On Total Functions, Existence Theorems and Computational Complexity
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Computational Complexity: a Modern Approach
i Computational Complexity: A Modern Approach Draft of a book: Dated January 2007 Comments welcome! Sanjeev Arora and Boaz Barak Princeton University [email protected] Not to be reproduced or distributed without the authors’ permission This is an Internet draft. Some chapters are more finished than others. References and attributions are very preliminary and we apologize in advance for any omissions (but hope you will nevertheless point them out to us). Please send us bugs, typos, missing references or general comments to [email protected] — Thank You!! DRAFT ii DRAFT Chapter 9 Complexity of counting “It is an empirical fact that for many combinatorial problems the detection of the existence of a solution is easy, yet no computationally efficient method is known for counting their number.... for a variety of problems this phenomenon can be explained.” L. Valiant 1979 The class NP captures the difficulty of finding certificates. However, in many contexts, one is interested not just in a single certificate, but actually counting the number of certificates. This chapter studies #P, (pronounced “sharp p”), a complexity class that captures this notion. Counting problems arise in diverse fields, often in situations having to do with estimations of probability. Examples include statistical estimation, statistical physics, network design, and more. Counting problems are also studied in a field of mathematics called enumerative combinatorics, which tries to obtain closed-form mathematical expressions for counting problems. To give an example, in the 19th century Kirchoff showed how to count the number of spanning trees in a graph using a simple determinant computation. Results in this chapter will show that for many natural counting problems, such efficiently computable expressions are unlikely to exist. -
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]. -
Paradigms of Combinatorial Optimization
W657-Paschos 2.qxp_Layout 1 01/07/2014 14:05 Page 1 MATHEMATICS AND STATISTICS SERIES Vangelis Th. Paschos Vangelis Edited by This updated and revised 2nd edition of the three-volume Combinatorial Optimization series covers a very large set of topics in this area, dealing with fundamental notions and approaches as well as several classical applications of Combinatorial Optimization. Combinatorial Optimization is a multidisciplinary field, lying at the interface of three major scientific domains: applied mathematics, theoretical computer science, and management studies. Its focus is on finding the least-cost solution to a mathematical problem in which each solution is associated with a numerical cost. In many such problems, exhaustive search is not feasible, so the approach taken is to operate within the domain of optimization problems, in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution. Some common problems involving combinatorial optimization are the traveling salesman problem and the Combinatorial Optimization minimum spanning tree problem. Combinatorial Optimization is a subset of optimization that is related to operations research, algorithm theory, and computational complexity theory. It 2 has important applications in several fields, including artificial intelligence, nd mathematics, and software engineering. Edition Revised and Updated This second volume, which addresses the various paradigms and approaches taken in Combinatorial Optimization, is divided into two parts: Paradigms of - Paradigmatic Problems, which discusses several famous combinatorial optimization problems, such as max cut, min coloring, optimal satisfiability TSP, Paradigms of etc., the study of which has largely contributed to the development, the legitimization and the establishment of Combinatorial Optimization as one of the most active current scientific domains. -
Solving Non-Boolean Satisfiability Problems with Stochastic Local Search
Solving Non-Boolean Satisfiability Problems with Stochastic Local Search: A Comparison of Encodings ALAN M. FRISCH †, TIMOTHY J. PEUGNIEZ, ANTHONY J. DOGGETT and PETER W. NIGHTINGALE§ Artificial Intelligence Group, Department of Computer Science, University of York, York YO10 5DD, UK. e-mail: [email protected], [email protected] Abstract. Much excitement has been generated by the success of stochastic local search procedures at finding solutions to large, very hard satisfiability problems. Many of the problems on which these procedures have been effective are non-Boolean in that they are most naturally formulated in terms of variables with domain sizes greater than two. Approaches to solving non-Boolean satisfiability problems fall into two categories. In the direct approach, the problem is tackled by an algorithm for non-Boolean problems. In the transformation approach, the non-Boolean problem is reformulated as an equivalent Boolean problem and then a Boolean solver is used. This paper compares four methods for solving non-Boolean problems: one di- rect and three transformational. The comparison first examines the search spaces confronted by the four methods then tests their ability to solve random formulas, the round-robin sports scheduling problem and the quasigroup completion problem. The experiments show that the relative performance of the methods depends on the domain size of the problem, and that the direct method scales better as domain size increases. Along the route to performing these comparisons we make three other contri- butions. First, we generalise Walksat, a highly-successful stochastic local search procedure for Boolean satisfiability problems, to work on problems with domains of any finite size. -
User's Guide for Complexity: a LATEX Package, Version 0.80
