DOCSLIB.ORG

Advanced Complexity Theory Lecture Notes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Advanced Complexity Theory Lecture Notes COMP 531 Advanced Complexity Theory Lecture Notes Hamed Hatami 2 1 Background: Before we start5 10 Randomized Complexity Classes 45 Decidable and Undecidable Languages:......5 Probabilistic Turing machines........... 45 Exercises......................6 Polynomial time Randomized complexity classes 46 The complexity class ZPP:.......... 46 2 Background: Basic Analysis7 The complexity class RP ........... 47 Some basic inequalities...............7 The complexity class BPP .......... 47 Measure spaces...................9 The complexity class PP:............. 48 Probability Spaces.............. 10 The relation between the classes......... 49 Normed spaces................... 11 How large is PP?............... 49 Hilbert Spaces................ 12 Randomness as advice and RP ⊆ NP .... 50 How large is BPP?.............. 50 The L spaces................. 13 p Exercises...................... 51 Exercises...................... 13 11 Non-uniform complexity: Circuits 53 3 Time complexity 15 What is a Boolean circuit?............ 53 Our computational model............. 15 The class P=Poly .................. 54 Time complexity.................. 16 Karp-Lipton Theorem............... 55 Time Hierarchy theorems............. 17 Shannon-Muller Lower bound........... 55 Exercises...................... 18 Adleman's theorem and advice: BPP ⊆ P=Poly . 56 Exercises...................... 57 4 Polynomial Hierarchy 19 Exercises...................... 21 12 AC0: Bounded Depth Alternating Circuits 59 Lower-bounds for AC circuits........... 60 5 Space complexity 23 Decision tree complexity........... 61 Time and Space.................. 23 Hastad's switching lemma............. 61 Logarithmic Space L and NL ........... 24 Influences in bounded depth circuits....... 64 Exercises...................... 25 Exercises...................... 65 6 Savitch's Theorem 27 13 Razborov-Smolensky 67 0 Savitch's theorem.................. 28 The class AC (⊕): Algebraic techniques..... 68 Exercises...................... 29 Razborov-Smolensky over F2 ........... 69 concluding remarks................. 71 7 NL versus coNL: Immerman-Szelepcsenyis Exercises...................... 71 Theorem 31 14 Razborov's monotone circuit lower-bound 75 NL versus coNL: Immerman-Szelepcsenyis Theo- Approximating the monotone circuit.... 76 rem...................... 31 The preliminary lemmas.............. 77 Exercises...................... 32 Approximation by a small number of cliques... 78 Concluding remarks................ 79 8 Ladner's Theorem: NP-completeness 35 Exercises...................... 79 Warm-up: Proof assuming the Exponential Time Hypothesis.................. 35 15 Razborov-Rudich: Natural Proof Barrier 81 The full proof of Ladner's Theorem........ 36 What are natural proofs?............. 81 Exercises...................... 37 The Razborov-Rudich Theorem.......... 82 Preparing for the proof: Pseudo-random Gener- 9 Oracles and relativization 39 ators...................... 83 Relativization................. 40 The proof of Razborov-Rudich Theorem..... 84 Relativization as a proof-barrier......... 41 Concluding Remarks................ 84 Exercises...................... 42 Exercises...................... 85 3 16 Fourier analysis and Polynomial Represen- Rank Lower-Bound................. 107 tations 87 Non-deterministic Communication complexity.. 107 Polynomial representations of f : {−1; 1gn ! R . 88 Exercises...................... 107 Fourier analysis of Finite Abelian Groups.... 90 Basic Fourier Theory................ 90 19 Randomized Communication complexity 109 Infinite Ableian groups and beyond..... 96 Yao's Min-Max Theorem.............. 110 Concluding remarks:................ 97 Exercises...................... 112 Exercises...................... 97 20 Discrepancy and Sign-rank 113 Discrepancy..................... 113 17 Learning and AC0 circuits 99 Other formulations of discrepancy..... 115 PAC learning from uniform samples.... 99 Unbounded-Error model.............. 115 Decision Trees................... 100 Sign-rank................... 116 Linial-Mansour-Nisan............... 101 Exercises...................... 117 Mansour-Nisan conjecture............. 102 Final remarks about Circuits........... 102 21 Machine Learning: Dimension and Margin 119 Exercises...................... 