DOCSLIB.ORG

Models and Algorithms for Metabolic Networks: Elementary Modes and Precursor Sets Vicente Acuña

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Models and Algorithms for Metabolic Networks: Elementary Modes and Precursor Sets Vicente Acuña Models and algorithms for metabolic networks: elementary modes and precursor sets Vicente Acuña To cite this version: Vicente Acuña. Models and algorithms for metabolic networks: elementary modes and precursor sets. Algorithme et structure de données [cs.DS]. Université Claude Bernard - Lyon I, 2010. Français. tel-00850705 HAL Id: tel-00850705 https://tel.archives-ouvertes.fr/tel-00850705 Submitted on 8 Aug 2013 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. No d’ordre: 218-2008 Année 2009 Thèse Présentée devant l’Université Claude Bernard - Lyon 1 pour l’obtention du Diplôme de Doctorat (arrêté du 7 août 2006) et soutenue publiquement le 4 Juin 2010 par Vicente Acuña Models and algorithms for metabolic networks: elementary modes and precursor sets Directrice de thèse: Marie-France Sagot Co-directeur de thèse: Christian Gautier Jury: Pierluigi Crescenzi, Rapporteur Khaled Elbassioni, Rapporteur Christian Gautier, Directeur Dominique Perrin, Examinateur Marie-France Sagot, Directrice Alain Viari, Président UNIVERSITÉ CLAUDE BERNARD-LYON 1 Président de l’Université M. le Professeur L. COLLET Vice-Président du Conseil Scientifique M. le Professeur J. F. MORNEX Vice-Président du Conseil d’Administration M. le Professeur G. ANNAT Vice-Président du Conseil des Etudes et M. le Professeur D. SIMON de la Vie Universitaire Secrétaire Général M. G. GAY COMPOSANTES SANTÉ UFR de Médecine Lyon-Est – Claude Bernard Directeur: M. le Professeur J. ETIENNE UFR de Médecine Lyon-Sud – Charles Mérieux Directeur: M. le Professeur F-N. GILLY UFR d’Ontologie Directeur: M. le professeur D. BOURGEOIS Institut des Sciences Pharmaceutiques et Bi- Directeur: M. le Professeur F. LOCHER ologiques Institut des Sciences et Techniques de Réadapta- Directeur: M. le Professeur Y. MATILLON tion Département de Formation et Centre de Recherche Directeur: M. le Professeur P. FARGE en Biologie Humaine COMPOSANTES SCIENCES ET TECHNOLOGIE Faculté des Sciences et Technologies Directeur: M. le Professeur F. GIERES UFR Sciences et Techniques des Activités Directeur: M. C. Collignon Physiques et Sportives Observatoire de Lyon Directeur: M. B. Guiderdoni Institut des Sciences et des Techniques de Directeur: M. le Professeur J. LIETO l’Ingénieur de Lyon Institut Universitaire de Technologie A Directeur: M. le Professeur C. COULET Institut Universitaire de Technologie B Directeur: M. le Professeur R. LAMARTINE Institut de Science Financière et d’Assurances Directeur: M. le Professeur J-C. AUGROS Abstract In this PhD, we present some algorithms and complexity results for two general prob- lems that arise in the analysis of a metabolic network: the search for elementary modes of a network and the search for minimal precursors sets. Elementary modes is a common tool in the study of the cellular characteristic of a metabolic network. An elementary mode can be seen as a minimal set of reactions that can work in steady state independently of the rest of the network. It has there- fore served as a mathematical model for the possible metabolic pathways of a cell. Their computation is not trivial and poses computational challenges. We show that some problems, like checking consistency of a network, finding one elementary mode or checking that a set of reactions constitutes a cut are easy problems, giving polynomial algorithms based on LP formulations. We also prove the hardness of central problems like finding a minimum size elementary mode, finding an elementary mode containing two given reactions, counting the number of elementary modes or finding a minimum reaction cut. On the enumeration problem, we show that enumerating all reactions containing one given reaction cannot be done in polynomial total time unless P=NP. This result provides some idea about the complexity of enumerating all the elementary modes. The search for precursor sets is motivated by discovering which external metabolites are sufficient