P, NP, and NP-Completeness: the Basics of Computational Complexity
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Complexity Theory Lecture 9 Co-NP Co-NP-Complete
Complexity Theory 1 Complexity Theory 2 co-NP Complexity Theory Lecture 9 As co-NP is the collection of complements of languages in NP, and P is closed under complementation, co-NP can also be characterised as the collection of languages of the form: ′ L = x y y <p( x ) R (x, y) { |∀ | | | | → } Anuj Dawar University of Cambridge Computer Laboratory NP – the collection of languages with succinct certificates of Easter Term 2010 membership. co-NP – the collection of languages with succinct certificates of http://www.cl.cam.ac.uk/teaching/0910/Complexity/ disqualification. Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 3 Complexity Theory 4 NP co-NP co-NP-complete P VAL – the collection of Boolean expressions that are valid is co-NP-complete. Any language L that is the complement of an NP-complete language is co-NP-complete. Any of the situations is consistent with our present state of ¯ knowledge: Any reduction of a language L1 to L2 is also a reduction of L1–the complement of L1–to L¯2–the complement of L2. P = NP = co-NP • There is an easy reduction from the complement of SAT to VAL, P = NP co-NP = NP = co-NP • ∩ namely the map that takes an expression to its negation. P = NP co-NP = NP = co-NP • ∩ VAL P P = NP = co-NP ∈ ⇒ P = NP co-NP = NP = co-NP • ∩ VAL NP NP = co-NP ∈ ⇒ Anuj Dawar May 14, 2010 Anuj Dawar May 14, 2010 Complexity Theory 5 Complexity Theory 6 Prime Numbers Primality Consider the decision problem PRIME: Another way of putting this is that Composite is in NP. -
If Np Languages Are Hard on the Worst-Case, Then It Is Easy to Find Their Hard Instances
IF NP LANGUAGES ARE HARD ON THE WORST-CASE, THEN IT IS EASY TO FIND THEIR HARD INSTANCES Dan Gutfreund, Ronen Shaltiel, and Amnon Ta-Shma Abstract. We prove that if NP 6⊆ BPP, i.e., if SAT is worst-case hard, then for every probabilistic polynomial-time algorithm trying to decide SAT, there exists some polynomially samplable distribution that is hard for it. That is, the algorithm often errs on inputs from this distribution. This is the ¯rst worst-case to average-case reduction for NP of any kind. We stress however, that this does not mean that there exists one ¯xed samplable distribution that is hard for all probabilistic polynomial-time algorithms, which is a pre-requisite assumption needed for one-way func- tions and cryptography (even if not a su±cient assumption). Neverthe- less, we do show that there is a ¯xed distribution on instances of NP- complete languages, that is samplable in quasi-polynomial time and is hard for all probabilistic polynomial-time algorithms (unless NP is easy in the worst case). Our results are based on the following lemma that may be of independent interest: Given the description of an e±cient (probabilistic) algorithm that fails to solve SAT in the worst case, we can e±ciently generate at most three Boolean formulae (of increasing lengths) such that the algorithm errs on at least one of them. Keywords. Average-case complexity, Worst-case to average-case re- ductions, Foundations of cryptography, Pseudo classes Subject classi¯cation. 68Q10 (Modes of computation (nondetermin- istic, parallel, interactive, probabilistic, etc.) 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q17 Compu- tational di±culty of problems (lower bounds, completeness, di±culty of approximation, etc.) 94A60 Cryptography 2 Gutfreund, Shaltiel & Ta-Shma 1. -
Randomised Computation 1 TM Taking Advices 2 Karp-Lipton Theorem
INFR11102: Computational Complexity 29/10/2019 Lecture 13: More on circuit models; Randomised Computation Lecturer: Heng Guo 1 TM taking advices An alternative way to characterize P=poly is via TMs that take advices. Definition 1. For functions F : N ! N and A : N ! N, the complexity class DTime[F ]=A consists of languages L such that there exist a TM with time bound F (n) and a sequence fangn2N of “advices” satisfying: • janj ≤ A(n); • for jxj = n, x 2 L if and only if M(x; an) = 1. The following theorem explains the notation P=poly, namely “polynomial-time with poly- nomial advice”. S c Theorem 1. P=poly = c;d2N DTime[n ]=nd . Proof. If L 2 P=poly, then it can be computed by a family C = fC1;C2; · · · g of Boolean circuits. Let an be the description of Cn, andS the polynomial time machine M just reads 2 c this description and simulates it. Hence L c;d2N DTime[n ]=nd . For the other direction, if a language L can be computed in polynomial-time with poly- nomial advice, say by TM M with advices fang, then we can construct circuits fDng to simulate M, as in the theorem P ⊂ P=poly in the last lecture. Hence, Dn(x; an) = 1 if and only if x 2 L. The final circuit Cn just does exactly what Dn does, except that Cn “hardwires” the advice an. Namely, Cn(x) := Dn(x; an). Hence, L 2 P=poly. 2 Karp-Lipton Theorem Dick Karp and Dick Lipton showed that NP is unlikely to be contained in P=poly [KL80]. -
