DOCSLIB.ORG

3-Satisfiability Is NP-Complete ∗

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
3-Satisfiability Is NP-Complete ∗ 3-Satisfiability Is NP-complete ∗ Peter Cappello Department of Computer Science University of California, Santa Barbara Santa Barbara, CA 93106 [email protected] • These notes are not intended to be complete. They are supplemented with figures, and material that arises during the lecture period in response to questions. ∗Based on Computers and Intractability, Michael R. Garey and David S. Johnson, Freeman, 1979. 1 3-Satisfiability, aka 3SAT, is the restricted form of the Satisfiability problem, where each clause has exactly 3 literals. Theorem: 3-Satisfiability is NP-complete. Proof • It is easy to check whether or not a given truth assignment satisfies a problem instance's clauses in polynomial time: 3SAT 2 NP. • Let { U = fu1; u2; : : : ; ung be a set of variables { C = fc1; c2; : : : ; cmg be a set of clauses making up an arbitrary instance of SAT. 2 • We construct a set C0 of 3-literal clauses on a set U 0 of variables such that C is satisfiable () C0 is satisfiable. • The construction of C0 replaces each clause in C with a set of \equiv- 0 0 alent" clauses in C , based on U and additional variables Uj used only 0 in clauses Cj. m U 0 = U [ 0 U 0 1 B [ j C @ j=1 A and m 0 0 C = [ Cj j=1 • Let an arbitrary clause in C be cj = fz1; z2; : : : ; zkg. 0 0 • The construction of Cj and Uj depends on k, the number of literals in cj. k = 1: 3 0 1 2 Uj = fyj ; yj g 0 1 2 1 2 1 2 1 2 Cj = ffz1; yj ; yj g; fz1; yj ; yj g; fz1; yj ; yj g; fz1; yj ; yj gg 0 All of the clauses in Cj are satisfied if and only if z1 evaluates to T . k = 2: 0 1 Uj = fyj g 0 1 1 Cj = ffz1; z2; yj g; fz1; z2; yj gg 0 All of the clauses in Cj are satisfied if and only if z1 or z2 evaluate to T . k = 3: 0 Uj = ; 0 Cj = fcjg 4 k > 3: 0 i Uj = fyj; 1 ≤ i ≤ k − 3g 0 1 Cj = ffz1; z2; yj gg i i+1 [ ffyj; zi+2; yj g; 1 ≤ i ≤ k − 4g k−3 [ ffyj ; zk−1; zkgg For example, if cj = fz1; z2; z3; z4; z5; z6; z7g then 0 1 1 2 2 3 3 4 4 Cj = ffz1; z2; yj g; fyj ; z3; yj g; fyj ; z4; yj g; fyj ; z5; yj g; fyj ; z6; z7g { 0 If cj is not satisfiable, then Cj is not satisfiable. To see this, eliminate the zis from the clauses. They then cannot all be satisfied. For the example, the reduced set of clauses looks like: 1 1 2 2 3 3 4 4 ffyj g; fyj ; yj g; fyj ; yj g; fyj ; yj g; fyj g: 5 { 0 If cj is satisfiable, then Cj is satisfiable. To see this, let l be the smallest index such that zl evaluates to T . i ∗ If l is 1 or 2, set yj = F . In the example, this looks like 0 Cj = ffz1; z2; F g; fT; z3; F g; fT; z4; F g; fT; z5; F g; fT; z6; z7g i ∗ If l is k − 1 or k, set yj = T . In the example, this looks like 0 Cj = ffz1; z2; T g; fF; z3; T g; fF; z4; T g; fF; z5; T g; fF; z6; z7g ∗ Otherwise, set: i · yj = T for 1 ≤ i ≤ l − 2. i · yj = F for l − 1 ≤ i ≤ k − 3. In the example, if 4 is the smallest index of a literal eval- uating to true, then clauses look like 6 0 Cj = ffz1; z2; T g; fF; z3; T g; fF; z4; F g; fT; z5; F g; fT; z6; z7g The clauses before zl occurs are satisfiable, because they all i contain a yj that is assigned to T ; The clauses after zl occurs are satisfiable, because they all i i contain a yj, where yj is assigned to F . • Thus, the transformation does the right thing. • The transformation clearly can be constructed in polynomial time, given U and C 7.