User’s Guide for complexity: a LATEX package, Version 0.80 Chris Bourke April 12, 2007 Contents 1 Introduction 2 1.1 What is complexity? ......................... 2 1.2 Why a complexity package? ..................... 2 2 Installation 2 3 Package Options 3 3.1 Mode Options .............................. 3 3.2 Font Options .............................. 4 3.2.1 The small Option ....................... 4 4 Using the Package 6 4.1 Overridden Commands ......................... 6 4.2 Special Commands ........................... 6 4.3 Function Commands .......................... 6 4.4 Language Commands .......................... 7 4.5 Complete List of Class Commands .................. 8 5 Customization 15 5.1 Class Commands ............................ 15 1 5.2 Language Commands .......................... 16 5.3 Function Commands .......................... 17 6 Extended Example 17 7 Feedback 18 7.1 Acknowledgements ........................... 19 1 Introduction 1.1 What is complexity? complexity is a LATEX package that typesets computational complexity classes such as P (deterministic polynomial time) and NP (nondeterministic polynomial time) as well as sets (languages) such as SAT (satisfiability). In all, over 350 commands are defined for helping you to typeset Computational Complexity con- structs. 1.2 Why a complexity package? A better question is why not? Complexity theory is a more recent, though mature area of Theoretical Computer Science. Each researcher seems to have his or her own preferences as to how to typeset Complexity Classes and has built up their own personal LATEX commands file. This can be frustrating, to say the least, when it comes to collaborations or when one has to go through an entire series of files changing commands for compatibility or to get exactly the look they want (or what may be required). -
Random Θ(Log N)-Cnfs Are Hard for Cutting Planes
Random Θ(log n)-CNFs are Hard for Cutting Planes Noah Fleming Denis Pankratov Toniann Pitassi Robert Robere University of Toronto University of Toronto University of Toronto University of Toronto noahfl[email protected] [email protected] [email protected] [email protected] Abstract—The random k-SAT model is the most impor- lower bounds for random k-SAT formulas in a particular tant and well-studied distribution over k-SAT instances. It is proof system show that any complete and efficient algorithm closely connected to statistical physics and is a benchmark for based on the proof system will perform badly on random satisfiability algorithms. We show that when k = Θ(log n), any Cutting Planes refutation for random k-SAT requires k-SAT instances. Furthermore, since the proof complexity exponential size in the interesting regime where the number of lower bounds hold in the unsatisfiable regime, they are clauses guarantees that the formula is unsatisfiable with high directly connected to Feige’s hypothesis. probability. Remarkably, determining whether or not a random SAT Keywords-Proof complexity; random k-SAT; Cutting Planes; instance from the distribution F(m; n; k) is satisfiable is controlled quite precisely by the ratio ∆ = m=n, which is called the clause density. A simple counting argument shows I. INTRODUCTION that F(m; n; k) is unsatisfiable with high probability for The Satisfiability (SAT) problem is perhaps the most ∆ > 2k ln 2. The famous satisfiability threshold conjecture famous problem in theoretical computer science, and sig- asserts that there is a constant ck such that random k-SAT nificant effort has been devoted to understanding randomly formulas of clause density ∆ are almost certainly satisfiable generated SAT instances. -
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 -
Lecture: Complexity of Finding a Nash Equilibrium 1 Computational
Algorithmic Game Theory Lecture Date: September 20, 2011 Lecture: Complexity of Finding a Nash Equilibrium Lecturer: Christos Papadimitriou Scribe: Miklos Racz, Yan Yang In this lecture we are going to talk about the complexity of finding a Nash Equilibrium. In particular, the class of problems NP is introduced and several important subclasses are identified. Most importantly, we are going to prove that finding a Nash equilibrium is PPAD-complete (defined in Section 2). For an expository article on the topic, see [4], and for a more detailed account, see [5]. 1 Computational complexity For us the best way to think about computational complexity is that it is about search problems.Thetwo most important classes of search problems are P and NP. 1.1 The complexity class NP NP stands for non-deterministic polynomial. It is a class of problems that are at the core of complexity theory. The classical definition is in terms of yes-no problems; here, we are concerned with the search problem form of the definition. Definition 1 (Complexity class NP). The class of all search problems. A search problem A is a binary predicate A(x, y) that is efficiently (in polynomial time) computable and balanced (the length of x and y do not differ exponentially). Intuitively, x is an instance of the problem and y is a solution. The search problem for A is this: “Given x,findy such that A(x, y), or if no such y exists, say “no”.” The class of all search problems is called NP. Examples of NP problems include the following two. -
The Relative Complexity of NP Search Problems
Journal of Computer and System Sciences 57, 319 (1998) Article No. SS981575 The Relative Complexity of NP Search Problems Paul Beame* Computer Science and Engineering, University of Washington, Box 352350, Seattle, Washington 98195-2350 E-mail: beameÄcs.washington.edu Stephen Cook- Computer Science Department, University of Toronto, Canada M5S 3G4 E-mail: sacookÄcs.toronto.edu Jeff Edmonds Department of Computer Science, York University, Toronto, Ontario, Canada M3J 1P3 E-mail: jeffÄcs.yorku.ca Russell Impagliazzo9 Computer Science and Engineering, UC, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0114 E-mail: russellÄcs.ucsd.edu and Toniann PitassiÄ Department of Computer Science, University of Arizona, Tucson, Arizona 85721-0077 E-mail: toniÄcs.arizona.edu Received January 1998 1. INTRODUCTION Papadimitriou introduced several classes of NP search problems based on combinatorial principles which guarantee the existence of In the study of computational complexity, there are many solutions to the problems. Many interesting search problems not known to be solvable in polynomial time are contained in these classes, problems that are naturally expressed as problems ``to find'' and a number of them are complete problems. We consider the question but are converted into decision problems to fit into standard of the relative complexity of these search problem classes. We prove complexity classes. For example, a more natural problem several separations which show that in a generic relativized world the than determining whether or not a graph is 3-colorable search classes are distinct and there is a standard search problem in might be that of finding a 3-coloring of the graph if it exists. -
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. -
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.