103 Sign-rank in Machine Learning: Dimension 119 Margin and Discrepancy.............. 121 18 Communication complexity 105 The Matrix Theory Background......... 122 Monochromatic rectangles, and Fooling sets... 105 Forster's Sign-rank Lower-bound...... 124 Rectangle Size Bounds............... 106 Exercises...................... 125 4 Chapter 1 Background: Before we start This is an advanced course on complexity theory, and thus we will assume that the reader is familiar with the basics of the theory of computation; topics such as Turing Machines, Non-determinism, Decidability, Undecidability, Turing reductions. We will also assume that the reader has some familiarity with the basic complexity theory: the complexity classes P, NP, coNP, the definition of NP through efficient verifiers, the notion of NP-completeness, and the Cook-Levin theorem. Similar to any theoretical computer science course, we shall use the asymptotic notations of O(·); Θ(·); Ω(·); o(·);!(·). In this short lecture, we quickly remind the reader of some of these concepts. Decidable and Undecidable Languages: An alphabet Σ is just a finite set, where the elements are interpreted as \symbols". In this course, almost always Σ = f0; 1g. We denote by Σ∗ the set of all finite strings constructed from the symbols in Σ. For example, if Σ = f0; 1g, then Σ∗ = f"; 0; 1; 00; 01; 10; 11;:::g. Here " denotes the empty string of length zero. We emphasis that all the elements in Σ∗ are of finite length. Every subset L ⊆ Σ∗ is called a language. Languages correspond to decision problems, i.e. YES/NO problems in a natural way: Given an input x, we would like to know whether x 2 L or x 62 L. A language is called decidable if and only if there is a Turing Machine that always halts and accepts x if x 2 L and rejects x if x 62 L. The Turing Machines that always halt are called algorithms. Since the set of the languages is uncountable, and the set of Turing Machines over any fixed alphabet is countable, there are languages that are undecidable. Recall that P is the class of the languages that can be decided in polynomial time, and NP is the class of the problems that can be decided in polynomial non-deterministic time. It is also useful to define NP through the notions of the verifiers. A verifier for a language L is an algorithm V that takes a pair (x; y) as input, and has the property that x 2 L () 9y; V (x; y) = Accept: Here we can think of y as a certificate/proof for the fact that x 2 L. That is if x 2 L, then there is a certificate for it and our algorithm can verify it. However if x 62 L, then such a certificate should not exist, and V should reject (x; y) for every y. A verifier is called efficient if its running time is bounded by a polynomial p in the length of x. Note that this is stronger than saying that V runs in polynomial time because the input of V is (x; y), but we bound its running time by p(jxj) rather than p(j(x; y)j). It is easy to see that NP is the set of the languages with efficient verifiers. An important problem in NP is the SAT problem, which in a certain sense is as hard as any other problem in the class NP. First we define the notion of a polynomial time reduction. Definition 1.1. A language A is polynomial time mapping reducible, or simply polynomial time reducible to a language ∗ ∗ B, written as A 6p B, if a polynomial time computable function f :Σ ! Σ exists, where for every w, w 2 A () f(w) 2 B: A Boolean formula over variables x1; : : : ; xn, is a formula consisting of variables and logical operators ^; _; : corresponding to AND, OR, and negation, respectively. For example φ(x1; x2; x3) := (x1 ^ x2) _ (x1 ^ x3) _ (x2 ^ x3) is a Boolean formula over the variables x1; x2; x3, and one can easily check that φ evaluates to True if and only if majority of the variables x1; x2; x3 are True. We will identify 1 with True and 0 with False, so for example φ(1; 1; 0) = 1. 