Load more

Recommended publications
  • Group Developed Weighing Matrices∗
    AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 55 (2013), Pages 205–233 Group developed weighing matrices∗ K. T. Arasu Department of Mathematics & Statistics Wright State University 3640 Colonel Glenn Highway, Dayton, OH 45435 U.S.A. Jeffrey R. Hollon Department of Mathematics Sinclair Community College 444 W 3rd Street, Dayton, OH 45402 U.S.A. Abstract A weighing matrix is a square matrix whose entries are 1, 0 or −1,such that the matrix times its transpose is some integer multiple of the identity matrix. We examine the case where these matrices are said to be devel- oped by an abelian group. Through a combination of extending previous results and by giving explicit constructions we will answer the question of existence for 318 such matrices of order and weight both below 100. At the end, we are left with 98 open cases out of a possible 1,022. Further, some of the new results provide insight into the existence of matrices with larger weights and orders. 1 Introduction 1.1 Group Developed Weighing Matrices A weighing matrix W = W (n, k) is a square matrix, of order n, whose entries are in t the set wi,j ∈{−1, 0, +1}. This matrix satisfies WW = kIn, where t denotes the matrix transpose, k is a positive integer known as the weight, and In is the identity matrix of size n. Definition 1.1. Let G be a group of order n.Ann×n matrix A =(agh) indexed by the elements of the group G (such that g and h belong to G)issaidtobeG-developed if it satisfies the condition agh = ag+k,h+k for all g, h, k ∈ G.
    [Show full text]
  • Operating Manual R&S NRP-Z22
    Operating Manual Average Power Sensor R&S NRP-Z22 1137.7506.02 R&S NRP-Z23 1137.8002.02 R&S NRP-Z24 1137.8502.02 Test and Measurement 1137.7870.12-07- 1 Dear Customer, R&S® is a registered trademark of Rohde & Schwarz GmbH & Co. KG Trade names are trademarks of the owners. 1137.7870.12-07- 2 Basic Safety Instructions Always read through and comply with the following safety instructions! All plants and locations of the Rohde & Schwarz group of companies make every effort to keep the safety standards of our products up to date and to offer our customers the highest possible degree of safety. Our products and the auxiliary equipment they require are designed, built and tested in accordance with the safety standards that apply in each case. Compliance with these standards is continuously monitored by our quality assurance system. The product described here has been designed, built and tested in accordance with the EC Certificate of Conformity and has left the manufacturer’s plant in a condition fully complying with safety standards. To maintain this condition and to ensure safe operation, you must observe all instructions and warnings provided in this manual. If you have any questions regarding these safety instructions, the Rohde & Schwarz group of companies will be happy to answer them. Furthermore, it is your responsibility to use the product in an appropriate manner. This product is designed for use solely in industrial and laboratory environments or, if expressly permitted, also in the field and must not be used in any way that may cause personal injury or property damage.
