Knowledge, Creativity and P Versus NP
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
The Top Eight Misconceptions About NP- Hardness
The top eight misconceptions about NP- hardness Zoltán Ádám Mann Budapest University of Technology and Economics Department of Computer Science and Information Theory Budapest, Hungary Abstract: The notions of NP-completeness and NP-hardness are used frequently in the computer science literature, but unfortunately, often in a wrong way. The aim of this paper is to show the most wide-spread misconceptions, why they are wrong, and why we should care. Keywords: F.1.3 Complexity Measures and Classes; F.2 Analysis of Algorithms and Problem Complexity; G.2.1.a Combinatorial algorithms; I.1.2.c Analysis of algorithms Published in IEEE Computer, volume 50, issue 5, pages 72-79, May 2017 1. Introduction Recently, I have been working on a survey about algorithmic approaches to virtual machine allocation in cloud data centers [11]. While doing so, I was shocked (again) to see how often the notions of NP-completeness or NP-hardness are used in a faulty, inadequate, or imprecise way in the literature. Here is an example from an – otherwise excellent – paper that appeared recently in IEEE Transactions on Computers, one of the most prestigious journals of the field: “Since the problem is NP-hard, no polynomial time optimal algorithm exists.” This claim, if it were shown to be true, would finally settle the famous P versus NP problem, which is one of the seven Millenium Prize Problems [10]; thus correctly solving it would earn the authors 1 million dollars. But of course, the authors do not attack this problem; the above claim is just an error in the paper. -
The Complexity Zoo
The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/. -
LSPACE VS NP IS AS HARD AS P VS NP 1. Introduction in Complexity
LSPACE VS NP IS AS HARD AS P VS NP FRANK VEGA Abstract. The P versus NP problem is a major unsolved problem in com- puter science. This consists in knowing the answer of the following question: Is P equal to NP? Another major complexity classes are LSPACE, PSPACE, ESPACE, E and EXP. Whether LSPACE = P is a fundamental question that it is as important as it is unresolved. We show if P = NP, then LSPACE = NP. Consequently, if LSPACE is not equal to NP, then P is not equal to NP. According to Lance Fortnow, it seems that LSPACE versus NP is easier to be proven. However, with this proof we show this problem is as hard as P versus NP. Moreover, we prove the complexity class P is not equal to PSPACE as a direct consequence of this result. Furthermore, we demonstrate if PSPACE is not equal to EXP, then P is not equal to NP. In addition, if E = ESPACE, then P is not equal to NP. 1. Introduction In complexity theory, a function problem is a computational problem where a single output is expected for every input, but the output is more complex than that of a decision problem [6]. A functional problem F is defined as a binary relation (x; y) 2 R over strings of an arbitrary alphabet Σ: R ⊂ Σ∗ × Σ∗: A Turing machine M solves F if for every input x such that there exists a y satisfying (x; y) 2 R, M produces one such y, that is M(x) = y [6]. -
QP Versus NP
QP versus NP Frank Vega Joysonic, Belgrade, Serbia [email protected] Abstract. Given an instance of XOR 2SAT and a positive integer 2K , the problem exponential exclusive-or 2-satisfiability consists in deciding whether this Boolean formula has a truth assignment with at leat K satisfiable clauses. We prove exponential exclusive-or 2-satisfiability is in QP and NP-complete. In this way, we show QP ⊆ NP . Keywords: Complexity Classes · Completeness · Logarithmic Space · Nondeterministic · XOR 2SAT. 1 Introduction The P versus NP problem is a major unsolved problem in computer science [4]. This is considered by many to be the most important open problem in the field [4]. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$1,000,000 prize for the first correct solution [4]. It was essentially mentioned in 1955 from a letter written by John Nash to the United States National Security Agency [1]. However, the precise statement of the P = NP problem was introduced in 1971 by Stephen Cook in a seminal paper [4]. In 1936, Turing developed his theoretical computational model [14]. The deterministic and nondeterministic Turing machines have become in two of the most important definitions related to this theoretical model for computation [14]. A deterministic Turing machine has only one next action for each step defined in its program or transition function [14]. A nondeterministic Turing machine could contain more than one action defined for each step of its program, where this one is no longer a function, but a relation [14]. -