to allow the production of a given set of target metabolites. In contrast with previous proposals, we present a new approach which is the first to formally consider the use of cycles in the way to produce the target. We present a polynomial algorithm to decide whether a set is a precursor set of a given target. We also show that, given a target set, finding a minimal precursor set is easy but finding a precursor set of minimum size is NP-hard. We further show that finding a solution with minimum size internal supply is NP-hard. We give a simple characterisation of precursors sets by the existence of hyperpaths between the solutions and the target. If we consider the enumeration of all the minimal precursor sets of a given target, we find that this problem cannot be solved in polynomial total time unless P=NP. Despite this result, we present two algorithms that have good performance for medium-size networks. 6 Contents Introduction 11 1 Some Basic Mathematical Definitions 19 1.1 Graphs, digraphs and hypergraphs .................... 19 1.1.1 Graph and digraph definitions ................... 20 1.1.2 Graphical representation and labels . 20 1.1.3 Walks, paths, cycles and hamiltonian cycles . 20 1.1.4 Adjacency and incidence matrix . 21 1.1.5 Induced subgraph, bipartite graphs and trees . 21 1.1.6 Directed hypergraphs ........................ 22 1.2 Hitting set .................................. 23 1.3 Boolean functions .............................. 24 1.3.1 Monotone Boolean functions .................... 25 2 Basic Concepts of Time Complexity Analysis 27 2.1 Defining a problem ............................. 28 2.1.1 Decision problems .......................... 28 2.1.2 Optimisation problems ....................... 29 2.1.3 Enumeration and counting problems . 29 2.2 Analysis of algorithms ........................... 30 2.2.1 Input size .............................. 31 2.2.2 Worst case analysis ......................... 31 2.2.3 Asymptotic analysis ........................ 31 2.3 Complexity classes of decision problems . 32 2.3.1 The class P ............................. 32 2.3.2 The class Np ............................ 32 2.3.3 Reducibility among problems ................... 33 2.3.4 Np-complete problems ....................... 33 2.4 Complexity classes of optimisation problems . 34 2.4.1 The classes Po and Npo ...................... 34 2.4.2 Np-hard optimisation problems . 35 2.4.3 Approximation algorithms ..................... 35 2.4.4 The classes Apx and Apx-hard . 36 2.5 Complexity of counting solutions ..................... 36 8 CONTENTS 2.5.1 ♯P and ♯P-complete ......................... 36 2.6 Complexity of enumerating all the solutions . 37 2.6.1 Time delay, incremental time and total time . 37 3 Metabolic Networks 39 3.1 Entities involved in metabolism ...................... 40 3.1.1 Biochemical reactions and metabolites . 40 3.1.2 Enzymes and genes ......................... 41 3.1.3 Metabolism regulation ....................... 41 3.1.4 Reconstructing a metabolic network . 42 3.2 Modelling metabolic networks ....................... 43 3.2.1 Graph and hypergraphs models . 43 3.2.2 Including stoichiometry ....................... 44 3.2.3 Assuming steady state ....................... 44 4 Complexity of Computing Elementary Modes 47 4.1 Modelling metabolic network in steady state . 48 4.1.1 Definitions .............................. 48 4.1.2 Relation between elementary modes and extreme rays . 50 4.1.3 Reversibility of reactions ...................... 50 4.2 Checking consistency of the stoichiometric matrix . 51 4.3 Finding elementary modes ......................... 53 4.3.1 Finding an elementary mode .................... 53 4.3.2 Finding elementary modes with support containing a given set of reactions ............................. 54 4.3.3 Finding the shortest elementary modes . 56 4.4 Counting elementary modes ........................ 59 4.5 Enumerating elementary modes ...................... 60 4.5.1 Enumerating elementary modes with a given reaction in its support 61 4.5.2 Analysis of the complexity result . 61 4.5.3 Case when all reactions are reversible . 62 4.6 Reaction cuts ................................ 