    [Show full text]
  • Z1 Z2 Z4 Z5 Z6 Z7 Z3 Z8
    Illinois State Police Division of Criminal Investigation JO DAVIESS STEPHENSON WINNEBAGO B McHENRY LAKE DIVISION OF CRIMINAL INVESTIGATION - O O Colonel Mark R. Peyton Pecatonica 16 N E Chicago Lieutenant Colonel Chris Trame CARROLL OGLE Chief of Staff DE KALB KANE Elgin COOK Lieutenant Jonathan Edwards NORTH 2 Z1DU PAGE C WHITESIDE LEE 15 1 Z2 NORTH COMMAND - Sterling KENDALL WILL Major Michael Witt LA SALLE 5 Zone 1 - (Districts Chicago and 2) East Moline HENRY BUREAU Captain Matthew Gainer 7 17 Joliet ROCK ISLAND GRUNDY Zone 2 - (Districts 1, 7, 16) LaSalle MERCER Captain Christopher Endress KANKAKEE PUTNAM Z3 Zone 3 - (Districts 5, 17, 21) KNOX STARK Captain Richard Wilk H MARSHALL LIVINGSTON E WARREN Statewide Gaming Command IROQUOIS N PEORIA 6 Captain Sean Brannon D WOODFORD E Pontiac Ashkum CENTRAL R 8 Metamora S 21 O McLEAN N FULTON CENTRAL COMMAND - McDONOUGH HANCOCK TAZEWELL Captain Calvin Brown, Interim Macomb FORD VERMILION Zone 4 - (Districts 8, 9, 14, 20) 14 MASON CHAMPAIGN Captain Don Payton LOGAN DE WITT SCHUYLER Zone 5 - (Districts 6, 10) PIATT ADAMS Captain Jason Henderson MENARD Investigative Support Command CASS MACON Pesotum BROWN Captain Aaron Fullington Z4 10 SANGAMON Medicaid Fraud Control Bureau MORGAN DOUGLAS EDGAR PIKE 9 Z5 Captain William Langheim Pittsfield SCOTT MOULTRIE Springfield 20 CHRISTIAN COLES SHELBY GREENE MACOUPIN SOUTH COMMAND - CLARK Major William Sons C CUMBERLAND A MONTGOMERY L Zone 6 - (Districts 11, 18) H Litchfield 18 O Lieutenant Abigail Keller, Interim JERSEY FAYETTE EFFINGHAM JASPER U Zone 7 - (Districts 13, 22) N Effingham 12 CRAWFORD Z6 BOND Captain Nicholas Dill MADISON Zone 8 - (Districts 12, 19) CLAY Collinsville RICHLAND LAWRENCE Captain Ryan Shoemaker MARION 11 Special Operations Command ST.
    [Show full text]
  • Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles
    Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles by Reza Olfati-Saber Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2001 © Massachusetts Institute of Technology 2001. All rights reserved. Author ................................................... Department of Electrical Engineering and Computer Science January 15, 2000 C ertified by.......................................................... Alexandre Megretski, Associate Professor of Electrical Engineering Thesis Supervisor Accepted by............................................ Arthur C. Smith Chairman, Department Committee on Graduate Students Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles by Reza Olfati-Saber Submitted to the Department of Electrical Engineering and Computer Science on January 15, 2000, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering and Computer Science Abstract This thesis is devoted to nonlinear control, reduction, and classification of underac- tuated mechanical systems. Underactuated systems are mechanical control systems with fewer controls than the number of configuration variables. Control of underactu- ated systems is currently an active field of research due to their broad applications in Robotics, Aerospace Vehicles, and Marine Vehicles. The examples of underactuated systems include flexible-link robots, nobile robots, walking robots, robots on mo- bile platforms, cars, locomotive systems, snake-type and swimming robots, acrobatic robots, aircraft, spacecraft, helicopters, satellites, surface vessels, and underwater ve- hicles. Based on recent surveys, control of general underactuated systems is a major open problem. Almost all real-life mechanical systems possess kinetic symmetry properties, i.e.
