LSPACE VS NP IS AS HARD AS P VS NP 1. Introduction in Complexity
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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/. -
Algorithms & Computational Complexity
Computational Complexity O ΩΘ Russell Buehler January 5, 2015 2 Chapter 1 Preface What follows are my personal notes created during my undergraduate algorithms course and later an independent study under Professor Holliday as a graduate student. I am not a complexity theorist; I am a graduate student with some knowledge who is–alas–quite fallible. Accordingly, this text is made available as a convenient reference, set of notes, and summary, but without even the slightest hint of a guarantee that everything contained within is factual and correct. This said, if you find an error, I would much appreciate it if you let me know so that it can be corrected. 3 4 CHAPTER 1. PREFACE Contents 1 Preface 3 2 A Brief Review of Algorithms 1 2.1 Algorithm Analysis............................................1 2.1.1 O-notation............................................1 2.2 Common Structures............................................3 2.2.1 Graphs..............................................3 2.2.2 Trees...............................................4 2.2.3 Networks.............................................4 2.3 Algorithms................................................6 2.3.1 Greedy Algorithms........................................6 2.3.2 Dynamic Programming...................................... 10 2.3.3 Divide and Conquer....................................... 12 2.3.4 Network Flow.......................................... 15 2.4 Data Structures.............................................. 17 3 Deterministic Computation 19 3.1 O-Notation............................................... -
Complexity Theory in Computer Science Pdf
Complexity theory in computer science pdf Continue Complexity is often used to describe an algorithm. One could hear something like my sorting algorithm working in n2n'2n2 time in complexity, it's written as O (n2)O (n'2)O (n2) and polynomial work time. Complexity is used to describe the use of resources in algorithms. In general, the resources of concern are time and space. The complexity of the algorithm's time is the number of steps it must take to complete. The cosmic complexity of the algorithm represents the amount of memory the algorithm needs to work. The complexity of the algorithm's time describes how many steps the algorithm should take with regard to input. If, for example, each of the nnn elements entered only works once, this algorithm will take time O'n)O(n)O(n). If, for example, the algorithm has to work on one input element (regardless of input size), it is a constant time, or O(1)O(1)O(1), the algorithm, because regardless of the size of the input, only one is done. If the algorithm performs nnn-operations for each of the nnn elements injected into the algorithm, then this algorithm is performed in time O'n2)O (n2). In the design and analysis of algorithms, there are three types of complexities that computer scientists think: the best case, the worst case, and the complexity of the average case. The best, worst and medium complexity complexity can describe the time and space this wiki will talk about in terms of the complexity of time, but the same concepts can be applied to the complexity of space. -
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 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. -
The Classes FNP and TFNP
Outline The classes FNP and TFNP C. Wilson1 1Lane Department of Computer Science and Electrical Engineering West Virginia University Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Outline Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Function Problems What are Function Problems? Function Problems FSAT Defined Total Functions TSP Defined Outline 1 Function Problems defined What are Function Problems? FSAT Defined TSP Defined 2 Relationship between Function and Decision Problems RL Defined Reductions between Function Problems 3 Total Functions Defined Total Functions Defined FACTORING HAPPYNET ANOTHER HAMILTON CYCLE Christopher Wilson Function Problems Function -
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. -
Total Functions in the Polynomial Hierarchy 11
Electronic Colloquium on Computational Complexity, Report No. 153 (2020) TOTAL FUNCTIONS IN THE POLYNOMIAL HIERARCHY ROBERT KLEINBERG∗, DANIEL MITROPOLSKYy, AND CHRISTOS PAPADIMITRIOUy Abstract. We identify several genres of search problems beyond NP for which existence of solutions is guaranteed. One class that seems especially rich in such problems is PEPP (for “polynomial empty pigeonhole principle”), which includes problems related to existence theorems proved through the union bound, such as finding a bit string that is far from all codewords, finding an explicit rigid matrix, as well as a problem we call Complexity, capturing Complexity Theory’s quest. When the union bound is generous, in that solutions constitute at least a polynomial fraction of the domain, we have a family of seemingly weaker classes α-PEPP, which are inside FPNPjpoly. Higher in the hierarchy, we identify the constructive version of the Sauer- Shelah lemma and the appropriate generalization of PPP that contains it. The resulting total function hierarchy turns out to be more stable than the polynomial hierarchy: it is known that, under oracles, total functions within FNP may be easy, but total functions a level higher may still be harder than FPNP. 1. Introduction The complexity of total functions has emerged over the past three decades as an intriguing and productive branch of Complexity Theory. Subclasses of TFNP, the set of all total functions in FNP, have been defined and studied: PLS, PPP, PPA, PPAD, PPADS, and CLS. These classes are replete with natural problems, several of which turned out to be complete for the corresponding class, see e.g. -
