5.2 Intractable Problems -- NP Problems
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Jonas Kvarnström Automated Planning Group Department of Computer and Information Science Linköping University
Jonas Kvarnström Automated Planning Group Department of Computer and Information Science Linköping University What is the complexity of plan generation? . How much time and space (memory) do we need? Vague question – let’s try to be more specific… . Assume an infinite set of problem instances ▪ For example, “all classical planning problems” . Analyze all possible algorithms in terms of asymptotic worst case complexity . What is the lowest worst case complexity we can achieve? . What is asymptotic complexity? ▪ An algorithm is in O(f(n)) if there is an algorithm for which there exists a fixed constant c such that for all n, the time to solve an instance of size n is at most c * f(n) . Example: Sorting is in O(n log n), So sorting is also in O( ), ▪ We can find an algorithm in O( ), and so on. for which there is a fixed constant c such that for all n, If we can do it in “at most ”, the time to sort n elements we can do it in “at most ”. is at most c * n log n Some problem instances might be solved faster! If the list happens to be sorted already, we might finish in linear time: O(n) But for the entire set of problems, our guarantee is O(n log n) So what is “a planning problem of size n”? Real World + current problem Abstraction Language L Approximation Planning Problem Problem Statement Equivalence P = (, s0, Sg) P=(O,s0,g) The input to a planner Planner is a problem statement The size of the problem statement depends on the representation! Plan a1, a2, …, an PDDL: How many characters are there in the domain and problem files? Now: The complexity of PLAN-EXISTENCE . -
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. -
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/. -
Sharp Lower Bounds for the Dimension of Linearizations of Matrix Polynomials∗
Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 17, pp. 518-531, November 2008 ELA SHARP LOWER BOUNDS FOR THE DIMENSION OF LINEARIZATIONS OF MATRIX POLYNOMIALS∗ FERNANDO DE TERAN´ † AND FROILAN´ M. DOPICO‡ Abstract. A standard way of dealing with matrixpolynomial eigenvalue problems is to use linearizations. Byers, Mehrmann and Xu have recently defined and studied linearizations of dimen- sions smaller than the classical ones. In this paper, lower bounds are provided for the dimensions of linearizations and strong linearizations of a given m × n matrixpolynomial, and particular lineariza- tions are constructed for which these bounds are attained. It is also proven that strong linearizations of an n × n regular matrixpolynomial of degree must have dimension n × n. Key words. Matrixpolynomials, Matrixpencils, Linearizations, Dimension. AMS subject classifications. 15A18, 15A21, 15A22, 65F15. 1. Introduction. We will say that a matrix polynomial of degree ≥ 1 −1 P (λ)=λ A + λ A−1 + ···+ λA1 + A0, (1.1) m×n where A0,A1,...,A ∈ C and A =0,is regular if m = n and det P (λ)isnot identically zero as a polynomial in λ. We will say that P (λ)issingular otherwise. A linearization of P (λ)isamatrix pencil L(λ)=λX + Y such that there exist unimod- ular matrix polynomials, i.e., matrix polynomials with constant nonzero determinant, of appropriate dimensions, E(λ)andF (λ), such that P (λ) 0 E(λ)L(λ)F (λ)= , (1.2) 0 Is where Is denotes the s × s identity matrix. Classically s =( − 1) min{m, n}, but recently linearizations of smaller dimension have been considered [3]. -
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 . -
Delegating Computation: Interactive Proofs for Muggles∗
Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 108 (2017) Delegating Computation: Interactive Proofs for Muggles∗ Shafi Goldwasser Yael Tauman Kalai Guy N. Rothblum MIT and Weizmann Institute Microsoft Research Weizmann Institute [email protected] [email protected] [email protected] Abstract In this work we study interactive proofs for tractable languages. The (honest) prover should be efficient and run in polynomial time, or in other words a \muggle".1 The verifier should be super-efficient and run in nearly-linear time. These proof systems can be used for delegating computation: a server can run a computation for a client and interactively prove the correctness of the result. The client can verify the result's correctness in nearly-linear time (instead of running the entire computation itself). Previously, related questions were considered in the Holographic Proof setting by Babai, Fortnow, Levin and Szegedy, in the argument setting under computational assumptions by Kilian, and in the random oracle model by Micali. Our focus, however, is on the original inter- active proof model where no assumptions are made on the computational power or adaptiveness of dishonest provers. Our main technical theorem gives a public coin interactive proof for any language computable by a log-space uniform boolean circuit with depth d and input length n. The verifier runs in time n · poly(d; log(n)) and space O(log(n)), the communication complexity is poly(d; log(n)), and the prover runs in time poly(n). In particular, for languages computable by log-space uniform NC (circuits of polylog(n) depth), the prover is efficient, the verifier runs in time n · polylog(n) and space O(log(n)), and the communication complexity is polylog(n). -
