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Notes for Lecture 2
Notes on Complexity Theory Last updated: December, 2011 Lecture 2 Jonathan Katz 1 Review The running time of a Turing machine M on input x is the number of \steps" M takes before it halts. Machine M is said to run in time T (¢) if for every input x the running time of M(x) is at most T (jxj). (In particular, this means it halts on all inputs.) The space used by M on input x is the number of cells written to by M on all its work tapes1 (a cell that is written to multiple times is only counted once); M is said to use space T (¢) if for every input x the space used during the computation of M(x) is at most T (jxj). We remark that these time and space measures are worst-case notions; i.e., even if M runs in time T (n) for only a fraction of the inputs of length n (and uses less time for all other inputs of length n), the running time of M is still T . (Average-case notions of complexity have also been considered, but are somewhat more di±cult to reason about. We may cover this later in the semester; or see [1, Chap. 18].) Recall that a Turing machine M computes a function f : f0; 1g¤ ! f0; 1g¤ if M(x) = f(x) for all x. We will focus most of our attention on boolean functions, a context in which it is more convenient to phrase computation in terms of languages. A language is simply a subset of f0; 1g¤. -
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/. -
When Subgraph Isomorphism Is Really Hard, and Why This Matters for Graph Databases Ciaran Mccreesh, Patrick Prosser, Christine Solnon, James Trimble
When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases Ciaran Mccreesh, Patrick Prosser, Christine Solnon, James Trimble To cite this version: Ciaran Mccreesh, Patrick Prosser, Christine Solnon, James Trimble. When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases. Journal of Artificial Intelligence Research, Association for the Advancement of Artificial Intelligence, 2018, 61, pp.723 - 759. 10.1613/jair.5768. hal-01741928 HAL Id: hal-01741928 https://hal.archives-ouvertes.fr/hal-01741928 Submitted on 26 Mar 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Journal of Artificial Intelligence Research 61 (2018) 723-759 Submitted 11/17; published 03/18 When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases Ciaran McCreesh [email protected] Patrick Prosser [email protected] University of Glasgow, Glasgow, Scotland Christine Solnon [email protected] INSA-Lyon, LIRIS, UMR5205, F-69621, France James Trimble [email protected] University of Glasgow, Glasgow, Scotland Abstract The subgraph isomorphism problem involves deciding whether a copy of a pattern graph occurs inside a larger target graph. -
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. -
Ex. 8 Complexity 1. by Savich Theorem NL ⊆ DSPACE (Log2n)
Ex. 8 Complexity 1. By Savich theorem NL ⊆ DSP ACE(log2n) ⊆ DSP ASE(n). we saw in one of the previous ex. that DSP ACE(n2f(n)) 6= DSP ACE(f(n)) we also saw NL ½ P SP ASE so we conclude NL ½ DSP ACE(n) ½ DSP ACE(n3) ½ P SP ACE but DSP ACE(n) 6= DSP ACE(n3) as needed. 2. (a) We emulate the running of the s ¡ t ¡ conn algorithm in parallel for two paths (that is, we maintain two pointers, both starts at s and guess separately the next step for each of them). We raise a flag as soon as the two pointers di®er. We stop updating each of the two pointers as soon as it gets to t. If both got to t and the flag is raised, we return true. (b) As A 2 NL by Immerman-Szelepsc¶enyi Theorem (NL = vo ¡ NL) we know that A 2 NL. As C = A \ s ¡ t ¡ conn and s ¡ t ¡ conn 2 NL we get that A 2 NL. 3. If L 2 NICE, then taking \quit" as reject we get an NP machine (when x2 = L we always reject) for L, and by taking \quit" as accept we get coNP machine for L (if x 2 L we always reject), and so NICE ⊆ NP \ coNP . For the other direction, if L is in NP \coNP , then there is one NP machine M1(x; y) solving L, and another NP machine M2(x; z) solving coL. We can therefore de¯ne a new machine that guesses the guess y for M1, and the guess z for M2. -
Glossary of Complexity Classes
