it started, how it evolved to the present day, and how it will develop in the future. The author, a well-known astronomer himself, describes the evolution of astronomical ideas, briefly discussing most of the instrumental developments. Using numerous figures to elucidate the mechanisms involved, the book starts with the astronomical ideas of the Egyptian and Mesopotamian philosophers, moves on to the Greek period, and then to the golden age of astronomy, i.e. to Copernicus, Galileo, Kepler, and Newton, and ends with modern theories of cosmology. Written with undergraduate students in mind, this book gives a fascinating survey of astronomical thinking.

Algorithms - ESA 2002

Author : Rolf Möhring
Publisher : Springer
ISBN 13 : 3540457496
Total Pages : 919 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Algorithms - ESA 2002 by : Rolf Möhring

Download or read book Algorithms - ESA 2002 written by Rolf Möhring and published by Springer. This book was released on 2003-08-02 with total page 919 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the 74 contributed papers and abstracts of 4 of the 5 invited talks presented at the 10th Annual European Symposium on Algorithms (ESA 2002), held at the University of Rome “La Sapienza”, Rome, Italy, 17-21 September, 2002. For the ?rst time, ESA had two tracks, with separate program committees, which dealt respectively with: – the design and mathematical analysis of algorithms (the “Design and An- ysis” track); – real-world applications, engineering and experimental analysis of algorithms (the “Engineering and Applications” track). Previous ESAs were held in Bad Honnef, Germany (1993); Utrecht, The Neth- lands (1994); Corfu, Greece (1995); Barcelona, Spain (1996); Graz, Austria (1997); Venice, Italy (1998); Prague, Czech Republic (1999); Saarbruc ̈ ken, Ger- ? many (2000), and Arhus, Denmark (2001). The predecessor to the Engineering and Applications track of ESA was the Annual Workshop on Algorithm En- neering (WAE). Previous WAEs were held in Venice, Italy (1997), Saarbruc ̈ ken, ? Germany (1998), London, UK (1999), Saarbru ̈cken, Germany (2000), and Arhus, Denmark (2001). The proceedings of the previous ESAs were published as Springer LNCS volumes 726, 855, 979, 1284, 1461, 1643, 1879, and 2161. The proceedings of WAEs from 1999 onwards were published as Springer LNCS volumes 1668, 1982, and 2161.

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

Author : Maria Serna
Publisher : Springer
ISBN 13 : 3642153690
Total Pages : 794 pages
Book Rating : 4.6/5 (421 download)

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Book Synopsis Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques by : Maria Serna

Download or read book Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques written by Maria Serna and published by Springer. This book was released on 2010-08-27 with total page 794 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the papers presented at the 13th International Wo- shop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2010) and the 14th International Workshop on Randomization and Computation (RANDOM 2010), which took place concurrently in Universitat Politècnica de Catalunya (UPC) Barcelona, Spain, during September 1-3, 2010. APPROX focuses on algorithmic and complexity issues surrounding the dev- opment of e?cient approximate solutions to computationally di?cult problems, and was the 13th in the series after Aalborg (1998), Berkeley (1999), Sa- brücken (2000), Berkeley (2001), Rome (2002), Princeton (2003), Cambridge (2004), Berkeley (2005), Barcelona (2006), Princeton (2007), Boston (2008) and Berkeley (2009). RANDOM is concerned with applications of randomness to computational and combinatorial problems, and was the 14th workshop in the - ries following Bologna (1997), Barcelona (1998), Berkeley (1999), Geneva (2000), Berkeley (2001), Harvard (2002), Princeton (2003), Cambridge (2004), Berkeley (2005), Barcelona (2006), Princeton (2007), Boston (2008), and Berkeley (2009).

Tractability

Author : Lucas Bordeaux
Publisher : Cambridge University Press
ISBN 13 : 110772922X
Total Pages : 401 pages
Book Rating : 4.1/5 (77 download)

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Book Synopsis Tractability by : Lucas Bordeaux

Download or read book Tractability written by Lucas Bordeaux and published by Cambridge University Press. This book was released on 2014-02-06 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classical computer science textbooks tell us that some problems are 'hard'. Yet many areas, from machine learning and computer vision to theorem proving and software verification, have defined their own set of tools for effectively solving complex problems. Tractability provides an overview of these different techniques, and of the fundamental concepts and properties used to tame intractability. This book will help you understand what to do when facing a hard computational problem. Can the problem be modelled by convex, or submodular functions? Will the instances arising in practice be of low treewidth, or exhibit another specific graph structure that makes them easy? Is it acceptable to use scalable, but approximate algorithms? A wide range of approaches is presented through self-contained chapters written by authoritative researchers on each topic. As a reference on a core problem in computer science, this book will appeal to theoreticians and practitioners alike.