5 Definition 1.2 (Conjunctive Normal Form (CNF)). A Boolean formula over variables x1; : : : ; xn is in CNF form if it is an AND of ORs of variables or their negations. We call formulas that are in CNF form, simply CNF formulas. The terms xi or :xi appearing in a formula are called terms or literals. For a CNF formula m 0 k 1 ^ _i φ = @ tij A ; i=1 j=1 ki where tij are terms, we call each (_j=1tij ) a clause of the formula. For example (x1 _:x2 _ x3) ^ (x2 _ x3) is a CNF formula with two clauses (x1 _:x2 _ x3) and (x2 _ x3). Theorem 1.3 (Cook-Levin Theorem, Cook [Coo71], Levin [Lev73]). The language of all satisfiable CNF formulas SAT = fφ j φ is a CNF, and there exists y such that φ(y) = Trueg; is NP-complete. That is it belongs to NP and furthermore X 6p SAT for every X 2 NP. It is easy to see that SAT is in fact in NP. To see this, note that a y that satisfies φ(y) = True can be used as a certificate to verify that φ is satisfiable. In other words an efficient certifier V takes the pair (φ, y), plugs y into φ and checks to see if φ is satisfied. If it is, then it accepts the pair, otherwise it rejects. Note that φ 2 SAT () 9y; V (φ, y) = Accept: The interesting and difficult part of the Cook-Levin theorem is the proof that every language in NP reduces to SAT. The proof is by encoding the computation of a Turing machine as a CNF. Big O notation : Finally let us finish this lecture by recalling the asymptotic notations. Let f; g : N ! N. We say • f(n) = O(g(n) if there exists n0; c > 0 such that f(n) < cg(n) for all n > n0. • f(n) = Ω(g(n) if there exists n0; c > 0 such that f(n) > cg(n) for all n > n0. • f(n) = Θ(g(n)) if there exists n0; c1; c2 > 0 such that c1g(n) < f(n) < c2g(n) for all n > n0. f(n) • f(n) = o(g(n)) if limn!1 g(n) = 0. f(n) • f(n) = !(g(n)) if limn!1 g(n) = 1.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs*
    On the Randomness Complexity of Interactive Proofs and Statistical Zero-Knowledge Proofs* Benny Applebaum† Eyal Golombek* Abstract We study the randomness complexity of interactive proofs and zero-knowledge proofs. In particular, we ask whether it is possible to reduce the randomness complexity, R, of the verifier to be comparable with the number of bits, CV , that the verifier sends during the interaction. We show that such randomness sparsification is possible in several settings. Specifically, unconditional sparsification can be obtained in the non-uniform setting (where the verifier is modelled as a circuit), and in the uniform setting where the parties have access to a (reusable) common-random-string (CRS). We further show that constant-round uniform protocols can be sparsified without a CRS under a plausible worst-case complexity-theoretic assumption that was used previously in the context of derandomization. All the above sparsification results preserve statistical-zero knowledge provided that this property holds against a cheating verifier. We further show that randomness sparsification can be applied to honest-verifier statistical zero-knowledge (HVSZK) proofs at the expense of increasing the communica- tion from the prover by R−F bits, or, in the case of honest-verifier perfect zero-knowledge (HVPZK) by slowing down the simulation by a factor of 2R−F . Here F is a new measure of accessible bit complexity of an HVZK proof system that ranges from 0 to R, where a maximal grade of R is achieved when zero- knowledge holds against a “semi-malicious” verifier that maliciously selects its random tape and then plays honestly.
    [Show full text]
  • Hierarchy Theorems
    Hierarchy Theorems Peter Mayr Computability Theory, April 14, 2021 So far we proved the following inclusions: L ⊆ NL ⊆ P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ EXPSPACE Question Which are proper? Space constructible functions In simulations we often want to fix s(n) space on the working tape before starting the actual computation without accruing any overhead in space complexity. Definition s(n) ≥ log(n) is space constructible if there exists N 2 N and a DTM with input tape that on any input of length n ≥ N uses and marks off s(n) cells on its working tape and halts. Note Equivalently, s(n) is space constructible iff there exists a DTM with input and output that on input 1n computes s(n) in space O(s(n)). Example log n; nk ; 2n are space constructible. E.g. log2 n space is constructed by counting the length n of the input in binary. Little o-notation Definition + For f ; g : N ! R we say f = o(g) (read f is little-o of g) if f (n) lim = 0: n!1 g(n) Intuitively: g grows much faster than f . Note I f = o(g) ) f = O(g), but not conversely. I f 6= o(f ) Separating space complexity classes Space Hierarchy Theorem Let s(n) be space constructible and r(n) = o(s(n)). Then DSPACE(r(n)) ( DSPACE(s(n)). Proof. Construct L 2 DSPACE(s(n))nDSPACE(r(n)) by diagonalization. Define DTM D such that on input x of length n: 1. D marks s(n) space on working tape.