App endix A Glossary of Complexity Classes Summary This glossary includes selfcontained denitions of most complexity classes mentioned in the b o ok Needless to say the glossary oers a very minimal discussion of these classes and the reader is re ferred to the main text for further discussion The items are organized by topics rather than by alphab etic order Sp ecically the glossary is partitioned into two parts dealing separately with complexity classes that are dened in terms of algorithms and their resources ie time and space complexity of Turing machines and complexity classes de ned in terms of nonuniform circuits and referring to their size and depth The algorithmic classes include timecomplexity based classes such as P NP coNP BPP RP coRP PH E EXP and NEXP and the space complexity classes L NL RL and P S P AC E The non k uniform classes include the circuit classes P p oly as well as NC and k AC Denitions and basic results regarding many other complexity classes are available at the constantly evolving Complexity Zoo A Preliminaries Complexity classes are sets of computational problems where each class contains problems that can b e solved with sp ecic computational resources To dene a complexity class one sp ecies a mo del of computation a complexity measure like time or space which is always measured as a function of the input length and a b ound on the complexity of problems in the class We follow the tradition of fo cusing on decision problems but refer to these problems using the terminology of promise problems -
ZPP (Zero Probabilistic Polynomial)
ZPPa (Zero Probabilistic Polynomial) • The class ZPP is defined as RP ∩ coRP. • A language in ZPP has two Monte Carlo algorithms, one with no false positives (RP) and the other with no false negatives (coRP). • If we repeatedly run both Monte Carlo algorithms, eventually one definite answer will come (unlike RP). – A positive answer from the one without false positives. – A negative answer from the one without false negatives. aGill (1977). c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 579 The ZPP Algorithm (Las Vegas) 1: {Suppose L ∈ ZPP.} 2: {N1 has no false positives, and N2 has no false negatives.} 3: while true do 4: if N1(x)=“yes”then 5: return “yes”; 6: end if 7: if N2(x) = “no” then 8: return “no”; 9: end if 10: end while c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 580 ZPP (concluded) • The expected running time for the correct answer to emerge is polynomial. – The probability that a run of the 2 algorithms does not generate a definite answer is 0.5 (why?). – Let p(n) be the running time of each run of the while-loop. – The expected running time for a definite answer is ∞ 0.5iip(n)=2p(n). i=1 • Essentially, ZPP is the class of problems that can be solved, without errors, in expected polynomial time. c 2016 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 581 Large Deviations • Suppose you have a biased coin. • One side has probability 0.5+ to appear and the other 0.5 − ,forsome0<<0.5. -
The Weakness of CTC Qubits and the Power of Approximate Counting
The weakness of CTC qubits and the power of approximate counting Ryan O'Donnell∗ A. C. Cem Sayy April 7, 2015 Abstract We present results in structural complexity theory concerned with the following interre- lated topics: computation with postselection/restarting, closed timelike curves (CTCs), and approximate counting. The first result is a new characterization of the lesser known complexity class BPPpath in terms of more familiar concepts. Precisely, BPPpath is the class of problems that can be efficiently solved with a nonadaptive oracle for the Approximate Counting problem. Similarly, PP equals the class of problems that can be solved efficiently with nonadaptive queries for the related Approximate Difference problem. Another result is concerned with the compu- tational power conferred by CTCs; or equivalently, the computational complexity of finding stationary distributions for quantum channels. Using the above-mentioned characterization of PP, we show that any poly(n)-time quantum computation using a CTC of O(log n) qubits may as well just use a CTC of 1 classical bit. This result essentially amounts to showing that one can find a stationary distribution for a poly(n)-dimensional quantum channel in PP. ∗Department of Computer Science, Carnegie Mellon University. Work performed while the author was at the Bo˘gazi¸ciUniversity Computer Engineering Department, supported by Marie Curie International Incoming Fellowship project number 626373. yBo˘gazi¸ciUniversity Computer Engineering Department. 1 Introduction It is well known that studying \non-realistic" augmentations of computational models can shed a great deal of light on the power of more standard models. The study of nondeterminism and the study of relativization (i.e., oracle computation) are famous examples of this phenomenon. -
Group, Graphs, Algorithms: the Graph Isomorphism Problem
