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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. -
The Shrinking Property for NP and Conp✩
Theoretical Computer Science 412 (2011) 853–864 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs The shrinking property for NP and coNPI Christian Glaßer a, Christian Reitwießner a,∗, Victor Selivanov b a Julius-Maximilians-Universität Würzburg, Germany b A.P. Ershov Institute of Informatics Systems, Siberian Division of the Russian Academy of Sciences, Russia article info a b s t r a c t Article history: We study the shrinking and separation properties (two notions well-known in descriptive Received 18 August 2009 set theory) for NP and coNP and show that under reasonable complexity-theoretic Received in revised form 11 November assumptions, both properties do not hold for NP and the shrinking property does not hold 2010 for coNP. In particular we obtain the following results. Accepted 15 November 2010 Communicated by A. Razborov P 1. NP and coNP do not have the shrinking property unless PH is finite. In general, Σn and P Πn do not have the shrinking property unless PH is finite. This solves an open question Keywords: posed by Selivanov (1994) [33]. Computational complexity 2. The separation property does not hold for NP unless UP ⊆ coNP. Polynomial hierarchy 3. The shrinking property does not hold for NP unless there exist NP-hard disjoint NP-pairs NP-pairs (existence of such pairs would contradict a conjecture of Even et al. (1984) [6]). P-separability Multivalued functions 4. The shrinking property does not hold for NP unless there exist complete disjoint NP- pairs. Moreover, we prove that the assumption NP 6D coNP is too weak to refute the shrinking property for NP in a relativizable way. -
The Randomized Quicksort Algorithm
Outline The Randomized Quicksort Algorithm K. Subramani1 1Lane Department of Computer Science and Electrical Engineering West Virginia University 7 February, 2012 Subramani Sample Analyses Outline Outline 1 The Randomized Quicksort Algorithm Subramani Sample Analyses The Randomized Quicksort Algorithm The Sorting Problem Problem Statement Given an array A of n distinct integers, in the indices A[1] through A[n], Subramani Sample Analyses The Randomized Quicksort Algorithm The Sorting Problem Problem Statement Given an array A of n distinct integers, in the indices A[1] through A[n], permute the elements of A, so that Subramani Sample Analyses The Randomized Quicksort Algorithm The Sorting Problem Problem Statement Given an array A of n distinct integers, in the indices A[1] through A[n], permute the elements of A, so that A[1] < A[2]... A[n]. Subramani Sample Analyses The Randomized Quicksort Algorithm The Sorting Problem Problem Statement Given an array A of n distinct integers, in the indices A[1] through A[n], permute the elements of A, so that A[1] < A[2]... A[n]. Note The assumption of distinctness simplifies the analysis. Subramani Sample Analyses The Randomized Quicksort Algorithm The Sorting Problem Problem Statement Given an array A of n distinct integers, in the indices A[1] through A[n], permute the elements of A, so that A[1] < A[2]... A[n]. Note The assumption of distinctness simplifies the analysis. It has no bearing on the running time. Subramani Sample Analyses The Randomized Quicksort Algorithm The Partition subroutine Function PARTITION(A,p,q) 1: {We partition the sub-array A[p,p + 1,...,q] about A[p].} 2: for (i = (p + 1) to q) do 3: if (A[i] < A[p]) then 4: Insert A[i] into bucket L. -
Week 1: an Overview of Circuit Complexity 1 Welcome 2
Topics in Circuit Complexity (CS354, Fall’11) Week 1: An Overview of Circuit Complexity Lecture Notes for 9/27 and 9/29 Ryan Williams 1 Welcome The area of circuit complexity has a long history, starting in the 1940’s. It is full of open problems and frontiers that seem insurmountable, yet the literature on circuit complexity is fairly large. There is much that we do know, although it is scattered across several textbooks and academic papers. I think now is a good time to look again at circuit complexity with fresh eyes, and try to see what can be done. 2 Preliminaries An n-bit Boolean function has domain f0; 1gn and co-domain f0; 1g. At a high level, the basic question asked in circuit complexity is: given a collection of “simple functions” and a target Boolean function f, how efficiently can f be computed (on all inputs) using