DOCSLIB.ORG

A. Definitions of Reductions and Complexity Classes, and Notation List

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A. Definitions of Reductions and Complexity Classes, and Notation List A. Definitions of Reductions and Complexity Classes, and N otation List A.I Reductions Below, in naming the reductions, "polynomial-time" is implicit unless ot her­ wise noted, e.g., "many-one reduction" means "many-one polynomial-time reduction." This list sorts among t he :Sb first by b, and in the case of identical b's, bya. :S -y - Gamma reduction. -A :S -y B if there is a nondeterministic polynomial-time machine N such that (a) for each st ring x, N(x) has at least one accepting path, and (b) for each string x and each accepting path p of N(x) it holds that: x E A if and only if the output of N(x) on path p is an element of B. :S ~ - Many-one reduction. -A :S~ B if (3f E FP)(Vx)[x E A {=} f(x) EB] . :S~ o s - (Globally) Positive Turing reduction. -A :S~o s B if there is a deterministic Turing machine Mand a polyno­ mial q such that 1. (VD)[runtimeMD(x):s q(lxl)], B 2. A = L(M ), and 3. (VC, D)[C ç D =} L(MC) ç L(MD)]. :S:08 - Locally positive Turing reduction. -A :S:os B if there is a deterministic polynomial-time Turing machine M such that 1. A = L(MB), 2. (VC)[L(M BUC) :2 L(MB)] , and 3. (VC)[L(M B- C) ç L(MB)] . :ST - Exponential-time Turing reduction. -A :ST B if A E EB. If A E EB via an exponential-time machine, M, having the property that on each input of length n it holds that M makes at most g(n) queries to its orade, we write A :S~ ( n ) - T B. If for some polynomial g(n) it holds that A :S~( n)-T B, we say A :S~olY-T B. :S ~ - Turing reduction. -A :S~ B if A E pB. 116 A. Definitions of Reductions and Complexity Classes, and Notation List ~Tn - Strong nondeterministic 'Iuring reduction. -A ~Tn B if A E NpB ncoNpB. <p <p <p <p -tt, -ptt' -btt' -k-tt - Various types of truth-table reductions. - Truth-table reductions are often defined in terms of separate "query generation" and "answer interpretation" machines. In this book, we will more commonly use a more intuitive, though equivalent, formu­ lation. We say that A ~rt B if there is a 'Iuring machine, having a designated query tape, that on each input x will enter a special query state at most once . When it does, our model is that the content of the query tape is interpreted as a collection of strings (say, each separated by the character "#"), and in one time step the content of the query tape is replaced by the bitstring bsbz:': bz , where bi is 1 if the ith string on the tape is an element of Band is 0 otherwise, and where z is the number of strings that were on the query tape. The Turing machine in this same step has its state moved to a special query-was­ just-answered state. (It is henceforth not allowed to query.) It then continues on , eventually accepting or rejecting. Note in particular that the query tape is readabie, and so our Turing machine may read the bitstring b1b2 ... bz , and these bits may affect the acceptance. For each natural number k, if A ~rt B via a Turing machine M of the form described above then we say that A ~~-tt B if on no input does the machine ever enter the query state with more than k - 1 "#" characters on its query tape. If there is a natural number k such that A ~~-tt B, then we say that A ~~tt B. The reducibility ~~tt is referred to as bounded-truth-table reducibility. Suppose that A ~rt B via a Turing machine M of the form described in the above definition of ~rt . We will, when this use is clear from content (i.e., when the machine is a truth-table machine), use MB to denote the computation of M when B is used to answer the query list written to the oracle tape. We say that A ~~tt B (A positive-truth­ table reduces to B) if there exists a polynomial-time Turing machine M (that functions as a truth-table machine) such that A ~rt B via M, and (VC, D)[L(Mc ) ç L(Mc UD)]. A.2 Complexity Classes This list is, with some exceptions for clarity, alphabetically "word-ordered." Also, it orders Greek letters under their Romanized versions (e.g., I;'s are alphabetized as if they were "Sigma"s). co· C - Set-wise complements of complexity class C. A.2 Complexity Classes 117 -A E co . C if A E C. C-:=:;r-complete - Let :=:;r be any reducibility and let C be any complexity class. A set A is said to be C-:=:;r-complete if A E C and A is C-:=:;r-hard. C-:=:;r-hard - Let :=:;r be any reducibility and let C be any complexity class . A set A is said to be C-:=:;r-hard if, for each B E C, B :=:;r A. C-complete - Let C be any complexity class. A set is said to be C-complete if it is C-:=:;~ -complete. C-hard - Let C be any complexity class, A set is said to be C-hard if it is C-:=:;~-hard. C/{F} -A token-based advice class. - Let F be any col1ection of (total) functions from N to N+. Let C be any col1ection of sets. Define C/{F} = {A I(3g E F)[A E C/{g}J). CIU} -A token-based advice class . - Let i : N ---. N+. Assume that natural numbers have their standard encoding over binary strings. Let C be any col1ection of sets. Define CIU} = {A I(3B E C)(3h : N ---. N+) [(\ln)[h(n) E {I, .. ., i(n)}] and (\Ix E I;*)[x E A {=} (x, h(lxl)) E B]]}. ClF - A length-based advice class. - Let F be any class of (total) functions mapping from N to No Define CIF = {A I(3i E F)[A E Clij). - One common value of F is "poly," which denotes the class of polyno­ mials. CIi - A length-based advice class . - Let i : N ---. N be any (total) function. Let C be any col1ection of sets. Define Cli = {A I (3B E C)(3h : N ---. {a, I }*) [(\ln)[lh(n)I i(n)]/\ (\Ix E I;*)[x E A {=} (x, h(lxl)) EB]]}. coNP - Co-nondeterministic polynomial time. -A E coNP if A E NP. ó~, k ~ 0 - The kth "ó" level of the polynomial hierarchy. _ I\P _ I\p,0 Uk - uk . Óp,A k > 0 k' - - The kth "ó" level of the polynomial hierarchy relativized via oracle A. _ uaI\p,A -.'