    [Show full text]
  • Konrad Zuse the Computer- My Life
    Konrad Zuse The Computer- My Life Konrad Zuse The Computer- My Life With Forewords byEL.Bauer and H. Zemanek Springer-Verlag Berlin Heidelberg GmbH Professor Dr. Ing. E. h. Dr. mult. rer. nat. h.c. Konrad Zuse 1m Haselgrund 21, D-36088 Hunfeld, Germany Editor: Dr. Hans Wossner, Springer-Verlag Heidelberg Translators: Patricia McKenna, New York J.Andrew Ross, Springer-Verlag Heidelberg Titl e of the original German edition: Der Computer - Mein Lebenswerk, 1993 © Springer-Verlag Berlin Heidelberg 1984, 1986, 1990, 1993 Computing Reviews Classification (1991) : K. 2, A. 0 With 68 Figures ISBN 978-3-642-08 151 -4 ISBN 978-3-662-02931-2 (eBook) DOI 10.1007/978-3-662-02931-2 Libary of Congress Cataloging-in-Publication Data . Zuse, Konr ad . (Computer. mein Lebensw erk . English) Th e computer, my life / Konrad Zuse;with for eword s bv F.L. Bauer and H. Zemanek. p. cm. Includes bibliographical references and index. I. Zuse, Konrad. 2. Computers -Germany - History . 3. Computer engineers - Germany - Biography. I. Titl e. TK7885.22.Z87A3 1993 62I.39'092-dc20 [B] 93-18574 This work is subject to copyright. All rights are reserved , whether the whole world or part for the mat erial is concerned , specifically the rights of translation, reprinting, reuse ofillustrati- ons, recitation, broadcasting , reproduction on microfilm or in any ot her way, and storage in data banks. Dupli cation of this publication or parts thereof is permitted only under the pro- visions of German Copyr ight Law of September 9, 1965, in its current version , and permissi- on for use must always be obtained from Springer-Verlag.
    [Show full text]
  • MVME6100 Single Board Computer Installation and Use P/N: 6806800D58E March 2009
    Embedded Computing for Business-Critical ContinuityTM MVME6100 Single Board Computer Installation and Use P/N: 6806800D58E March 2009 © 2009 Emerson All rights reserved. Trademarks Emerson, Business-Critical Continuity, Emerson Network Power and the Emerson Network Power logo are trademarks and service marks of Emerson Electric Co. © 2008 Emerson Electric Co. All other product or service names are the property of their respective owners. Intel® is a trademark or registered trademark of Intel Corporation or its subsidiaries in the United States and other countries. Java™ and all other Java-based marks are trademarks or registered trademarks of Sun Microsystems, Inc. in the U.S. and other countries. Microsoft®, Windows® and Windows Me® are registered trademarks of Microsoft Corporation; and Windows XP™ is a trademark of Microsoft Corporation. PICMG®, CompactPCI®, AdvancedTCA™ and the PICMG, CompactPCI and AdvancedTCA logos are registered trademarks of the PCI Industrial Computer Manufacturers Group. UNIX® is a registered trademark of The Open Group in the United States and other countries. Notice While reasonable efforts have been made to assure the accuracy of this document, Emerson assumes no liability resulting from any omissions in this document, or from the use of the information obtained therein. Emerson reserves the right to revise this document and to make changes from time to time in the content hereof without obligation of Emerson to notify any person of such revision or changes. Electronic versions of this material may be read online, downloaded for personal use, or referenced in another document as a URL to a Emerson website. The text itself may not be published commercially in print or electronic form, edited, translated, or otherwise altered without the permission of Emerson, It is possible that this publication may contain reference to or information about Emerson products (machines and programs), programming, or services that are not available in your country.
    [Show full text]
  • The Use of Computer Competencies of Students in the Departments of Physical Education and Sport Teaching, and School Teaching
    INTERNATIONAL JOURNAL OF ENVIRONMENTAL & SCIENCE EDUCATION 2016, VOL. 11, NO. 15, 8470-8490 OPEN ACCESS The Use of Computer Competencies of Students in the Departments of Physical Education and Sport Teaching, and School Teaching Ilyas Okana Gazi University, TURKEY ABSTRACT This study aims to reveal the levels of the use of computer, which is nowadays one of the most important technologies, of teacher candidate studying in the departments of Physical Education and Sport Teaching, and School teaching; also aims to research whether there is differences according to various criteria or not. In research, data were collected via the survey “Specifying The Use of Computer Competencies of Students in the Departments of Physical Education and Sport Teaching, and School Teaching” with researcher adapting this to Departments of Physical Education and Sport Teaching, and School Teaching. The sample of this study consists of 210 students studying in Gazi University Faculty of Education School Teaching and Physical Education and Sport Teaching Departments in 2014-2015 academic years. Statistical package program was used for research data. Statistic technics such as ANOVA for class variable, t-test for the other variables and mean standard deviation were used. As a result of analysis conducted, the use of computer competencies of students in the departments of physical education and sport teaching, and school teaching was researched and it was figured out that students who have computers at home or who benefits from the computers in school lab use many computer program menus meaningfully a lot at the level of p<0,05. Based on the research results, suggestions for using computer effectively in Physical Education and Sport classes were presented.