A Decade of Lattice Cryptography
Full text available at: http://dx.doi.org/10.1561/0400000074 A Decade of Lattice Cryptography Chris Peikert Computer Science and Engineering University of Michigan, United States Boston — Delft Full text available at: http://dx.doi.org/10.1561/0400000074 Foundations and Trends R in Theoretical Computer Science Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 United States Tel. +1-781-985-4510 www.nowpublishers.com [email protected] Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 The preferred citation for this publication is C. Peikert. A Decade of Lattice Cryptography. Foundations and Trends R in Theoretical Computer Science, vol. 10, no. 4, pp. 283–424, 2014. R This Foundations and Trends issue was typeset in LATEX using a class file designed by Neal Parikh. Printed on acid-free paper. ISBN: 978-1-68083-113-9 c 2016 C. Peikert All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. Photocopying. In the USA: This journal is registered at the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923. Authorization to photocopy items for in- ternal or personal use, or the internal or personal use of specific clients, is granted by now Publishers Inc for users registered with the Copyright Clearance Center (CCC). The ‘services’ for users can be found on the internet at: www.copyright.com For those organizations that have been granted a photocopy license, a separate system of payment has been arranged. -
The Best Nurturers in Computer Science Research
The Best Nurturers in Computer Science Research Bharath Kumar M. Y. N. Srikant IISc-CSA-TR-2004-10 http://archive.csa.iisc.ernet.in/TR/2004/10/ Computer Science and Automation Indian Institute of Science, India October 2004 The Best Nurturers in Computer Science Research Bharath Kumar M.∗ Y. N. Srikant† Abstract The paper presents a heuristic for mining nurturers in temporally organized collaboration networks: people who facilitate the growth and success of the young ones. Specifically, this heuristic is applied to the computer science bibliographic data to find the best nurturers in computer science research. The measure of success is parameterized, and the paper demonstrates experiments and results with publication count and citations as success metrics. Rather than just the nurturer’s success, the heuristic captures the influence he has had in the indepen- dent success of the relatively young in the network. These results can hence be a useful resource to graduate students and post-doctoral can- didates. The heuristic is extended to accurately yield ranked nurturers inside a particular time period. Interestingly, there is a recognizable deviation between the rankings of the most successful researchers and the best nurturers, which although is obvious from a social perspective has not been statistically demonstrated. Keywords: Social Network Analysis, Bibliometrics, Temporal Data Mining. 1 Introduction Consider a student Arjun, who has finished his under-graduate degree in Computer Science, and is seeking a PhD degree followed by a successful career in Computer Science research. How does he choose his research advisor? He has the following options with him: 1. Look up the rankings of various universities [1], and apply to any “rea- sonably good” professor in any of the top universities. -
Constraint Based Dimension Correlation and Distance
Preface The papers in this volume were presented at the Fourteenth Annual IEEE Conference on Computational Complexity held from May 4-6, 1999 in Atlanta, Georgia, in conjunction with the Federated Computing Research Conference. This conference was sponsored by the IEEE Computer Society Technical Committee on Mathematical Foundations of Computing, in cooperation with the ACM SIGACT (The special interest group on Algorithms and Complexity Theory) and EATCS (The European Association for Theoretical Computer Science). The call for papers sought original research papers in all areas of computational complexity. A total of 70 papers were submitted for consideration of which 28 papers were accepted for the conference and for inclusion in these proceedings. Six of these papers were accepted to a joint STOC/Complexity session. For these papers the full conference paper appears in the STOC proceedings and a one-page summary appears in these proceedings. The program committee invited two distinguished researchers in computational complexity - Avi Wigderson and Jin-Yi Cai - to present invited talks. These proceedings contain survey articles based on their talks. The program committee thanks Pradyut Shah and Marcus Schaefer for their organizational and computer help, Steve Tate and the SIGACT Electronic Publishing Board for the use and help of the electronic submissions server, Peter Shor and Mike Saks for the electronic conference meeting software and Danielle Martin of the IEEE for editing this volume. The committee would also like to thank the following people for their help in reviewing the papers: E. Allender, V. Arvind, M. Ajtai, A. Ambainis, G. Barequet, S. Baumer, A. Berthiaume, S. -