Logic and Complexity
o| f\ Richard Lassaigne and Michel de Rougemont Logic and Complexity Springer Contents Introduction Part 1. Basic model theory and computability 3 Chapter 1. Prepositional logic 5 1.1. Propositional language 5 1.1.1. Construction of formulas 5 1.1.2. Proof by induction 7 1.1.3. Decomposition of a formula 7 1.2. Semantics 9 1.2.1. Tautologies. Equivalent formulas 10 1.2.2. Logical consequence 11 1.2.3. Value of a formula and substitution 11 1.2.4. Complete systems of connectives 15 1.3. Normal forms 15 1.3.1. Disjunctive and conjunctive normal forms 15 1.3.2. Functions associated to formulas 16 1.3.3. Transformation methods 17 1.3.4. Clausal form 19 1.3.5. OBDD: Ordered Binary Decision Diagrams 20 1.4. Exercises 23 Chapter 2. Deduction systems 25 2.1. Examples of tableaux 25 2.2. Tableaux method 27 2.2.1. Trees 28 2.2.2. Construction of tableaux 29 2.2.3. Development and closure 30 2.3. Completeness theorem 31 2.3.1. Provable formulas 31 2.3.2. Soundness 31 2.3.3. Completeness 32 2.4. Natural deduction 33 2.5. Compactness theorem 36 2.6. Exercices 38 vi CONTENTS Chapter 3. First-order logic 41 3.1. First-order languages 41 3.1.1. Construction of terms 42 3.1.2. Construction of formulas 43 3.1.3. Free and bound variables 44 3.2. Semantics 45 3.2.1. Structures and languages 45 3.2.2. Structures and satisfaction of formulas 46 3.2.3. -
Optimisation Function Problems FNP and FP
Complexity Theory 99 Complexity Theory 100 Optimisation The Travelling Salesman Problem was originally conceived of as an This is still reasonable, as we are establishing the difficulty of the optimisation problem problems. to find a minimum cost tour. A polynomial time solution to the optimisation version would give We forced it into the mould of a decision problem { TSP { in order a polynomial time solution to the decision problem. to fit it into our theory of NP-completeness. Also, a polynomial time solution to the decision problem would allow a polynomial time algorithm for finding the optimal value, CLIQUE Similar arguments can be made about the problems and using binary search, if necessary. IND. Anuj Dawar May 21, 2007 Anuj Dawar May 21, 2007 Complexity Theory 101 Complexity Theory 102 Function Problems FNP and FP Still, there is something interesting to be said for function problems A function which, for any given Boolean expression φ, gives a arising from NP problems. satisfying truth assignment if φ is satisfiable, and returns \no" Suppose otherwise, is a witness function for SAT. L = fx j 9yR(x; y)g If any witness function for SAT is computable in polynomial time, where R is a polynomially-balanced, polynomial time decidable then P = NP. relation. If P = NP, then for every language in NP, some witness function is A witness function for L is any function f such that: computable in polynomial time, by a binary search algorithm. • if x 2 L, then f(x) = y for some y such that R(x; y); P = NP if, and only if, FNP = FP • f(x) = \no" otherwise. -
On Total Functions, Existence Theorems and Computational Complexity
Theoretical Computer Science 81 (1991) 317-324 Elsevier Note On total functions, existence theorems and computational complexity Nimrod Megiddo IBM Almaden Research Center, 650 Harry Road, Sun Jose, CA 95120-6099, USA, and School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel Christos H. Papadimitriou* Computer Technology Institute, Patras, Greece, and University of California at Sun Diego, CA, USA Communicated by M. Nivat Received October 1989 Abstract Megiddo, N. and C.H. Papadimitriou, On total functions, existence theorems and computational complexity (Note), Theoretical Computer Science 81 (1991) 317-324. wondeterministic multivalued functions with values that are polynomially verifiable and guaran- teed to exist form an interesting complexity class between P and NP. We show that this class, which we call TFNP, contains a host of important problems, whose membership in P is currently not known. These include, besides factoring, local optimization, Brouwer's fixed points, a computa- tional version of Sperner's Lemma, bimatrix equilibria in games, and linear complementarity for P-matrices. 1. The class TFNP Let 2 be an alphabet with two or more symbols, and suppose that R G E*x 2" is a polynomial-time recognizable relation which is polynomially balanced, that is, (x,y) E R implies that lyl sp(lx()for some polynomial p. * Research supported by an ESPRIT Basic Research Project, a grant to the Universities of Patras and Bonn by the Volkswagen Foundation, and an NSF Grant. Research partially performed while the author was visiting the IBM Almaden Research Center. 0304-3975/91/$03.50 @ 1991-Elsevier Science Publishers B.V.