Computational Complexity and Intractability: an Introduction to the Theory of NP Chapter 9 2 Objectives
1 Computational Complexity and Intractability: An Introduction to the Theory of NP Chapter 9 2 Objectives . Classify problems as tractable or intractable . Define decision problems . Define the class P . Define nondeterministic algorithms . Define the class NP . Define polynomial transformations . Define the class of NP-Complete 3 Input Size and Time Complexity . Time complexity of algorithms: . Polynomial time (efficient) vs. Exponential time (inefficient) f(n) n = 10 30 50 n 0.00001 sec 0.00003 sec 0.00005 sec n5 0.1 sec 24.3 sec 5.2 mins 2n 0.001 sec 17.9 mins 35.7 yrs 4 “Hard” and “Easy” Problems . “Easy” problems can be solved by polynomial time algorithms . Searching problem, sorting, Dijkstra’s algorithm, matrix multiplication, all pairs shortest path . “Hard” problems cannot be solved by polynomial time algorithms . 0/1 knapsack, traveling salesman . Sometimes the dividing line between “easy” and “hard” problems is a fine one. For example, . Find the shortest path in a graph from X to Y (easy) . Find the longest path (with no cycles) in a graph from X to Y (hard) 5 “Hard” and “Easy” Problems . Motivation: is it possible to efficiently solve “hard” problems? Efficiently solve means polynomial time solutions. Some problems have been proved that no efficient algorithms for them. For example, print all permutation of a number n. However, many problems we cannot prove there exists no efficient algorithms, and at the same time, we cannot find one either. 6 Traveling Salesperson Problem . No algorithm has ever been developed with a Worst-case time complexity better than exponential . -
On the NP-Completeness of the Minimum Circuit Size Problem
On the NP-Completeness of the Minimum Circuit Size Problem John M. Hitchcock∗ A. Pavany Department of Computer Science Department of Computer Science University of Wyoming Iowa State University Abstract We study the Minimum Circuit Size Problem (MCSP): given the truth-table of a Boolean function f and a number k, does there exist a Boolean circuit of size at most k computing f? This is a fundamental NP problem that is not known to be NP-complete. Previous work has studied consequences of the NP-completeness of MCSP. We extend this work and consider whether MCSP may be complete for NP under more powerful reductions. We also show that NP-completeness of MCSP allows for amplification of circuit complexity. We show the following results. • If MCSP is NP-complete via many-one reductions, the following circuit complexity amplifi- Ω(1) cation result holds: If NP\co-NP requires 2n -size circuits, then ENP requires 2Ω(n)-size circuits. • If MCSP is NP-complete under truth-table reductions, then EXP 6= NP \ SIZE(2n ) for some > 0 and EXP 6= ZPP. This result extends to polylog Turing reductions. 1 Introduction Many natural NP problems are known to be NP-complete. Ladner's theorem [14] tells us that if P is different from NP, then there are NP-intermediate problems: problems that are in NP, not in P, but also not NP-complete. The examples arising out of Ladner's theorem come from diagonalization and are not natural. A canonical candidate example of a natural NP-intermediate problem is the Graph Isomorphism (GI) problem. -
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. -
NP-Completeness Part I
NP-Completeness Part I Outline for Today ● Recap from Last Time ● Welcome back from break! Let's make sure we're all on the same page here. ● Polynomial-Time Reducibility ● Connecting problems together. ● NP-Completeness ● What are the hardest problems in NP? ● The Cook-Levin Theorem ● A concrete NP-complete problem. Recap from Last Time The Limits of Computability EQTM EQTM co-RE R RE LD LD ADD HALT ATM HALT ATM 0*1* The Limits of Efficient Computation P NP R P and NP Refresher ● The class P consists of all problems solvable in deterministic polynomial time. ● The class NP consists of all problems solvable in nondeterministic polynomial time. ● Equivalently, NP consists of all problems for which there is a deterministic, polynomial-time verifier for the problem. Reducibility Maximum Matching ● Given an undirected graph G, a matching in G is a set of edges such that no two edges share an endpoint. ● A maximum matching is a matching with the largest number of edges. AA maximummaximum matching.matching. Maximum Matching ● Jack Edmonds' paper “Paths, Trees, and Flowers” gives a polynomial-time algorithm for finding maximum matchings. ● (This is the same Edmonds as in “Cobham- Edmonds Thesis.) ● Using this fact, what other problems can we solve? Domino Tiling Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling Solving Domino Tiling The Setup ● To determine whether you can place at least k dominoes on a crossword grid, do the following: ● Convert the grid into a graph: each empty cell is a node, and any two adjacent empty cells have an edge between them. -
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/. -
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Lecture 9 1 Interactive Proof Systems/Protocols