A Short History of Computational Complexity
The Computational Complexity Column by Lance FORTNOW NEC Laboratories America 4 Independence Way, Princeton, NJ 08540, USA [email protected] http://www.neci.nj.nec.com/homepages/fortnow/beatcs Every third year the Conference on Computational Complexity is held in Europe and this summer the University of Aarhus (Denmark) will host the meeting July 7-10. More details at the conference web page http://www.computationalcomplexity.org This month we present a historical view of computational complexity written by Steve Homer and myself. This is a preliminary version of a chapter to be included in an upcoming North-Holland Handbook of the History of Mathematical Logic edited by Dirk van Dalen, John Dawson and Aki Kanamori. A Short History of Computational Complexity Lance Fortnow1 Steve Homer2 NEC Research Institute Computer Science Department 4 Independence Way Boston University Princeton, NJ 08540 111 Cummington Street Boston, MA 02215 1 Introduction It all started with a machine. In 1936, Turing developed his theoretical com- putational model. He based his model on how he perceived mathematicians think. As digital computers were developed in the 40's and 50's, the Turing machine proved itself as the right theoretical model for computation. Quickly though we discovered that the basic Turing machine model fails to account for the amount of time or memory needed by a computer, a critical issue today but even more so in those early days of computing. The key idea to measure time and space as a function of the length of the input came in the early 1960's by Hartmanis and Stearns. -
Interactions of Computational Complexity Theory and Mathematics
Interactions of Computational Complexity Theory and Mathematics Avi Wigderson October 22, 2017 Abstract [This paper is a (self contained) chapter in a new book on computational complexity theory, called Mathematics and Computation, whose draft is available at https://www.math.ias.edu/avi/book]. We survey some concrete interaction areas between computational complexity theory and different fields of mathematics. We hope to demonstrate here that hardly any area of modern mathematics is untouched by the computational connection (which in some cases is completely natural and in others may seem quite surprising). In my view, the breadth, depth, beauty and novelty of these connections is inspiring, and speaks to a great potential of future interactions (which indeed, are quickly expanding). We aim for variety. We give short, simple descriptions (without proofs or much technical detail) of ideas, motivations, results and connections; this will hopefully entice the reader to dig deeper. Each vignette focuses only on a single topic within a large mathematical filed. We cover the following: • Number Theory: Primality testing • Combinatorial Geometry: Point-line incidences • Operator Theory: The Kadison-Singer problem • Metric Geometry: Distortion of embeddings • Group Theory: Generation and random generation • Statistical Physics: Monte-Carlo Markov chains • Analysis and Probability: Noise stability • Lattice Theory: Short vectors • Invariant Theory: Actions on matrix tuples 1 1 introduction The Theory of Computation (ToC) lays out the mathematical foundations of computer science. I am often asked if ToC is a branch of Mathematics, or of Computer Science. The answer is easy: it is clearly both (and in fact, much more). Ever since Turing's 1936 definition of the Turing machine, we have had a formal mathematical model of computation that enables the rigorous mathematical study of computational tasks, algorithms to solve them, and the resources these require. -
Notes for Lecture 1