I Huvudet På Bender Futurama, Parodi, Satir Och Konsten Att Se På Tv
Lunds universitet Oscar Jansson Avd. för litteraturvetenskap, SOL-centrum LIVR07 Handledare: Paul Tenngart 2012-05-30 I huvudet på Bender Futurama, parodi, satir och konsten att se på tv Innehållsförteckning Förord ......................................................................................................................................... 3 Inledning ..................................................................................................................................... 4 Tidigare forskning och utmärkelser ................................................................................... 7 Bender’s Head, urval och disposition ................................................................................ 9 Teoretiska ramverk och utgångspunkter .................................................................................. 11 Förförståelser, genre och tolkning .................................................................................... 12 Parodi, intertextualitet och implicita agenter ................................................................... 18 Parodi och satir i Futurama ...................................................................................................... 23 Ett inoriginellt medium? ................................................................................................... 26 Animerad sitcom vs. parodi ............................................................................................. 31 ”Try this, kids at home!”: parodins sammanblandade världar ........................................ -
Computational Complexity Theory
500 COMPUTATIONAL COMPLEXITY THEORY A. M. Turing, Collected Works: Mathematical Logic, R. O. Gandy and C. E. M. Yates (eds.), Amsterdam, The Nethrelands: Elsevier, Amsterdam, New York, Oxford, 2001. S. BARRY COOPER University of Leeds Leeds, United Kingdom COMPUTATIONAL COMPLEXITY THEORY Complexity theory is the part of theoretical computer science that attempts to prove that certain transformations from input to output are impossible to compute using a reasonable amount of resources. Theorem 1 below illus- trates the type of ‘‘impossibility’’ proof that can sometimes be obtained (1); it talks about the problem of determining whether a logic formula in a certain formalism (abbreviated WS1S) is true. Theorem 1. Any circuit of AND,OR, and NOT gates that takes as input a WS1S formula of 610 symbols and outputs a bit that says whether the formula is true must have at least 10125gates. This is a very compelling argument that no such circuit will ever be built; if the gates were each as small as a proton, such a circuit would fill a sphere having a diameter of 40 billion light years! Many people conjecture that somewhat similar intractability statements hold for the problem of factoring 1000-bit integers; many public-key cryptosys- tems are based on just such assumptions. It is important to point out that Theorem 1 is specific to a particular circuit technology; to prove that there is no efficient way to compute a function, it is necessary to be specific about what is performing the computation. Theorem 1 is a compelling proof of intractability precisely because every deterministic computer that can be pur- Yu. -
Lecture 10: Space Complexity III
Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Lecture 10: Space Complexity III Arijit Bishnu 27.03.2010 Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Outline 1 Space Complexity Classes: NL and L 2 Reductions 3 NL-completeness 4 The Relation between NL and coNL 5 A Relation Among the Complexity Classes Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Outline 1 Space Complexity Classes: NL and L 2 Reductions 3 NL-completeness 4 The Relation between NL and coNL 5 A Relation Among the Complexity Classes Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). The class NPSPACE is an analog of the class NP. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition L = SPACE(log n). Definition NL = NSPACE(log n). Space Complexity Classes: NL and L Reductions NL-completeness The Relation between NL and coNL A Relation Among the Complexity Classes Space Complexity Classes Definition for Recapitulation S c PSPACE = c>0 SPACE(n ). The class PSPACE is an analog of the class P. Definition for Recapitulation S c NPSPACE = c>0 NSPACE(n ). -
BACCALAURÉAT-Session 2015