CS 282A/MATH 209A: Foundations of Cryptography Prof. Rafail Ostrovsky Lecture 9 Lecture date: March 7-9, 2005 Scribe: S. Bhattacharyya, R. Deak, P. Mirzadeh 1 Interactive Proof Systems/Protocols 1.1 Introduction The traditional mathematical notion of a proof is a simple passive protocol in which a prover P outputs a complete proof to a verifier V who decides on its validity. The interaction in this traditional sense is minimal and one-way, prover → verifier. The observation has been made that allowing the verifier to interact with the prover can have advantages, for example proving the assertion faster or proving more expressive languages. This extension allows for the idea of interactive proof systems (protocols). The general framework of the interactive proof system (protocol) involves a prover P with an exponential amount of time (computationally unbounded) and a verifier V with a polyno- mial amount of time. Both P and V exchange multiple messages (challenges and responses), usually dependent upon outcomes of fair coin tosses which they may or may not share. It is easy to see that since V is a poly-time machine (PPT), only a polynomial number of messages may be exchanged between the two. P ’s objective is to convince (prove to) the verifier the truth of an assertion, e.g., claimed knowledge of a proof that x ∈ L. V either accepts or rejects the interaction with the P . 1.2 Definition of Interactive Proof Systems An interactive proof system for a language L is a protocol PV for communication between a computationally unbounded (exponential time) machine P and a probabilistic poly-time (PPT) machine V such that the protocol satisfies the properties of completeness and sound- ness. -
The Correlation Among Software Complexity Metrics with Case Study
International Journal of Advanced Computer Research (ISSN (print): 2249-7277 ISSN (online): 2277-7970) Volume-4 Number-2 Issue-15 June-2014 The Correlation among Software Complexity Metrics with Case Study Yahya Tashtoush1, Mohammed Al-Maolegi2, Bassam Arkok3 Abstract software product attributes such as functionality, quality, complexity, efficiency, reliability or People demand for software quality is growing maintainability. For example, a higher number of increasingly, thus different scales for the software code lines will lead to greater software complexity are growing fast to handle the quality of software. and so on. The software complexity metric is one of the measurements that use some of the internal The complexity of software effects on maintenance attributes or characteristics of software to know how activities like software testability, reusability, they effect on the software quality. In this paper, we understandability and modifiability. Software cover some of more efficient software complexity complexity is defined as ―the degree to which a metrics such as Cyclomatic complexity, line of code system or component has a design or implementation and Hallstead complexity metric. This paper that is difficult to understand and verify‖ [1]. All the presents their impacts on the software quality. It factors that make program difficult to understand are also discusses and analyzes the correlation between responsible for complexity. So it is necessary to find them. It finally reveals their relation with the measurements for software to reduce the impacts of number of errors using a real dataset as a case the complexity and guarantee the quality at the same study. time as much as possible. -
Lecture 10: Learning DNF, AC0, Juntas Feb 15, 2007 Lecturer: Ryan O’Donnell Scribe: Elaine Shi
Analysis of Boolean Functions (CMU 18-859S, Spring 2007) Lecture 10: Learning DNF, AC0, Juntas Feb 15, 2007 Lecturer: Ryan O’Donnell Scribe: Elaine Shi 1 Learning DNF in Almost Polynomial Time From previous lectures, we have learned that if a function f is ǫ-concentrated on some collection , then we can learn the function using membership queries in poly( , 1/ǫ)poly(n) log(1/δ) time.S |S| O( w ) In the last lecture, we showed that a DNF of width w is ǫ-concentrated on a set of size n ǫ , and O( w ) concluded that width-w DNFs are learnable in time n ǫ . Today, we shall improve this bound, by showing that a DNF of width w is ǫ-concentrated on O(w log 1 ) a collection of size w ǫ . We shall hence conclude that poly(n)-size DNFs are learnable in almost polynomial time. Recall that in the last lecture we introduced H˚astad’s Switching Lemma, and we showed that 1 DNFs of width w are ǫ-concentrated on degrees up to O(w log ǫ ). Theorem 1.1 (Hastad’s˚ Switching Lemma) Let f be computable by a width-w DNF, If (I, X) is a random restriction with -probability ρ, then d N, ∗ ∀ ∈ d Pr[DT-depth(fX→I) >d] (5ρw) I,X ≤ Theorem 1.2 If f is a width-w DNF, then f(U)2 ǫ ≤ |U|≥OX(w log 1 ) ǫ b O(w log 1 ) To show that a DNF of width w is ǫ-concentrated on a collection of size w ǫ , we also need the following theorem: Theorem 1.3 If f is a width-w DNF, then 1 |U| f(U) 2 20w | | ≤ XU b Proof: Let (I, X) be a random restriction with -probability 1 .