    [Show full text]
  • EXPSPACE-Hardness of Behavioural Equivalences of Succinct One
    EXPSPACE-hardness of behavioural equivalences of succinct one-counter nets Petr Janˇcar1 Petr Osiˇcka1 Zdenˇek Sawa2 1Dept of Comp. Sci., Faculty of Science, Palack´yUniv. Olomouc, Czech Rep. [email protected], [email protected] 2Dept of Comp. Sci., FEI, Techn. Univ. Ostrava, Czech Rep. [email protected] Abstract We note that the remarkable EXPSPACE-hardness result in [G¨oller, Haase, Ouaknine, Worrell, ICALP 2010] ([GHOW10] for short) allows us to answer an open complexity ques- tion for simulation preorder of succinct one counter nets (i.e., one counter automata with no zero tests where counter increments and decrements are integers written in binary). This problem, as well as bisimulation equivalence, turn out to be EXPSPACE-complete. The technique of [GHOW10] was referred to by Hunter [RP 2015] for deriving EXPSPACE-hardness of reachability games on succinct one-counter nets. We first give a direct self-contained EXPSPACE-hardness proof for such reachability games (by adjust- ing a known PSPACE-hardness proof for emptiness of alternating finite automata with one-letter alphabet); then we reduce reachability games to (bi)simulation games by using a standard “defender-choice” technique. 1 Introduction arXiv:1801.01073v1 [cs.LO] 3 Jan 2018 We concentrate on our contribution, without giving a broader overview of the area here. A remarkable result by G¨oller, Haase, Ouaknine, Worrell [2] shows that model checking a fixed CTL formula on succinct one-counter automata (where counter increments and decre- ments are integers written in binary) is EXPSPACE-hard. Their proof is interesting and nontrivial, and uses two involved results from complexity theory.
    [Show full text]
  • Nitin Saurabh the Institute of Mathematical Sciences, Chennai
    ALGEBRAIC MODELS OF COMPUTATION By Nitin Saurabh The Institute of Mathematical Sciences, Chennai. A thesis submitted to the Board of Studies in Mathematical Sciences In partial fulllment of the requirements For the Degree of Master of Science of HOMI BHABHA NATIONAL INSTITUTE April 2012 CERTIFICATE Certied that the work contained in the thesis entitled Algebraic models of Computation, by Nitin Saurabh, has been carried out under my supervision and that this work has not been submitted elsewhere for a degree. Meena Mahajan Theoretical Computer Science Group The Institute of Mathematical Sciences, Chennai ACKNOWLEDGEMENTS I would like to thank my advisor Prof. Meena Mahajan for her invaluable guidance and continuous support since my undergraduate days. Her expertise and ideas helped me comprehend new techniques. Her guidance during the preparation of this thesis has been invaluable. I also thank her for always being there to discuss and clarify any matter. I am extremely grateful to all the faculty members of theory group at IMSc and CMI for their continuous encouragement and giving me an opportunity to learn from them. I would like to thank all my friends, at IMSc and CMI, for making my stay in Chennai a memorable one. Most of all, I take this opportunity to thank my parents, my uncle and my brother. Abstract Valiant [Val79, Val82] had proposed an analogue of the theory of NP-completeness in an entirely algebraic framework to study the complexity of polynomial families. Artihmetic circuits form the most standard model for studying the complexity of polynomial computations. In a note [Val92], Valiant argued that in order to prove lower bounds for boolean circuits, obtaining lower bounds for arithmetic circuits should be a rst step.
    [Show full text]
  • Probabilistically Checkable Proofs Over the Reals
    Electronic Notes in Theoretical Computer Science 123 (2005) 165–177 www.elsevier.com/locate/entcs Probabilistically Checkable Proofs Over the Reals Klaus Meer1 ,2 Department of Mathematics and Computer Science Syddansk Universitet, Campusvej 55, 5230 Odense M, Denmark Abstract Probabilistically checkable proofs (PCPs) have turned out to be of great importance in complexity theory. On the one hand side they provide a new characterization of the complexity class NP, on the other hand they show a deep connection to approximation results for combinatorial optimization problems. In this paper we study the notion of PCPs in the real number model of Blum, Shub, and Smale. The existence of transparent long proofs for the real number analogue NPR of NP is discussed. Keywords: PCP, real number model, self-testing over the reals. 1 Introduction One of the most important and influential results in theoretical computer science within the last decade is the PCP theorem proven by Arora et al. in 1992, [1,2]. Here, PCP stands for probabilistically checkable proofs, a notion that was further developed out of interactive proofs around the end of the 1980’s. The PCP theorem characterizes the class NP in terms of languages accepted by certain so-called verifiers, a particular version of probabilistic Turing machines. It allows to stabilize verification procedures for problems in NP in the following sense. Suppose for a problem L ∈ NP and a problem 1 partially supported by the EU Network of Excellence PASCAL Pattern Analysis, Statis- tical Modelling and Computational Learning and by the Danish Natural Science Research Council SNF. 2 Email: [email protected] 1571-0661/$ – see front matter © 2005 Elsevier B.V.