_ pA 118 A. Definitions of Reductions and Complexity Classes, and Notation List pA EP.A - For k 2 1, /:),.k' = P k-l . - See, however, the discussion in footnote 6 (on page 43), since the analog of that holds here . Disjunctively self-reducible sets -A set A is said to be disjunctively self-reducible if there is a deterrnin­ istic polynomial-time Turing machine M such that 1. A = L(MA), 2. for each x, MA(x) queries only strings of lengths strictly less than lxi , and 3. the acceptance behavior of M is such that on each input M accepts exactly when either (a) M asks at least one query that is in the oracle set, or (b) M asks no queries and halts in an accepting state. If A is disjunctively self-reducible via a machine M that on each input asks at most two oracle questions, then we say that A is 2-disjunctively self-reducible. DSPACE[f(n)] - Deterministic space f (n). -A E DSPACE[f(n)] if A is accepted by a deterministic Turing machine whose running space is O(J(n)). DTIME[J(n)] - Deterministic time f (n). -A E DTIME[f(n)] if A is accepted by a deterministic Turing machine whose running time is O(J(n)). E- Deterministic exponential time. -E = Uk>o DTIME[2 kn]. EL6.~' k22 - The /:),.1 extended low sets. -A E EL6.~ if /:),.1 ,A ç /:),.1~lEBSAT . EL~p, k > 2 k - - The /:),.1 extended low sets in relativized world W. -A E EL~~ if /:),.1,AEBW ç /:),.1~lEBSATEBW . ELH - The extended low hierarchy. - ELH = Uk~2 ELE~' ELE~' k 2 2 - The E1 extended low sets. -A E ELE~ if E1,A ç E1~lEBSAT. ELe~, k 2 2 - The 81 extended low sets. -A E ELe~ if 81,A ç 81~lEBSAT. Er(C) - The sets :::::>equivalent to some set in C. -A E Er(C) if (:JB E C)[A :::;r B 1\ B :::;r A]. - Note: :::;r must he a defined reduction type. A.2 Complexity Classes 119 EXP - Deterministic "polynomial exponential" time. nkj. - EXP = Uk>O DTIME[2 F-sel - Selectivity via general functions. - Let F be a class of functions. We say a set A is F-selective if there is an f E F such that, for each x and y, 1. set- f(x, y) ç {x, y}, and 2. ifAn{x,y}=f:.0then0=f:.set-f(x,y)ÇA. We say such a function f is an F-selector for A. Let F be a class of functions. F-sel denotes {A IA is F-selective}. FewP - Polynornial-ambiguity (nondeterministic) polynomial time.
Load more

Recommended publications
  • 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¤.
    [Show full text]
  • 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]
  • Computational Complexity Theory Introduction to Computational
    31/03/2012 Computational complexity theory Introduction to computational complexity theory Complexity (computability) theory theory deals with two aspects: Algorithm’s complexity. Problem’s complexity. References S. Cook, « The complexity of Theorem Proving Procedures », 1971. Garey and Johnson, « Computers and Intractability, A guide to the theory of NP-completeness », 1979. J. Carlier et Ph. Chrétienne « Problèmes d’ordonnancements : algorithmes et complexité », 1988. 1 31/03/2012 Basic Notions • Some problem is a “question” characterized by parameters and needs an answer. – Parameters description; – Properties that a solutions must satisfy; – An instance is obtained when the parameters are fixed to some values. • An algorithm: a set of instructions describing how some task can be achieved or a problem can be solved. • A program : the computational implementation of an algorithm. Algorithm’s complexity (I) • There may exists several algorithms for the same problem • Raised questions: – Which one to choose ? – How they are compared ? – How measuring the efficiency ? – What are the most appropriate measures, running time, memory space ? 2 31/03/2012 Algorithm’s complexity (II) • Running time depends on: – The data of the problem, – Quality of program..., – Computer type, – Algorithm’s efficiency, – etc. • Proceed by analyzing the algorithm: – Search for some n characterizing the data. – Compute the running time in terms of n. – Evaluating the number of elementary operations, (elementary operation = simple instruction of a programming language). Algorithm’s evaluation (I) • Any algorithm is composed of two main stages: initialization and computing one • The complexity parameter is the size data n (binary coding). Definition: Let be n>0 andT(n) the running time of an algorithm expressed in terms of the size data n, T(n) is of O(f(n)) iff n0 and some constant c such that: n n0, we have T(n) c f(n).