BACCALAURÉAT-Session 2015 Epreuve de Discipline Non Linguistique Mathématiques/Anglais The Simpsons and their mathematical secrets Without doubt, the most mathematically sophisticated television show in the history of primetime broadcasting is The Simpsons. Al Jean, the executive producer, went to Harvard University to study mathematics at the age of just 16. Others have similarly impressive degrees in mathematics, and Jeff Westbrook resigned from a senior research post at Yale University to write scripts. "Colonel Homer" features the local movie theatre, called the Springfield Googolplex. In order to appreciate this reference, it is necessary to go back to 1938, when the mathematician Edward Kasner casually mentioned that it would be useful to have a label to describe the number 10100. His nine-year-old nephew Milton Sirotta suggested the word googol. At the same time, he gave a name for a still larger number: 'googolplex'. It was first suggested that a googolplex should be 1 followed by writing zeros until you get tired." The uncle rightly felt that the googolplex would then be a somewhat arbitrary and subjective number, so he redefined the googolplex as 10googol (that is 1 followed by a googol zeros) which is far more zeros that you could fit on a piece of paper the size of the observable universe, even if you used the smallest font imaginable. From The Guardian Weekly, 4thOctober, 2013 Questions 1. Make a short presentation of The Simpsons and the show's writing team. 2. a. How can you write differently the number 10100 ? The number 10−3? b. Explain the rules of powers, the main uses and advantages of powers of ten. -
COMP 355 Advanced Algorithms NP-Completeness: General Definitions Chapter 8 Intro (KT)
COMP 355 Advanced Algorithms NP-Completeness: General Definitions Chapter 8 Intro (KT) 1 Efficiency and Polynomial Time The notion of what we mean by efficient is quite vague. • If n is small, 2푛 run time may be just fine • When 푛 is huge, even 푛2 may be unacceptably slow 2 general classes of combinatorial problems • Those involving brute-force search of all feasible solutions, whose worst-case running time is an exponential function of the input size • Those that are based on a systematic solution, whose worst-case running time is a polynomial function of the input size. An algorithm is said to run in polynomial time if its worst-case running time is 푶(풏풄), where 푐 is a nonnegative constant. Higher worst-case running times, such as cn, 푛!, and 푛푛 are not polynomial time. 2 You can’t be serious! Arguments against using polynomial-time as the definition of efficiency • When n is small, run time of 2푛 much faster than 푂(푛20) • Many problems for which good average case solutions exist, but the worst case complexity may be very bad • On modern architectures, practical efficiency is a function of many issues that have to do with the machine’s internal architecture Mathematical advantages of defining “efficiently solvable” to be “worst- case polynomial time solvable” • The composition of two polynomials is a polynomial – Example: an algorithm that makes 푂(푛2) calls to a function that takes 푂(푛3) time runs in 푂(푛5) time, which is still polynomial. • No need to worry about the distribution of inputs When 푛 is sufficiently large, exponential time algorithms (2푛 time) are not efficient 3 The Emergence of Hard Problems “hard” problem = no known efficient algorithmic solutions exist for these problems. -
The Complexity of Debate Checking
Electronic Colloquium on Computational Complexity, Report No. 16 (2014) The Complexity of Debate Checking H. G¨okalp Demirci1, A. C. Cem Say2 and Abuzer Yakaryılmaz3,∗ 1University of Chicago, Department of Computer Science, Chicago, IL, USA 2Bo˘gazi¸ci University, Department of Computer Engineering, Bebek 34342 Istanbul,˙ Turkey 3University of Latvia, Faculty of Computing, Raina bulv. 19, R¯ıga, LV-1586 Latvia [email protected],[email protected],[email protected] Keywords: incomplete information debates, private alternation, games, probabilistic computation Abstract We study probabilistic debate checking, where a silent resource-bounded verifier reads a di- alogue about the membership of a given string in the language under consideration between a prover and a refuter. We consider debates of partial and zero information, where the prover is prevented from seeing some or all of the messages of the refuter, as well as those of com- plete information. This model combines and generalizes the concepts of one-way interactive proof systems, games of possibly incomplete information, and probabilistically checkable debate systems. We give full characterizations of the classes of languages with debates checkable by verifiers operating under simultaneous bounds of O(1) space and O(1) random bits. It turns out such verifiers are strictly more powerful than their deterministic counterparts, and allowing a logarithmic space bound does not add to this power. PSPACE and EXPTIME have zero- and partial-information debates, respectively, checkable by constant-space verifiers for any desired error bound when we omit the randomness bound. No amount of randomness can give a ver- ifier under only a fixed time bound a significant performance advantage over its deterministic counterpart.