Stanford University | CS254: Computational Complexity Notes 1 Luca Trevisan January 7, 2014 Notes for Lecture 1 This course assumes CS154, or an equivalent course on automata, computability and computational complexity, as a prerequisite. We will assume that the reader is familiar with the notions of algorithm, running time, and of polynomial time reduction, as well as with basic notions of discrete math and probability. We will occasionally refer to Turing machines. In this lecture we give an (incomplete) overview of the field of Computational Complexity and of the topics covered by this course. Computational complexity is the part of theoretical computer science that studies 1. Impossibility results (lower bounds) for computations. Eventually, one would like the theory to prove its major conjectured lower bounds, and in particular, to prove Conjecture 1P 6= NP, implying that thousands of natural combinatorial problems admit no polynomial time algorithm 2. Relations between the power of different computational resources (time, memory ran- domness, communications) and the difficulties of different modes of computations (exact vs approximate, worst case versus average-case, etc.). A representative open problem of this type is Conjecture 2P = BPP, meaning that every polynomial time randomized algorithm admits a polynomial time deterministic simulation. Ultimately, one would like complexity theory to not only answer asymptotic worst-case complexity questions such as the P versus NP problem, but also address the complexity of specific finite-size instances, on average as well as in worst-case. Ideally, the theory would be eventually able to prove results such as • The smallest boolean circuit that solves 3SAT on formulas with 300 variables has size more than 270, or • The smallest boolean circuit that can factor more than half of the 2000-digit integers, has size more than 280. -
Optimal Hitting Sets for Combinatorial Shapes
Optimal Hitting Sets for Combinatorial Shapes Aditya Bhaskara∗ Devendra Desai† Srikanth Srinivasan‡ November 5, 2018 Abstract We consider the problem of constructing explicit Hitting sets for Combinatorial Shapes, a class of statistical tests first studied by Gopalan, Meka, Reingold, and Zuckerman (STOC 2011). These generalize many well-studied classes of tests, including symmetric functions and combi- natorial rectangles. Generalizing results of Linial, Luby, Saks, and Zuckerman (Combinatorica 1997) and Rabani and Shpilka (SICOMP 2010), we construct hitting sets for Combinatorial Shapes of size polynomial in the alphabet, dimension, and the inverse of the error parame- ter. This is optimal up to polynomial factors. The best previous hitting sets came from the Pseudorandom Generator construction of Gopalan et al., and in particular had size that was quasipolynomial in the inverse of the error parameter. Our construction builds on natural variants of the constructions of Linial et al. and Rabani and Shpilka. In the process, we construct fractional perfect hash families and hitting sets for combinatorial rectangles with stronger guarantees. These might be of independent interest. 1 Introduction Randomness is a tool of great importance in Computer Science and combinatorics. The probabilistic method is highly effective both in the design of simple and efficient algorithms and in demonstrating the existence of combinatorial objects with interesting properties. But the use of randomness also comes with some disadvantages. In the setting of algorithms, introducing randomness adds to the number of resource requirements of the algorithm, since truly random bits are hard to come by. For combinatorial constructions, ‘explicit’ versions of these objects often turn out to have more structure, which yields advantages beyond the mere fact of their existence (e.g., we know of explicit arXiv:1211.3439v1 [cs.CC] 14 Nov 2012 error-correcting codes that can be efficiently encoded and decoded, but we don’t know if random codes can [5]). -
The P Versus Np Problem
THE P VERSUS NP PROBLEM STEPHEN COOK 1. Statement of the Problem The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a computer. The standard computer model in computability theory is the Turing machine, introduced by Alan Turing in 1936 [37]. Although the model was introduced before physical computers were built, it nevertheless continues to be accepted as the proper computer model for the purpose of defining the notion of computable function. Informally the class P is the class of decision problems solvable by some algorithm within a number of steps bounded by some fixed polynomial in the length of the input. Turing was not concerned with the efficiency of his machines, rather his concern was whether they can simulate arbitrary algorithms given sufficient time. It turns out, however, Turing machines can generally simulate more efficient computer models (for example, machines equipped with many tapes or an unbounded random access memory) by at most squaring or cubing the computation time. Thus P is a robust class and has equivalent definitions over a large class of computer models. Here we follow standard practice and define the class P in terms of Turing machines. Formally the elements of the class P are languages. Let Σ be a finite alphabet (that is, a finite nonempty set) with at least two elements, and let Σ∗ be the set of finite strings over Σ.