62 4.6.1 Finding minimal and minimum reaction cuts . 63 4.7 Proof of Theorem 4.11 and Theorem 4.15 ................ 65 5 Modelling Precursor Sets in Metabolic Networks 71 5.1 Definitions and Characterisations ..................... 71 5.1.1 Modelling a metabolic network . 71 5.1.2 Forward propagation ........................ 72 5.1.3 Definition of precursor sets considering cycles . 73 5.1.4 Alternative characterisation of precursor set . 74 5.1.5 Maximal target ........................... 75 5.1.6 Hyperpaths from sources to the target . 76 5.1.7 Precursor cut set .......................... 77 CONTENTS 9 5.2 Complexity results ............................. 77 5.2.1 Deciding if a set of sources is a precursor set . 78 5.2.2 Finding a minimal and a minimum precursor sets . 78 5.2.3 Enumerating all minimal precursor sets . 81 6 Algorithms to Enumerate All Minimal Precursor Sets 85 6.1 Preprocessing the network ......................... 85 6.2 The replacement tree ...........................
Load more

Recommended publications
  • CS 512, Spring 2017, Handout 10 [1Ex] Propositional Logic: [2Ex
    CS 512, Spring 2017, Handout 10 Propositional Logic: Conjunctive Normal Forms, Disjunctive Normal Forms, Horn Formulas, and other special forms Assaf Kfoury 5 February 2017 Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 1 of 28 CNF L ::= p j :p literal D ::= L j L _ D disjuntion of literals C ::= D j D ^ C conjunction of disjunctions DNF L ::= p j :p literal C ::= L j L ^ C conjunction of literals D ::= C j C _ D disjunction of conjunctions conjunctive normal form & disjunctive normal form Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 2 of 28 DNF L ::= p j :p literal C ::= L j L ^ C conjunction of literals D ::= C j C _ D disjunction of conjunctions conjunctive normal form & disjunctive normal form CNF L ::= p j :p literal D ::= L j L _ D disjuntion of literals C ::= D j D ^ C conjunction of disjunctions Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 3 of 28 conjunctive normal form & disjunctive normal form CNF L ::= p j :p literal D ::= L j L _ D disjuntion of literals C ::= D j D ^ C conjunction of disjunctions DNF L ::= p j :p literal C ::= L j L ^ C conjunction of literals D ::= C j C _ D disjunction of conjunctions Assaf Kfoury, CS 512, Spring 2017, Handout 10 page 4 of 28 I A disjunction of literals L1 _···_ Lm is valid (equivalently, is a tautology) iff there are 1 6 i; j 6 m with i 6= j such that Li is :Lj.
    [Show full text]
  • Advanced Complexity Theory
    Advanced Complexity Theory Markus Bl¨aser& Bodo Manthey Universit¨atdes Saarlandes Draft|February 9, 2011 and forever 2 1 Complexity of optimization prob- lems 1.1 Optimization problems The study of the complexity of solving optimization problems is an impor- tant practical aspect of complexity theory. A good textbook on this topic is the one by Ausiello et al. [ACG+99]. The book by Vazirani [Vaz01] is also recommend, but its focus is on the algorithmic side. Definition 1.1. An optimization problem P is a 4-tuple (IP ; SP ; mP ; goalP ) where ∗ 1. IP ⊆ f0; 1g is the set of valid instances of P , 2. SP is a function that assigns to each valid instance x the set of feasible ∗ 1 solutions SP (x) of x, which is a subset of f0; 1g . + 3. mP : f(x; y) j x 2 IP and y 2 SP (x)g ! N is the objective function or measure function. mP (x; y) is the objective value of the feasible solution y (with respect to x). 4. goalP 2 fmin; maxg specifies the type of the optimization problem. Either it is a minimization or a maximization problem. When the context is clear, we will drop the subscript P . Formally, an optimization problem is defined over the alphabet f0; 1g. But as usual, when we talk about concrete problems, we want to talk about graphs, nodes, weights, etc. In this case, we tacitly assume that we can always find suitable encodings of the objects we talk about. ∗ Given an instance x of the optimization problem P , we denote by SP (x) the set of all optimal solutions, that is, the set of all y 2 SP (x) such that mP (x; y) = goalfmP (x; z) j z 2 SP (x)g: (Note that the set of optimal solutions could be empty, since the maximum need not exist.