the simple functions? Of course, efficiency can be measured in many ways. The most natural measure is that of the “size” of computation: how many copies of these simple functions are necessary to compute f? Let B be a set of Boolean functions, which we call a basis set. The fan-in of a function g 2 B is the number of inputs that g takes. (Typical choices are fan-in 2, or unbounded fan-in, meaning that g can take any number of inputs.) We define a circuit C with n inputs and size s over a basis B, as follows. C consists of a directed acyclic graph (DAG) of s + n + 2 nodes, with n sources and one sink (the sth node in some fixed topological order on the nodes). -
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. -
Chapter 24 Conp, Self-Reductions
Chapter 24 coNP, Self-Reductions CS 473: Fundamental Algorithms, Spring 2013 April 24, 2013 24.1 Complementation and Self-Reduction 24.2 Complementation 24.2.1 Recap 24.2.1.1 The class P (A) A language L (equivalently decision problem) is in the class P if there is a polynomial time algorithm A for deciding L; that is given a string x, A correctly decides if x 2 L and running time of A on x is polynomial in jxj, the length of x. 24.2.1.2 The class NP Two equivalent definitions: (A) Language L is in NP if there is a non-deterministic polynomial time algorithm A (Turing Machine) that decides L. (A) For x 2 L, A has some non-deterministic choice of moves that will make A accept x (B) For x 62 L, no choice of moves will make A accept x (B) L has an efficient certifier C(·; ·). (A) C is a polynomial time deterministic algorithm (B) For x 2 L there is a string y (proof) of length polynomial in jxj such that C(x; y) accepts (C) For x 62 L, no string y will make C(x; y) accept 1 24.2.1.3 Complementation Definition 24.2.1. Given a decision problem X, its complement X is the collection of all instances s such that s 62 L(X) Equivalently, in terms of languages: Definition 24.2.2. Given a language L over alphabet Σ, its complement L is the language Σ∗ n L. 24.2.1.4 Examples (A) PRIME = nfn j n is an integer and n is primeg o PRIME = n n is an integer and n is not a prime n o PRIME = COMPOSITE . -
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. -
Randomized Algorithms, Quicksort and Randomized Selection Carola Wenk Slides Courtesy of Charles Leiserson with Additions by Carola Wenk
CMPS 2200 – Fall 2014 Randomized Algorithms, Quicksort and Randomized Selection Carola Wenk Slides courtesy of Charles Leiserson with additions by Carola Wenk CMPS 2200 Intro. to Algorithms 1 Deterministic Algorithms Runtime for deterministic algorithms with input size n: • Best-case runtime Attained by one input of size n • Worst-case runtime Attained by one input of size n • Average runtime Averaged over all possible inputs of size n CMPS 2200 Intro. to Algorithms 2 Deterministic Algorithms: Insertion Sort for j=2 to n { key = A[j] // insert A[j] into sorted sequence A[1..j-1] i=j-1 while(i>0 && A[i]>key){ A[i+1]=A[i] i-- } A[i+1]=key } • Best case runtime? • Worst case runtime? CMPS 2200 Intro. to Algorithms 3 Deterministic Algorithms: Insertion Sort Best-case runtime: O(n), input [1,2,3,…,n] Attained by one input of size n • Worst-case runtime: O(n2), input [n, n-1, …,2,1] Attained by one input of size n • Average runtime : O(n2) Averaged over all possible inputs of size n •What kind of inputs are there? • How many inputs are there? CMPS 2200 Intro. to Algorithms 4 Average Runtime • What kind of inputs are there? • Do [1,2,…,n] and [5,6,…,n+5] cause different behavior of Insertion Sort? • No. Therefore it suffices to only consider all permutations of [1,2,…,n] . • How many inputs are there? • There are n! different permutations of [1,2,…,n] CMPS 2200 Intro. to Algorithms 5 Average Runtime Insertion Sort: n=4 • Inputs: 4!=24 [1,2,3,4]0346 [4,1,2,3] [4,1,3,2] [4,3,2,1] [2,1,3,4]1235 [1,4,2,3] [1,4,3,2] [3,4,2,1] [1,3,2,4]1124 [1,2,4,3] [1,3,4,2] [3,2,4,1] [3,1,2,4]2455 [4,2,1,3] [4,3,1,2] [4,2,3,1] [3,2,1,4]3244 [2,1,4,3] [3,4,1,2] [2,4,3,1] [2,3,1,4]2333 [2,4,1,3] [3,1,4,2] [2,3,4,1] • Runtime is proportional to: 3 + #times in while loop • Best: 3+0, Worst: 3+6=9, Average: 3+72/24 = 6 CMPS 2200 Intro. -
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/. -
CS286.2 Lectures 5-6: Introduction to Hamiltonian Complexity, QMA-Completeness of the Local Hamiltonian Problem