Proc. Int. Cong. of Math. – 2018 Rio de Janeiro, Vol. 3 (3303–3320) GROUP, GRAPHS, ALGORITHMS: THE GRAPH ISOMORPHISM PROBLEM László Babai Abstract Graph Isomorphism (GI) is one of a small number of natural algorithmic problems with unsettled complexity status in the P / NP theory: not expected to be NP-complete, yet not known to be solvable in polynomial time. Arguably, the GI problem boils down to filling the gap between symmetry and regularity, the former being defined in terms of automorphisms, the latter in terms of equations satisfied by numerical parameters. Recent progress on the complexity of GI relies on a combination of the asymptotic theory of permutation groups and asymptotic properties of highly regular combinato- rial structures called coherent configurations. Group theory provides the tools to infer either global symmetry or global irregularity from local information, eliminating the symmetry/regularity gap in the relevant scenario; the resulting global structure is the subject of combinatorial analysis. These structural studies are melded in a divide- and-conquer algorithmic framework pioneered in the GI context by Eugene M. Luks (1980). 1 Introduction We shall consider finite structures only; so the terms “graph” and “group” will refer to finite graphs and groups, respectively. 1.1 Graphs, isomorphism, NP-intermediate status. A graph is a set (the set of ver- tices) endowed with an irreflexive, symmetric binary relation called adjacency. Isomor- phisms are adjacency-preseving bijections between the sets of vertices. The Graph Iso- morphism (GI) problem asks to determine whether two given graphs are isomorphic. It is known that graphs are universal among explicit finite structures in the sense that the isomorphism problem for explicit structures can be reduced in polynomial time to GI (in the sense of Karp-reductions1) Hedrlı́n and Pultr [1966] and Miller [1979]. -
Complexity, Part 2 —
EXPTIME-membership EXPTIME-hardness PSPACE-hardness harder Description Logics: a Nice Family of Logics — Complexity, Part 2 — Uli Sattler1 Thomas Schneider 2 1School of Computer Science, University of Manchester, UK 2Department of Computer Science, University of Bremen, Germany ESSLLI, 18 August 2016 Uli Sattler, Thomas Schneider DL: Complexity (2) 1 EXPTIME-membership EXPTIME-hardness PSPACE-hardness harder Goal for today Automated reasoning plays an important role for DLs. It allows the development of intelligent applications. The expressivity of DLs is strongly tailored towards this goal. Requirements for automated reasoning: Decidability of the relevant decision problems Low complexity if possible Algorithms that perform well in practice Yesterday & today: 1 & 3 Now: 2 Uli Sattler, Thomas Schneider DL: Complexity (2) 2 EXPTIME-membership EXPTIME-hardness PSPACE-hardness harder Plan for today 1 EXPTIME-membership 2 EXPTIME-hardness 3 PSPACE-hardness (not covered in the lecture) 4 Undecidability and a NEXPTIME lower bound ; Uli Thanks to Carsten Lutz for much of the material on these slides. Uli Sattler, Thomas Schneider DL: Complexity (2) 3 EXPTIME-membership EXPTIME-hardness PSPACE-hardness harder And now . 1 EXPTIME-membership 2 EXPTIME-hardness 3 PSPACE-hardness (not covered in the lecture) 4 Undecidability and a NEXPTIME lower bound ; Uli Uli Sattler, Thomas Schneider DL: Complexity (2) 4 EXPTIME-membership EXPTIME-hardness PSPACE-hardness harder EXPTIME-membership We start with an EXPTIME upper bound for concept satisfiability in ALC relative to TBoxes. Theorem The following problem is in EXPTIME. Input: an ALC concept C0 and an ALC TBox T I Question: is there a model I j= T with C0 6= ; ? We’ll use a technique known from modal logic: type elimination [Pratt 1978] The basis is a syntactic notion of a type. -
Physics 212B, Ken Intriligator Lecture 6, Jan 29, 2018 • Parity P Acts As P
Physics 212b, Ken Intriligator lecture 6, Jan 29, 2018 Parity P acts as P = P † = P −1 with P~vP = ~v and P~aP = ~a for vectors (e.g. ~x • − and ~p) and axial pseudo-vectors (e.g. L~ and B~ ) respectively. Note that it is not a rotation. Parity preserves all the basic commutation relations, but it is not necessarily a symmetry of the Hamiltonian. The names scalar and pseudo scalar are often used to denote parity even vs odd: P P = for “scalar” operators (like ~x2 or L~ 2, and P ′P = ′ for pseudo O O O −O scalar operators (like S~ ~x). · Nature respects parity aside from some tiny effects at the fundamental level. E.g. • in electricity and magnetism, we can preserve Maxwell’s equations via V~ ~ V and → − V~ +V~ . axial → axial In QM, parity is represented by a unitary operator such that P ~r = ~r and • | i | − i P 2 = 1. So the eigenvectors of P have eigenvalue 1. A state is even or odd under parity ± if P ψ = ψ ; in position space, ψ( ~x) = ψ(~x). In radial coordinates, parity takes | i ±| i − ± cos θ cos θ and φ φ + π, so P α,ℓ,m =( 1)ℓ α,ℓ,m . This is clear also from the → − → | i − | i ℓ = 1 case, any getting general ℓ from tensor products. If [H,P ] = 0, then the non-degenerate energy eigenstates can be written as P eigen- states. Parity in 1d: PxP = x and P pP = p. The Hamiltonian respects parity if V ( x) = − − − V (x).