    [Show full text]
  • APX Radio Family Brochure
    APX MISSION-CRITICAL P25 COMMUNICATIONS BROCHURE APX P25 COMMUNICATIONS THE BEST OF WHAT WE DO Whether you’re a state trooper, firefighter, law enforcement officer or highway maintenance technician, people count on you to get the job done. There’s no room for error. This is mission critical. APX™ radios exist for this purpose. They’re designed to be reliable and to optimize your communications, specifically in extreme environments and during life-threatening situations. Even with the widest portfolio in the industry, APX continues to evolve. The latest APX NEXT smart radio series delivers revolutionary new capabilities to keep you safer and more effective. WE’VE PUT EVERYTHING WE’VE LEARNED OVER THE LAST 90 YEARS INTO APX. THAT’S WHY IT REPRESENTS THE VERY BEST OF THE MOTOROLA SOLUTIONS PORTFOLIO. THERE IS NO BETTER. BROCHURE APX P25 COMMUNICATIONS OUTLAST AND OUTPERFORM RELIABLE COMMUNICATIONS ARE NON-NEGOTIABLE APX two-way radios are designed for extreme durability, so you can count on them to work under the toughest conditions. From the rugged aluminum endoskeleton of our portable radios to the steel encasement of our mobile radios, APX is built to last. Pressure-tested HEAR AND BE HEARD tempered glass display CLEAR COMMUNICATION CAN MAKE A DIFFERENCE The APX family is designed to help you hear and be heard with unparalleled clarity, so you’re confident your message will always get through. Multiple microphones and adaptive windporting technology minimize noise from wind, crowds and sirens. And the loud and clear speaker ensures you can hear over background sounds. KEEP INFORMATION PROTECTED EVERYDAY, SECURITY IS BEING PUT TO THE TEST With the APX family, you can be sure that your calls stay private, secure, and confidential.
    [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]
  • Boolean Algebra
    Boolean Algebra Definition: A Boolean Algebra is a math construct (B,+, . , ‘, 0,1) where B is a non-empty set, + and . are binary operations in B, ‘ is a unary operation in B, 0 and 1 are special elements of B, such that: a) + and . are communative: for all x and y in B, x+y=y+x, and x.y=y.x b) + and . are associative: for all x, y and z in B, x+(y+z)=(x+y)+z, and x.(y.z)=(x.y).z c) + and . are distributive over one another: x.(y+z)=xy+xz, and x+(y.z)=(x+y).(x+z) d) Identity laws: 1.x=x.1=x and 0+x=x+0=x for all x in B e) Complementation laws: x+x’=1 and x.x’=0 for all x in B Examples: • (B=set of all propositions, or, and, not, T, F) • (B=2A, U, ∩, c, Φ,A) Theorem 1: Let (B,+, . , ‘, 0,1) be a Boolean Algebra. Then the following hold: a) x+x=x and x.x=x for all x in B b) x+1=1 and 0.x=0 for all x in B c) x+(xy)=x and x.(x+y)=x for all x and y in B Proof: a) x = x+0 Identity laws = x+xx’ Complementation laws = (x+x).(x+x’) because + is distributive over . = (x+x).1 Complementation laws = x+x Identity laws x = x.1 Identity laws = x.(x+x’) Complementation laws = x.x +x.x’ because + is distributive over .