    [Show full text]
  • Submitted Thesis
    Saarland University Faculty of Mathematics and Computer Science Bachelor’s Thesis Synthetic One-One, Many-One, and Truth-Table Reducibility in Coq Author Advisor Felix Jahn Yannick Forster Reviewers Prof. Dr. Gert Smolka Prof. Dr. Markus Bläser Submitted: September 15th, 2020 ii Eidesstattliche Erklärung Ich erkläre hiermit an Eides statt, dass ich die vorliegende Arbeit selbstständig ver- fasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet habe. Statement in Lieu of an Oath I hereby confirm that I have written this thesis on my own and that I have not used any other media or materials than the ones referred to in this thesis. Einverständniserklärung Ich bin damit einverstanden, dass meine (bestandene) Arbeit in beiden Versionen in die Bibliothek der Informatik aufgenommen und damit veröffentlicht wird. Declaration of Consent I agree to make both versions of my thesis (with a passing grade) accessible to the public by having them added to the library of the Computer Science Department. Saarbrücken, September 15th, 2020 Abstract Reducibility is an essential concept for undecidability proofs in computability the- ory. The idea behind reductions was conceived by Turing, who introduced the later so-called Turing reduction based on oracle machines. In 1944, Post furthermore in- troduced with one-one, many-one, and truth-table reductions in comparison to Tur- ing reductions more specific reducibility notions. Post then also started to analyze the structure of the different reducibility notions and their computability degrees. Most undecidable problems were reducible from the halting problem, since this was exactly the method to show them undecidable. However, Post was able to con- struct also semidecidable but undecidable sets that do not one-one, many-one, or truth-table reduce from the halting problem.
    [Show full text]
  • On Uniformity Within NC
    On Uniformity Within NC David A Mix Barrington Neil Immerman HowardStraubing University of Massachusetts University of Massachusetts Boston Col lege Journal of Computer and System Science Abstract In order to study circuit complexity classes within NC in a uniform setting we need a uniformity condition which is more restrictive than those in common use Twosuch conditions stricter than NC uniformity RuCo have app eared in recent research Immermans families of circuits dened by rstorder formulas ImaImb and a unifor mity corresp onding to Buss deterministic logtime reductions Bu We show that these two notions are equivalent leading to a natural notion of uniformity for lowlevel circuit complexity classes Weshow that recent results on the structure of NC Ba still hold true in this very uniform setting Finallyweinvestigate a parallel notion of uniformity still more restrictive based on the regular languages Here we givecharacterizations of sub classes of the regular languages based on their logical expressibility extending recentwork of Straubing Therien and Thomas STT A preliminary version of this work app eared as BIS Intro duction Circuit Complexity Computer scientists have long tried to classify problems dened as Bo olean predicates or functions by the size or depth of Bo olean circuits needed to solve them This eort has Former name David A Barrington Supp orted by NSF grant CCR Mailing address Dept of Computer and Information Science U of Mass Amherst MA USA Supp orted by NSF grants DCR and CCR Mailing address Dept of
    [Show full text]
  • NP-Completeness: Reductions Tue, Nov 21, 2017
    CMSC 451 Dave Mount CMSC 451: Lecture 19 NP-Completeness: Reductions Tue, Nov 21, 2017 Reading: Chapt. 8 in KT and Chapt. 8 in DPV. Some of the reductions discussed here are not in either text. Recap: We have introduced a number of concepts on the way to defining NP-completeness: Decision Problems/Language recognition: are problems for which the answer is either yes or no. These can also be thought of as language recognition problems, assuming that the input has been encoded as a string. For example: HC = fG j G has a Hamiltonian cycleg MST = f(G; c) j G has a MST of cost at most cg: P: is the class of all decision problems which can be solved in polynomial time. While MST 2 P, we do not know whether HC 2 P (but we suspect not). Certificate: is a piece of evidence that allows us to verify in polynomial time that a string is in a given language. For example, the language HC above, a certificate could be a sequence of vertices along the cycle. (If the string is not in the language, the certificate can be anything.) NP: is defined to be the class of all languages that can be verified in polynomial time. (Formally, it stands for Nondeterministic Polynomial time.) Clearly, P ⊆ NP. It is widely believed that P 6= NP. To define NP-completeness, we need to introduce the concept of a reduction. Reductions: The class of NP-complete problems consists of a set of decision problems (languages) (a subset of the class NP) that no one knows how to solve efficiently, but if there were a polynomial time solution for even a single NP-complete problem, then every problem in NP would be solvable in polynomial time.