    [Show full text]
  • The Relay Computer of Carl Zeiss Jena
    Oprema – 2019AUG26 The Relay Computer of Carl Zeiss Jena Jürgen F. H. Winkler Friedrich Schiller University, Jena The Oprema (Optikrechenmaschine = computer for optical calculations) was a relay computer whose development was initiated by Herbert Kortum and which was designed and built by a team under the leadership of Wilhelm Kämmerer at Carl Zeiss Jena (CZJ) in 1954 and 1955. Basic experiments, de‐ sign and construction of machine‐1 were all done, partly concurrently, in the remarkably short time of about 14 months. Shortly after the electronic G 2 of Heinz Billing in Göttingen it was the 7th universal computer in Germa‐ ny and the 1st in the GDR. The Oprema consisted of two identical machines. One machine consisted of about 8,300 relays, 45,000 selenium rectifiers and 250 km cable. The main reason for the construction of the Oprema was the computational needs of CZJ, which was the leading company for optics and precision me‐ chanics in the GDR. During its lifetime (1955−1963) the Oprema was applied by CZJ and a number of other institutes and companies in the GDR. The paper presents new details of the Oprema project and of the arithmetic operations implemented in the Oprema. Additionally, it covers briefly the lives of the two protagonists, W. Kämmerer and H. Kortum, and draws some comparisons with other early projects, namely Colossus, ASCC/Mark 1 and ENIAC. Finally, it discusses the question, whether Kortum is a German com‐ puter pioneer. Pre-history in Jena (CZJ) and became a member of Kortum’s At the beginning of the 20th century, three people development lab3.
    [Show full text]
  • CSE331 : Computer Security Fundamentals Part I : Crypto (2)
    CSE331 : Computer Security Fundamentals Part I : Crypto (2) CSE331 - Part I Cryptography - Slides: Mark Stamp Chapter 3: Symmetric Key Crypto The chief forms of beauty are order and symmetry… ¾ Aristotle “You boil it in sawdust: you salt it in glue: You condense it with locusts and tape: Still keeping one principal object in view ¾ To preserve its symmetrical shape.” ¾ Lewis Carroll, The Hunting of the Snark CSE331 - Part I Cryptography - Slides: Mark Stamp Symmetric Key Crypto q Stream cipher ¾ generalize one-time pad o Except that key is relatively short o Key is stretched into a long keystream o Keystream is used just like a one-time pad q Block cipher ¾ generalized codebook o Block cipher key determines a codebook o Each key yields a different codebook o Employs both “confusion” and “diffusion” CSE331 - Part I Cryptography - Slides: Mark Stamp Stream Ciphers CSE331 - Part I Cryptography - Slides: Mark Stamp Stream Ciphers q Once upon a time, not so very long ago… stream ciphers were the king of crypto q Today, not as popular as block ciphers q We’ll discuss two stream ciphers: q A5/1 o Based on shift registers o Used in GSM mobile phone system q RC4 o Based on a changing lookup table o Used many places CSE331 - Part I Cryptography - Slides: Mark Stamp A5/1: Shift Registers q A5/1 uses 3 shift registers o X: 19 bits (x0,x1,x2, …,x18) o Y: 22 bits (y0,y1,y2, …,y21) o Z: 23 bits (z0,z1,z2, …,z22) CSE331 - Part I Cryptography - Slides: Mark Stamp A5/1: Keystream q At each iteration: m = maj(x8, y10, z10) o Examples: maj(0,1,0) = 0 and
    [Show full text]
  • The HEPGAME Project • a Little Bit of History • Original Targets
    The HEPGAME project J.A.M. Vermaseren • A little bit of history • Original targets of the project • Various failures • Success or failure? • Success • Where does this leave us? A little bit of history The HEPGAME project was born out of a sense that strategy in game theory and method- ology of computations, in particular solving IBP systems, have much in common. This was already pointed out in 2000 (60-th birthday of F.Yndurain, unpublished) and in Loops and Legs 2004 (”The rules of physics”). It was brought forward (and 1.732 MEuros awarded) in an ERC Advanced grant proposal in 2012 and the project could start on 1-July-2013. Because it is a 5 year project, the money will run out on 30-June-2018. Hence now seems the right moment to look at what has been done and in which direction it takes some of us. Before the project started I was in the fortunate condition of having Jan Kuipers as a postdoc to help me with Form. He is the one responsible for the initial implementations of polynomial facilities and expression optimization in Form (later upgraded and improved by Takahiro Ueda and Ben Ruijl). With him and the computer scientists Jaap van den Herik and Aske Plaat we designed the first algorithms for the expression simplification and put this as part of the HEPGAME objectives. Some of the people present at this workshop have directly benefitted from this work. Unfortunately Jan did not want to spend the rest of his life doing big calculations and hence he did not accept my offer to stay a few more years.
    [Show full text]
  • Groups with at Most Twelve Subgroups
    Groups With At Most Twelve Subgroups Michael C. Slattery Marquette University, Milwaukee WI 53201-1881 [email protected] July 9, 2020 In [2] I showed that the finite groups with a specified number of subgroups can always be described as a finite list of similarity classes. Neil Sloane suggested that I submit the corresponding sequence, the number of similarity classes with n subgroups, to his Online Encyclopedia of Integer Sequences [4]. I thought this would be a quick calculation until I discovered that my “example” case in the paper (n = 6) was wrong (as noted in [3]). I realized that producing a reliable count required a full proof. This note contains the proof behind my computation of the first 12 terms of the sequence. For completeness, we include a section of [2]: Now, we saw [above] that cyclic Sylow subgroups of G which are direct factors allow many non-isomorphic groups to have the same number of subgroups. [This lemma] tells us that these are precisely the cyclic Sylow subgroups which lie in the center of G. Consequently, if p1,...,pc are the primes which divide |G| such that a Sylow pi-subgroup is cyclic and central, then we can write G = P1 × P2 ×···× Pc × Oπ′ (G) where each Pi is a Sylow pi-subgroup, the set π = {p1,...,pc} and Oπ′ (G) is the largest normal subgroup of G with order not divisible by any prime in π. We e e will write G = O ′ (G). In other words, G is the unique subgroup of G left after arXiv:1607.01834v2 [math.GR] 7 Jul 2020 π factoring out the cyclic, central Sylow subgroups.
    [Show full text]
  • MVME55006E Single-Board Computer Installation and Use
    MVME55006E Single-Board Computer Installation and Use 6806800A37D March 2008 © Copyright 2008 Emerson All rights reserved. Trademarks Emerson, Business-Critical Continuity, Emerson Network Power and the Emerson Network Power logo are trademarks and service marks of Emerson Electric Co. © 2008 Emerson Electric Co. All other product or service names are the property of their respective owners. Intel® is a trademark or registered trademark of Intel Corporation or its subsidiaries in the United States and other countries. PICMG®, CompactPCI®, AdvancedTCA™ and the PICMG, CompactPCI and AdvancedTCA logos are registered trademarks of the PCI Industrial Computer Manufacturers Group. Notice While reasonable efforts have been made to assure the accuracy of this document, Emerson assumes no liability resulting from any omissions in this document, or from the use of the information obtained therein. Emerson reserves the right to revise this document and to make changes from time to time in the content hereof without obligation of Emerson to notify any person of such revision or changes. Electronic versions of this material may be read online, downloaded for personal use, or referenced in another document as a URL to a Emerson website. The text itself may not be published commercially in print or electronic form, edited, translated, or otherwise altered without the permission of Emerson, It is possible that this publication may contain reference to or information about Emerson products (machines and programs), programming, or services that are not available in your country. Such references or information must not be construed to mean that Emerson intends to announce such Emerson products, programming, or services in your country.
    [Show full text]