    [Show full text]
  • MODULES and ITS REVERSES 1. Introduction the Hölder Inequality
    Ann. Funct. Anal. A nnals of F unctional A nalysis ISSN: 2008-8752 (electronic) URL: www.emis.de/journals/AFA/ HOLDER¨ TYPE INEQUALITIES ON HILBERT C∗-MODULES AND ITS REVERSES YUKI SEO1 This paper is dedicated to Professor Tsuyoshi Ando Abstract. In this paper, we show Hilbert C∗-module versions of H¨older- McCarthy inequality and its complementary inequality. As an application, we obtain H¨oldertype inequalities and its reverses on a Hilbert C∗-module. 1. Introduction The H¨olderinequality is one of the most important inequalities in functional analysis. If a = (a1; : : : ; an) and b = (b1; : : : ; bn) are n-tuples of nonnegative numbers, and 1=p + 1=q = 1, then ! ! Xn Xn 1=p Xn 1=q ≤ p q aibi ai bi for all p > 1 i=1 i=1 i=1 and ! ! Xn Xn 1=p Xn 1=q ≥ p q aibi ai bi for all p < 0 or 0 < p < 1: i=1 i=1 i=1 Non-commutative versions of the H¨olderinequality and its reverses have been studied by many authors. T. Ando [1] showed the Hadamard product version of a H¨oldertype. T. Ando and F. Hiai [2] discussed the norm H¨olderinequality and the matrix H¨olderinequality. B. Mond and O. Shisha [15], M. Fujii, S. Izumino, R. Nakamoto and Y. Seo [7], and S. Izumino and M. Tominaga [11] considered the vector state version of a H¨oldertype and its reverses. J.-C. Bourin, E.-Y. Lee, M. Fujii and Y. Seo [3] showed the geometric operator mean version, and so on.
    [Show full text]
  • User's Guide for Complexity: a LATEX Package, Version 0.80
    User’s Guide for complexity: a LATEX package, Version 0.80 Chris Bourke April 12, 2007 Contents 1 Introduction 2 1.1 What is complexity? ......................... 2 1.2 Why a complexity package? ..................... 2 2 Installation 2 3 Package Options 3 3.1 Mode Options .............................. 3 3.2 Font Options .............................. 4 3.2.1 The small Option ....................... 4 4 Using the Package 6 4.1 Overridden Commands ......................... 6 4.2 Special Commands ........................... 6 4.3 Function Commands .......................... 6 4.4 Language Commands .......................... 7 4.5 Complete List of Class Commands .................. 8 5 Customization 15 5.1 Class Commands ............................ 15 1 5.2 Language Commands .......................... 16 5.3 Function Commands .......................... 17 6 Extended Example 17 7 Feedback 18 7.1 Acknowledgements ........................... 19 1 Introduction 1.1 What is complexity? complexity is a LATEX package that typesets computational complexity classes such as P (deterministic polynomial time) and NP (nondeterministic polynomial time) as well as sets (languages) such as SAT (satisfiability). In all, over 350 commands are defined for helping you to typeset Computational Complexity con- structs. 1.2 Why a complexity package? A better question is why not? Complexity theory is a more recent, though mature area of Theoretical Computer Science. Each researcher seems to have his or her own preferences as to how to typeset Complexity Classes and has built up their own personal LATEX commands file. This can be frustrating, to say the least, when it comes to collaborations or when one has to go through an entire series of files changing commands for compatibility or to get exactly the look they want (or what may be required).