    [Show full text]
  • Dspace 6.X Documentation
    DSpace 6.x Documentation DSpace 6.x Documentation Author: The DSpace Developer Team Date: 27 June 2018 URL: https://wiki.duraspace.org/display/DSDOC6x Page 1 of 924 DSpace 6.x Documentation Table of Contents 1 Introduction ___________________________________________________________________________ 7 1.1 Release Notes ____________________________________________________________________ 8 1.1.1 6.3 Release Notes ___________________________________________________________ 8 1.1.2 6.2 Release Notes __________________________________________________________ 11 1.1.3 6.1 Release Notes _________________________________________________________ 12 1.1.4 6.0 Release Notes __________________________________________________________ 14 1.2 Functional Overview _______________________________________________________________ 22 1.2.1 Online access to your digital assets ____________________________________________ 23 1.2.2 Metadata Management ______________________________________________________ 25 1.2.3 Licensing _________________________________________________________________ 27 1.2.4 Persistent URLs and Identifiers _______________________________________________ 28 1.2.5 Getting content into DSpace __________________________________________________ 30 1.2.6 Getting content out of DSpace ________________________________________________ 33 1.2.7 User Management __________________________________________________________ 35 1.2.8 Access Control ____________________________________________________________ 36 1.2.9 Usage Metrics _____________________________________________________________
    [Show full text]
  • 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/.
    [Show full text]
  • UNIONS of LINES in Fn 1. Introduction the Main Problem We
    UNIONS OF LINES IN F n RICHARD OBERLIN Abstract. We show that if a collection of lines in a vector space over a finite field has \dimension" at least 2(d−1)+β; then its union has \dimension" at least d + β: This is the sharp estimate of its type when no structural assumptions are placed on the collection of lines. We also consider some refinements and extensions of the main result, including estimates for unions of k-planes. 1. Introduction The main problem we will consider here is to give a lower bound for the dimension of the union of a collection of lines in terms of the dimension of the collection of lines, without imposing a structural hy- pothesis on the collection (in contrast to the Kakeya problem where one assumes that the lines are direction-separated, or perhaps satisfy the weaker \Wolff axiom"). Specifically, we are motivated by the following conjecture of D. Ober- lin (hdim denotes Hausdorff dimension). Conjecture 1.1. Suppose d ≥ 1 is an integer, that 0 ≤ β ≤ 1; and that L is a collection of lines in Rn with hdim(L) ≥ 2(d − 1) + β: Then [ (1) hdim( L) ≥ d + β: L2L The bound (1), if true, would be sharp, as one can see by taking L to be the set of lines contained in the d-planes belonging to a β- dimensional family of d-planes. (Furthermore, there is nothing to be gained by taking 1 < β ≤ 2 since the dimension of the set of lines contained in a d + 1-plane is 2(d − 1) + 2.) Standard Fourier-analytic methods show that (1) holds for d = 1, but the conjecture is open for d > 1: As a model problem, one may consider an analogous question where Rn is replaced by a vector space over a finite-field.
    [Show full text]
  • The Polynomial Hierarchy
    ij 'I '""T', :J[_ ';(" THE POLYNOMIAL HIERARCHY Although the complexity classes we shall study now are in one sense byproducts of our definition of NP, they have a remarkable life of their own. 17.1 OPTIMIZATION PROBLEMS Optimization problems have not been classified in a satisfactory way within the theory of P and NP; it is these problems that motivate the immediate extensions of this theory beyond NP. Let us take the traveling salesman problem as our working example. In the problem TSP we are given the distance matrix of a set of cities; we want to find the shortest tour of the cities. We have studied the complexity of the TSP within the framework of P and NP only indirectly: We defined the decision version TSP (D), and proved it NP-complete (corollary to Theorem 9.7). For the purpose of understanding better the complexity of the traveling salesman problem, we now introduce two more variants. EXACT TSP: Given a distance matrix and an integer B, is the length of the shortest tour equal to B? Also, TSP COST: Given a distance matrix, compute the length of the shortest tour. The four variants can be ordered in "increasing complexity" as follows: TSP (D); EXACTTSP; TSP COST; TSP. Each problem in this progression can be reduced to the next. For the last three problems this is trivial; for the first two one has to notice that the reduction in 411 j ;1 17.1 Optimization Problems 413 I 412 Chapter 17: THE POLYNOMIALHIERARCHY the corollary to Theorem 9.7 proving that TSP (D) is NP-complete can be used with DP.
    [Show full text]
  • 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 .
    [Show full text]