    [Show full text]
  • Module 34: Reductions and NP-Complete Proofs
    Module 34: Reductions and NP-Complete Proofs This module 34 focuses on reductions and proof of NP-Complete problems. The module illustrates the ways of reducing one problem to another. Then, the module illustrates the outlines of proof of NP- Complete problems with some examples. The objectives of this module are To understand the concept of Reductions among problems To Understand proof of NP-Complete problems To overview proof of some Important NP-Complete problems. Computational Complexity There are two types of complexity theories. One is related to algorithms, known as algorithmic complexity theory and another related to complexity of problems called computational complexity theory [2,3]. Algorithmic complexity theory [3,4] aims to analyze the algorithms in terms of the size of the problem. In modules 3 and 4, we had discussed about these methods. The size is the length of the input. It can be recollected from module 3 that the size of a number n is defined to be the number of binary bits needed to write n. For example, Example: b(5) = b(1012) = 3. In other words, the complexity of the algorithm is stated in terms of the size of the algorithm. Asymptotic Analysis [1,2] refers to the study of an algorithm as the input size reaches a limit and analyzing the behaviour of the algorithms. The asymptotic analysis is also science of approximation where the behaviour of the algorithm is expressed in terms of notations such as big-oh, Big-Omega and Big- Theta. Computational complexity theory is different. Computational complexity aims to determine complexity of the problems itself.
    [Show full text]
  • 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 .
    [Show full text]
  • Mapping Reducibility and Rice's Theorem
    6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 9 Nancy Lynch Today • Mapping reducibility and Rice’s Theorem • We’ve seen several undecidability proofs. • Today we’ll extract some of the key ideas of those proofs and present them as general, abstract definitions and theorems. • Two main ideas: – A formal definition of reducibility from one language to another. Captures many of the reduction arguments we have seen. – Rice’s Theorem, a general theorem about undecidability of properties of Turing machine behavior (or program behavior). Today • Mapping reducibility and Rice’s Theorem • Topics: – Computable functions. – Mapping reducibility, ≤m – Applications of ≤m to show undecidability and non- recognizability of languages. – Rice’s Theorem – Applications of Rice’s Theorem • Reading: – Sipser Section 5.3, Problems 5.28-5.30. Computable Functions Computable Functions • These are needed to define mapping reducibility, ≤m. • Definition: A function f: Σ1* →Σ2* is computable if there is a Turing machine (or program) such that, for every w in Σ1*, M on input w halts with just f(w) on its tape. • To be definite, use basic TM model, except replace qacc and qrej states with one qhalt state. • So far in this course, we’ve focused on accept/reject decisions, which let TMs decide language membership. • That’s the same as computing functions from Σ* to { accept, reject }. • Now generalize to compute functions that produce strings. Total vs. partial computability • We require f to be total = defined for every string. • Could also define partial computable (= partial recursive) functions, which are defined on some subset of Σ1*.
    [Show full text]
  • Succinctness of the Complement and Intersection of Regular Expressions Wouter Gelade, Frank Neven
    Succinctness of the Complement and Intersection of Regular Expressions Wouter Gelade, Frank Neven To cite this version: Wouter Gelade, Frank Neven. Succinctness of the Complement and Intersection of Regular Expres- sions. STACS 2008, Feb 2008, Bordeaux, France. pp.325-336. hal-00226864 HAL Id: hal-00226864 https://hal.archives-ouvertes.fr/hal-00226864 Submitted on 30 Jan 2008 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. Symposium on Theoretical Aspects of Computer Science 2008 (Bordeaux), pp. 325-336 www.stacs-conf.org SUCCINCTNESS OF THE COMPLEMENT AND INTERSECTION OF REGULAR EXPRESSIONS WOUTER GELADE AND FRANK NEVEN Hasselt University and Transnational University of Limburg, School for Information Technology E-mail address: [email protected] Abstract. We study the succinctness of the complement and intersection of regular ex- pressions. In particular, we show that when constructing a regular expression defining the complement of a given regular expression, a double exponential size increase cannot be avoided. Similarly, when constructing a regular expression defining the intersection of a fixed and an arbitrary number of regular expressions, an exponential and double expo- nential size increase, respectively, can in worst-case not be avoided.
    [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]
  • 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.
    [Show full text]