    [Show full text]
  • Lecture 1: Propositional Logic
    Lecture 1: Propositional Logic Syntax Semantics Truth tables Implications and Equivalences Valid and Invalid arguments Normal forms Davis-Putnam Algorithm 1 Atomic propositions and logical connectives An atomic proposition is a statement or assertion that must be true or false. Examples of atomic propositions are: “5 is a prime” and “program terminates”. Propositional formulas are constructed from atomic propositions by using logical connectives. Connectives false true not and or conditional (implies) biconditional (equivalent) A typical propositional formula is The truth value of a propositional formula can be calculated from the truth values of the atomic propositions it contains. 2 Well-formed propositional formulas The well-formed formulas of propositional logic are obtained by using the construction rules below: An atomic proposition is a well-formed formula. If is a well-formed formula, then so is . If and are well-formed formulas, then so are , , , and . If is a well-formed formula, then so is . Alternatively, can use Backus-Naur Form (BNF) : formula ::= Atomic Proposition formula formula formula formula formula formula formula formula formula formula 3 Truth functions The truth of a propositional formula is a function of the truth values of the atomic propositions it contains. A truth assignment is a mapping that associates a truth value with each of the atomic propositions . Let be a truth assignment for . If we identify with false and with true, we can easily determine the truth value of under . The other logical connectives can be handled in a similar manner. Truth functions are sometimes called Boolean functions. 4 Truth tables for basic logical connectives A truth table shows whether a propositional formula is true or false for each possible truth assignment.
    [Show full text]
  • Logic and Proof
    Logic and Proof Computer Science Tripos Part IB Michaelmas Term Lawrence C Paulson Computer Laboratory University of Cambridge [email protected] Copyright c 2007 by Lawrence C. Paulson Contents 1 Introduction and Learning Guide 1 2 Propositional Logic 3 3 Proof Systems for Propositional Logic 13 4 First-order Logic 20 5 Formal Reasoning in First-Order Logic 27 6 Clause Methods for Propositional Logic 33 7 Skolem Functions and Herbrand’s Theorem 41 8 Unification 50 9 Applications of Unification 58 10 BDDs, or Binary Decision Diagrams 65 11 Modal Logics 68 12 Tableaux-Based Methods 74 i ii 1 1 Introduction and Learning Guide This course gives a brief introduction to logic, with including the resolution method of theorem-proving and its relation to the programming language Prolog. Formal logic is used for specifying and verifying computer systems and (some- times) for representing knowledge in Artificial Intelligence programs. The course should help you with Prolog for AI and its treatment of logic should be helpful for understanding other theoretical courses. Try to avoid getting bogged down in the details of how the various proof methods work, since you must also acquire an intuitive feel for logical reasoning. The most suitable course text is this book: Michael Huth and Mark Ryan, Logic in Computer Science: Modelling and Reasoning about Systems, 2nd edition (CUP, 2004) It costs £35. It covers most aspects of this course with the exception of resolution theorem proving. It includes material (symbolic model checking) that should be useful for Specification and Verification II next year.
    [Show full text]
  • Solving QBF by Combining Conjunctive and Disjunctive Normal Forms
    Solving QBF with Combined Conjunctive and Disjunctive Normal Form Lintao Zhang Microsoft Research Silicon Valley Lab 1065 La Avenida, Mountain View, CA 94043, USA [email protected] Abstract of QBF solvers have been proposed based on a number of Similar to most state-of-the-art Boolean Satisfiability (SAT) different underlying principles such as Davis-Logemann- solvers, all contemporary Quantified Boolean Formula Loveland (DLL) search (Cadoli et al., 1998, Giunchiglia et (QBF) solvers require inputs to be in the Conjunctive al. 2002a, Zhang & Malik, 2002, Letz, 2002), resolution Normal Form (CNF). Most of them also store the QBF in and expansion (Biere 2004), Binary Decision Diagrams CNF internally for reasoning. In order to use these solvers, (Pan & Vardi, 2005) and symbolic Skolemization arbitrary Boolean formulas have to be transformed into (Benedetti, 2004). Unfortunately, unlike the SAT solvers, equi-satisfiable formulas in Conjunctive Normal Form by which enjoy huge success in solving problems generated introducing additional variables. In this paper, we point out from real world applications, QBF solvers are still only an inherent limitation of this approach, namely the able to tackle trivial problems and remain limited in their asymmetric treatment of satisfactions and conflicts. This deficiency leads to artificial increase of search space for usefulness in practice. QBF solving. To overcome the limitation, we propose to Even though the current QBF solvers are based on many transform a Boolean formula into a combination of an equi- different reasoning principles, they all require the input satisfiable CNF formula and an equi-tautological DNF QBF to be in Conjunctive Normal Form (CNF).
    [Show full text]
  • Approximation Hardness of Optimization Problems in Intersection Graphs of D-Dimensional Boxes
    Approximation Hardness of Optimization Problems in Intersection Graphs of d-dimensional Boxes Miroslav Chleb´ık¤ Janka Chleb´ıkov´a y Abstract Maximum Independent Set, in d-box intersection The Maximum Independent Set problem in d-box graphs, graphs for any fixed d ¸ 3. i.e., in the intersection graphs of axis-parallel rectangles in Rd, is a challenge open problem. For any fixed d ¸ 2 Overview. Many optimization problems like Max- the problem is NP-hard and no approximation algorithm imum Clique, Maximum Independent Set, and with ratio o(logd¡1 n) is known. In some restricted cases, Minimum (Vertex) Coloring are NP-hard for gen- e.g., for d-boxes with bounded aspect ratio, a PTAS exists eral graphs but solvable in polynomial time for interval [17]. In this paper we prove APX-hardness (and hence non- graphs [20]. Many of them are known to be NP-hard existence of a PTAS, unless P = NP), of the Maximum already in 2-dimensional models of geometric intersec- Independent Set problem in d-box graphs for any fixed tion graphs (e.g., in unit disk graphs). In most cases the 443 geometric restrictions allow us to obtain better approxi- d ¸ 3. We state also first explicit lower bound 442 on efficient approximability in such case. Additionally, we mation algorithms (or even in polynomial time solvabil- provide a generic method how to prove APX-hardness for ity) for problems that are in general graphs extremely many NP-hard graph optimization problems in d-box graphs hard to approximate. On the other hand, these geomet- for any fixed d ¸ 3.
    [Show full text]
  • Transformation of Boolean Expression Into Disjunctive Or Conjunctive Normal Form
    Central European Researchers Journal, Vol.3 Issue 1 Transformation of Boolean Expression into Disjunctive or Conjunctive Normal Form Patrik Rusnak Abstract—Reliability is an important characteristic of many systems. One of the current issues of reliability analysis is investigation of systems that are composed of many components. Structure of such systems can be defined in the form of a Boolean function. A Boolean function can be expressed in several ways. One of them is a symbolic expression. Several types of symbolic representations of Boolean functions exist. The most commonly known are disjunctive and conjunctive normal forms. These forms are often used not only in reliability analysis but also in other fields, such as game theory or logic design. Therefore, it is very important to have software that is able to transform any kind of Boolean expression into one of these normal forms. In this paper, an algorithm that allows such transformation is presented. Keywords—reliability, Boolean function, normal forms. I. INTRODUCTION Investigation of system reliability is a complex problem consisting of many steps. One of them is creation of a model of the system. Since the system is usually composed of several components, a special map defining the dependency between operation of the components and operation of the system has to be known. This map is known as structure function and, for a system consisting of 푛 components, it has the following form [1]: 푛 휙(푥1, 푥2, … , 푥푛) = 휙(풙): {0,1} → {0,1}, (1) where set {0,1} is a set of possible states at which the components and the system can operate (state 1 means that the component/system is functioning while state 0 agrees with a failure of the component/system), 푥푖 is a variable defining state of the 푖-th system component, for 푖 = 1,2, … , 푛, and 풙 = (푥1, 푥2, … , 푥푛) is a vector of components states (state vector).
    [Show full text]