CS286.2 Lectures 5-6: Introduction to Hamiltonian Complexity, QMA-completeness of the Local Hamiltonian problem Scribe: Jenish C. Mehta The Complexity Class BQP The complexity class BQP is the quantum analog of the class BPP. It consists of all languages that can be decided in quantum polynomial time. More formally, Definition 1. A language L 2 BQP if there exists a classical polynomial time algorithm A that ∗ maps inputs x 2 f0, 1g to quantum circuits Cx on n = poly(jxj) qubits, where the circuit is considered a sequence of unitary operators each on 2 qubits, i.e Cx = UTUT−1...U1 where each 2 2 Ui 2 L C ⊗ C , such that: 2 i. Completeness: x 2 L ) Pr(Cx accepts j0ni) ≥ 3 1 ii. Soundness: x 62 L ) Pr(Cx accepts j0ni) ≤ 3 We say that the circuit “Cx accepts jyi” if the first output qubit measured in Cxjyi is 0. More j0i specifically, letting P1 = j0ih0j1 be the projection of the first qubit on state j0i, j0i 2 Pr(Cx accepts jyi) =k (P1 ⊗ In−1)Cxjyi k2 The Complexity Class QMA The complexity class QMA (or BQNP, as Kitaev originally named it) is the quantum analog of the class NP. More formally, Definition 2. A language L 2 QMA if there exists a classical polynomial time algorithm A that ∗ maps inputs x 2 f0, 1g to quantum circuits Cx on n + q = poly(jxj) qubits, such that: 2q i. Completeness: x 2 L ) 9jyi 2 C , kjyik2 = 1, such that Pr(Cx accepts j0ni ⊗ 2 jyi) ≥ 3 2q 1 ii. -
Complexity Theory
Complexity Theory Course Notes Sebastiaan A. Terwijn Radboud University Nijmegen Department of Mathematics P.O. Box 9010 6500 GL Nijmegen the Netherlands [email protected] Copyright c 2010 by Sebastiaan A. Terwijn Version: December 2017 ii Contents 1 Introduction 1 1.1 Complexity theory . .1 1.2 Preliminaries . .1 1.3 Turing machines . .2 1.4 Big O and small o .........................3 1.5 Logic . .3 1.6 Number theory . .4 1.7 Exercises . .5 2 Basics 6 2.1 Time and space bounds . .6 2.2 Inclusions between classes . .7 2.3 Hierarchy theorems . .8 2.4 Central complexity classes . 10 2.5 Problems from logic, algebra, and graph theory . 11 2.6 The Immerman-Szelepcs´enyi Theorem . 12 2.7 Exercises . 14 3 Reductions and completeness 16 3.1 Many-one reductions . 16 3.2 NP-complete problems . 18 3.3 More decision problems from logic . 19 3.4 Completeness of Hamilton path and TSP . 22 3.5 Exercises . 24 4 Relativized computation and the polynomial hierarchy 27 4.1 Relativized computation . 27 4.2 The Polynomial Hierarchy . 28 4.3 Relativization . 31 4.4 Exercises . 32 iii 5 Diagonalization 34 5.1 The Halting Problem . 34 5.2 Intermediate sets . 34 5.3 Oracle separations . 36 5.4 Many-one versus Turing reductions . 38 5.5 Sparse sets . 38 5.6 The Gap Theorem . 40 5.7 The Speed-Up Theorem . 41 5.8 Exercises . 43 6 Randomized computation 45 6.1 Probabilistic classes . 45 6.2 More about BPP . 48 6.3 The classes RP and ZPP . -
Simple Doubly-Efficient Interactive Proof Systems for Locally
Electronic Colloquium on Computational Complexity, Revision 3 of Report No. 18 (2017) Simple doubly-efficient interactive proof systems for locally-characterizable sets Oded Goldreich∗ Guy N. Rothblumy September 8, 2017 Abstract A proof system is called doubly-efficient if the prescribed prover strategy can be implemented in polynomial-time and the verifier’s strategy can be implemented in almost-linear-time. We present direct constructions of doubly-efficient interactive proof systems for problems in P that are believed to have relatively high complexity. Specifically, such constructions are presented for t-CLIQUE and t-SUM. In addition, we present a generic construction of such proof systems for a natural class that contains both problems and is in NC (and also in SC). The proof systems presented by us are significantly simpler than the proof systems presented by Goldwasser, Kalai and Rothblum (JACM, 2015), let alone those presented by Reingold, Roth- blum, and Rothblum (STOC, 2016), and can be implemented using a smaller number of rounds. Contents 1 Introduction 1 1.1 The current work . 1 1.2 Relation to prior work . 3 1.3 Organization and conventions . 4 2 Preliminaries: The sum-check protocol 5 3 The case of t-CLIQUE 5 4 The general result 7 4.1 A natural class: locally-characterizable sets . 7 4.2 Proof of Theorem 1 . 8 4.3 Generalization: round versus computation trade-off . 9 4.4 Extension to a wider class . 10 5 The case of t-SUM 13 References 15 Appendix: An MA proof system for locally-chracterizable sets 18 ∗Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel.