Reductions & NPcompleteness as part of Foundations of Computer Science undergraduate course


 Wendy Ray
 2 years ago
 Views:
Transcription
1 Reductions & NPcompleteness as part of Foundations of Computer Science undergraduate course Alex Angelopoulos, NTUA January 22, 2015
2 Outline Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness 0/1 Integer Programming 2/26 0/1 Integer Programming Vertex Cover 3colorability Hamilton Path (HP) Traveling Salesman Problem (TSP)
3 Reducing 3SAT to 0/1 IP Definition 1 (0/1 IP). Input: an integer matrix C and vector b. Output: decide if there is a 0/1 vector x such that: Cx b. 0/1 IP NP (why?) We choose 3SAT as our known NPcomplete problem and consider the formula: φ = C 1 C 2... C m, literals x 1,..., x n We will construct our m n matrix C : 1, if x j C i c ij = 1, if x j C i 0, otherwise b i = 1 (the number of complemented variables in C i ) Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness 0/1 Integer Programming 3/26
4 Reducing 3SAT to 0/1 IP Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness 0/1 Integer Programming 4/26 Note that: Cx b actually means n j=1 c ijx j b i, i. If 3SAT is satisfiable, then every C i is True. Focus on a line of C and discard the zeros: c ij1 x j1 + c ij2 x j2 + c ij3 x j3 1 #(complemented) = 1x j1 + 1x j2 + 1x j3 1 1x j1 + 1x j2 1x j3 0 1x j1 1x j2 1x j3 1 1x j1 1x j2 1x j3 2 3SAT P 0/1 IP
5 Outline Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Vertex Cover 5/26 0/1 Integer Programming Vertex Cover 3colorability Hamilton Path (HP) Traveling Salesman Problem (TSP)
6 From 3SAT to Vertex Cover Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Vertex Cover 6/26 Definition 2 (Vertex Cover (VC)). Input: a graph G(V, E). Output: decide if there is a set S V of size S k such that every edge is adjacent to a vertex v S. VC NP (why?) We choose 3SAT as our known NPcomplete problem and consider (again) the formula: φ = C 1 C 2... C m, with literals x 1,..., x n
7 Intro to gadgets! Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Vertex Cover 7/26 A gadget is anything (in the context of the target problem) that can help you fix the desired instance... z y y x x x y z z Size of a VC here: 5 = 3 (literals) + 2 (2 clause)
8 G φ and the if and only if check Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Vertex Cover 8/26 φ = (x y z) ( y z w) x x y y z z w w z w x y y z ( ) {x, y, z, w} = {T, F, T, T } is a satisfying assignment. See that we need 2 nodes from each clause gadget in order to get a valid VC of size k = n + 2m.
9 Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Vertex Cover 9/26 G φ and the if and only if check x x y y z z w w z w x y y z ( ) For a VC we must get: At least 1 vertex of each literal gadget Clause gadget s vetrices cannot cover the literal gadget s edges. At least 2 vertices of each clause gadget (1 cannot cover the triangle). If there is a cover of size n + 2m then the literal gadgets indicate the satisfying assignment!
10 Outline Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness 3colorability 10/26 0/1 Integer Programming Vertex Cover 3colorability Hamilton Path (HP) Traveling Salesman Problem (TSP)
11 Reducing 3SAT to 3COLOR Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness 3colorability 11/26 Definition 3 (3COLOR). Input: a graph G(V, E). Output: decide if χ(g) 3? 3COLOR NP (why?) We choose 3SAT as our known NPcomplete problem and consider (yet again) the formula: φ = C 1 C 2... C m, with literals x 1,..., x n
12 Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness 3colorability 12/26 Constructing the graph G φ We ll consider the forumla φ = (x y z) ( y z w). Let s start with the vertices of the literals: for each x i we create v i and v i. In order to dictate an equivalent True/False coloring of v i, v i, we draw all edges v i v i plus we link all v i, v i with a base vertex b. Check that now we have n triangles, all having b in common. b x x y y z z w w
13 c Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness 3colorability 13/26 Constructing the graph G φ The gadget: a colordriven or gate C i s literals as input a b c C i evaluation If all a,b,c are colored False, the output vertex has to be False. If a or b or c is True, then the output vertex can also be True.
14 Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness 3colorability 14/26 Completing G φ φ = (x y z) ( y z w) t f b x x Let s satisfy φ... χ(g φ ) 3 y y G 1 z z w w G 2
15 Checking the if and only if Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness 3colorability 15/26 Now let G φ be 3colorable. And pay attention to the coloring of u i, ū i t f b 3SAT P 3COLOR u1 u1 u2 u2 G1 u3 u3 u4 u4. G2. Since the gadgets output orange, they must each have an orange input. So our true color is the orange, and an assignment that satisfies φ follows the orange unodes.
16 Outline Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Hamilton Path (HP) 16/26 0/1 Integer Programming Vertex Cover 3colorability Hamilton Path (HP) Traveling Salesman Problem (TSP)
17 Reducing 3SAT to Hamilton Path Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Hamilton Path (HP) 17/26 Definition 4 (Hamilton Path). Input: graph G. Output: decide whether G allows a path visiting all nodes excatly once. Hamilton Path NP. We can guess n 1 edges and verify if they add up to a Hamilton Path. We need 3 gadgets for this problem..
18 Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Hamilton Path (HP) 18/26 Gadgets (1/3) The choice gadget  one per literal T F Actually, the colored edges will become subgraphs that allow a path between the blue nodes. They sure translate to an evaluation True of False for the literal.
19 Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Hamilton Path (HP) 19/26 Gadgets (2/3) The consistency gadget  an xor gate A part o a Hamilton Path must either enter and exit this subgraph using both top vertices or both bottom vertices. That exclusive or functionality will be the hint for gadget 3 to prove useful.
20 Gadgets (3/3) Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Hamilton Path (HP) 20/26 The constraint gadget  one per clause x 3 x 3 x 1 x 1 C i x 2 x 2 Let s take C i = (x 1 x 2 x 3 ) We must force that the edges (paths) of the triangle are traversed by a Hamilton Path if and only if the corresponding literal is false. Then the clause is True, or else there would be no Hamilton Path!
21 Constructing the full R(φ) Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Hamilton Path (HP) 21/26 Let φ = (x 1 x 2 x 3 ) ( x 1 x 2 x 3 ) ( x 1 x 2 x 3 ) s t Orange nodes form a clique
22 Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Hamilton Path (HP) 22/26 φ is satisfiable R(φ) has a Hamilton Path Let φ = (x 1 x 2 x 3 ) ( x 1 x 2 x 3 ) ( x 1 x 2 x 3 ) s t
23 Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Hamilton Path (HP) 23/26 R(φ) has a Hamilton Path φ is satisfiable Remember the costraint gadget. If the red edge xor path belongs to the Hamilton Path, then both green edges do not belong to the path. But this defines a truth assignment, where no clause gets all 3 literals false. t3sat P Hamilton Path
24 Outline Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Traveling Salesman Problem (TSP) 24/26 0/1 Integer Programming Vertex Cover 3colorability Hamilton Path (HP) Traveling Salesman Problem (TSP)
25 The Traveling Salesman Problem Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Traveling Salesman Problem (TSP) 25/26 Definition 5 (TSP). Given a set of n cities and the distance between any two of them, find the shortest tour covering all cities. Definition 6 (TSP (Decision problem)). Input: a complete graph G with weighted edges, budget (target cost) B Output: is there a tour (cycle) visiting every vertex of G with total cost B? Verify that TSP(D) belongs to class NP... We shall use Hamilton Path as ou known NPcomplete problem.
26 Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness Traveling Salesman Problem (TSP) 26/26 Hamilton Path P TSP(D) Take any instance of Hamilton Path (i.e. any graph G with n vertices) and take a copy of it, Ḡ. Set all edges of Ḡ to have a weight equal to 1. Insert all missing edges of Ḡ with weight 2. To finalize the instance of TSP(D), take B = n + 1. G has a Hamilton Path Ḡ has a tour of cost n Hamilton Path P TSP(D) Ḡ has a tour of cost n + 1 G has a Hamilton Path...
Lecture 7: NPComplete Problems
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NPComplete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit
More informationOutline. NPcompleteness. When is a problem easy? When is a problem hard? Today. Euler Circuits
Outline NPcompleteness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2pairs sum vs. general Subset Sum Reducing one problem to another Clique
More informationNPCompleteness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University
NPCompleteness CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Hard Graph Problems Hard means no known solutions with
More informationOne last point: we started off this book by introducing another famously hard search problem:
S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani 261 Factoring One last point: we started off this book by introducing another famously hard search problem: FACTORING, the task of finding all prime factors
More informationWhy? A central concept in Computer Science. Algorithms are ubiquitous.
Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online
More informationDefinition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing:
1. GRAPHS AND COLORINGS Definition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing: 3 vertices 3 edges 4 vertices 4 edges 4 vertices 6 edges A graph
More informationComplexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar
Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples
More informationPage 1. CSCE 310J Data Structures & Algorithms. CSCE 310J Data Structures & Algorithms. P, NP, and NPComplete. PolynomialTime Algorithms
CSCE 310J Data Structures & Algorithms P, NP, and NPComplete Dr. Steve Goddard goddard@cse.unl.edu CSCE 310J Data Structures & Algorithms Giving credit where credit is due:» Most of the lecture notes
More informationIntroduction to Logic in Computer Science: Autumn 2006
Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Now that we have a basic understanding
More informationComputer Algorithms. NPComplete Problems. CISC 4080 Yanjun Li
Computer Algorithms NPComplete Problems NPcompleteness The quest for efficient algorithms is about finding clever ways to bypass the process of exhaustive search, using clues from the input in order
More informationIntroduction to Algorithms. Part 3: P, NP Hard Problems
Introduction to Algorithms Part 3: P, NP Hard Problems 1) Polynomial Time: P and NP 2) NPCompleteness 3) Dealing with Hard Problems 4) Lower Bounds 5) Books c Wayne Goddard, Clemson University, 2004 Chapter
More informationAlgorithm Design and Analysis Homework #6 Due: 1pm, Monday, January 9, === Homework submission instructions ===
Algorithm Design and Analysis Homework #6 Due: 1pm, Monday, January 9, 2012 === Homework submission instructions === Submit the answers for writing problems (including your programming report) through
More informationNPCompleteness I. Lecture 19. 19.1 Overview. 19.2 Introduction: Reduction and Expressiveness
Lecture 19 NPCompleteness I 19.1 Overview In the past few lectures we have looked at increasingly more expressive problems that we were able to solve using efficient algorithms. In this lecture we introduce
More information2.3 Scheduling jobs on identical parallel machines
2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed
More informationLecture 19: Introduction to NPCompleteness Steven Skiena. Department of Computer Science State University of New York Stony Brook, NY 11794 4400
Lecture 19: Introduction to NPCompleteness Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Reporting to the Boss Suppose
More informationCMPSCI611: Approximating MAXCUT Lecture 20
CMPSCI611: Approximating MAXCUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NPhard problems. Today we consider MAXCUT, which we proved to
More informationOHJ2306 Introduction to Theoretical Computer Science, Fall 2012 8.11.2012
276 The P vs. NP problem is a major unsolved problem in computer science It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a $ 1,000,000 prize for the
More informationProblem Set 7 Solutions
8 8 Introduction to Algorithms May 7, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Handout 25 Problem Set 7 Solutions This problem set is due in
More informationDiscuss the size of the instance for the minimum spanning tree problem.
3.1 Algorithm complexity The algorithms A, B are given. The former has complexity O(n 2 ), the latter O(2 n ), where n is the size of the instance. Let n A 0 be the size of the largest instance that can
More informationCSC 373: Algorithm Design and Analysis Lecture 16
CSC 373: Algorithm Design and Analysis Lecture 16 Allan Borodin February 25, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 17 Announcements and Outline Announcements
More information2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]
Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)
More informationThe Traveling Beams Optical Solutions for Bounded NPComplete Problems
The Traveling Beams Optical Solutions for Bounded NPComplete Problems Shlomi Dolev, Hen Fitoussi Abstract Architectures for optical processors designed to solve bounded instances of NPComplete problems
More informationCoNP and Function Problems
CoNP and Function Problems conp By definition, conp is the class of problems whose complement is in NP. NP is the class of problems that have succinct certificates. conp is therefore the class of problems
More informationGraph Theory. Euler tours and Chinese postmen. John Quinn. Week 5
Graph Theory Euler tours and Chinese postmen John Quinn Week 5 Recap: connectivity Connectivity and edgeconnectivity of a graph Blocks Kruskal s algorithm Königsberg, Prussia The Seven Bridges of Königsberg
More informationNotes on Matrix Multiplication and the Transitive Closure
ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.
More informationAnswers to some of the exercises.
Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the kcenter algorithm first to D and then for each center in D
More informationNotes on NP Completeness
Notes on NP Completeness Rich Schwartz November 10, 2013 1 Overview Here are some notes which I wrote to try to understand what NP completeness means. Most of these notes are taken from Appendix B in Douglas
More informationDO NOT REDISTRIBUTE THIS SOLUTION FILE
Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should
More informationOn the Unique Games Conjecture
On the Unique Games Conjecture Antonios Angelakis National Technical University of Athens June 16, 2015 Antonios Angelakis (NTUA) Theory of Computation June 16, 2015 1 / 20 Overview 1 Introduction 2 Preliminary
More informationCAD Algorithms. P and NP
CAD Algorithms The Classes P and NP Mohammad Tehranipoor ECE Department 6 September 2010 1 P and NP P and NP are two families of problems. P is a class which contains all of the problems we solve using
More information1. Nondeterministically guess a solution (called a certificate) 2. Check whether the solution solves the problem (called verification)
Some N P problems Computer scientists have studied many N P problems, that is, problems that can be solved nondeterministically in polynomial time. Traditionally complexity question are studied as languages:
More informationInformatique Fondamentale IMA S8
Informatique Fondamentale IMA S8 Cours 4 : graphs, problems and algorithms on graphs, (notions of) NP completeness Laure Gonnord http://laure.gonnord.org/pro/teaching/ Laure.Gonnord@polytechlille.fr Université
More informationMemoization/Dynamic Programming. The String reconstruction problem. CS125 Lecture 5 Fall 2016
CS125 Lecture 5 Fall 2016 Memoization/Dynamic Programming Today s lecture discusses memoization, which is a method for speeding up algorithms based on recursion, by using additional memory to remember
More informationGraph Theory Problems and Solutions
raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is
More informationGuessing Game: NPComplete?
Guessing Game: NPComplete? 1. LONGESTPATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTESTPATH: Given a graph G = (V, E), does there exists a simple
More informationA Working Knowledge of Computational Complexity for an Optimizer
A Working Knowledge of Computational Complexity for an Optimizer ORF 363/COS 323 Instructor: Amir Ali Ahmadi TAs: Y. Chen, G. Hall, J. Ye Fall 2014 1 Why computational complexity? What is computational
More informationChapter. NPCompleteness. Contents
Chapter 13 NPCompleteness Contents 13.1 P and NP......................... 593 13.1.1 Defining the Complexity Classes P and NP...594 13.1.2 Some Interesting Problems in NP.......... 597 13.2 NPCompleteness....................
More informationComplexity and Completeness of Finding Another Solution and Its Application to Puzzles
yato@is.s.utokyo.ac.jp seta@is.s.utokyo.ac.jp Π (ASP) Π x s x s ASP Ueda Nagao n nasp parsimonious ASP ASP NP Complexity and Completeness of Finding Another Solution and Its Application to Puzzles Takayuki
More informationExistence of Simple Tours of Imprecise Points
Existence of Simple Tours of Imprecise Points Maarten Löffler Department of Information and Computing Sciences, Utrecht University Technical Report UUCS00700 www.cs.uu.nl ISSN: 097 Existence of Simple
More informationApproximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai
Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization
More informationTheoretical Computer Science (Bridging Course) Complexity
Theoretical Computer Science (Bridging Course) Complexity Gian Diego Tipaldi A scenario You are a programmer working for a logistics company Your boss asks you to implement a program that optimizes the
More informationBasic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by
Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn
More informationCOLORED GRAPHS AND THEIR PROPERTIES
COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring
More informationTetris is Hard: An Introduction to P vs NP
Tetris is Hard: An Introduction to P vs NP Based on Tetris is Hard, Even to Approximate in COCOON 2003 by Erik D. Demaine (MIT) Susan Hohenberger (JHU) David LibenNowell (Carleton) What s Your Problem?
More informationLecture 4: The Chromatic Number
Introduction to Graph Theory Instructor: Padraic Bartlett Lecture 4: The Chromatic Number Week 1 Mathcamp 2011 In our discussion of bipartite graphs, we mentioned that one way to classify bipartite graphs
More informationGreat Theoretical Ideas in Computer Science
15251 Great Theoretical Ideas in Computer Science Complexity Theory: Efficient Reductions Between Computational Problems Lecture 26 (Nov 18, 2010) A Graph Named Gadget KColoring We define a kcoloring
More informationTutorial 8. NPComplete Problems
Tutorial 8 NPComplete Problems Decision Problem Statement of a decision problem Part 1: instance description defining the input Part 2: question stating the actual yesorno question A decision problem
More informationBoolean Representations and Combinatorial Equivalence
Chapter 2 Boolean Representations and Combinatorial Equivalence This chapter introduces different representations of Boolean functions. It then discuss the applications of these representations for proving
More informationFinding the Shortest MoveSequence in the GraphGeneralized 15Puzzle is NPHard
Finding the Shortest MoveSequence in the GraphGeneralized 15Puzzle is NPHard Oded Goldreich Abstract. Following Wilson (J. Comb. Th. (B), 1975), Johnson (J. of Alg., 1983), and Kornhauser, Miller and
More informationHomework 15 Solutions
PROBLEM ONE (Trees) Homework 15 Solutions 1. Recall the definition of a tree: a tree is a connected, undirected graph which has no cycles. Which of the following definitions are equivalent to this definition
More informationWe know a formula for and some properties of the determinant. Now we see how the determinant can be used.
Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we
More informationLattice Point Geometry: Pick s Theorem and Minkowski s Theorem. Senior Exercise in Mathematics. Jennifer Garbett Kenyon College
Lattice Point Geometry: Pick s Theorem and Minkowski s Theorem Senior Exercise in Mathematics Jennifer Garbett Kenyon College November 18, 010 Contents 1 Introduction 1 Primitive Lattice Triangles 5.1
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More informationSocial Media Mining. Graph Essentials
Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures
More informationV. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005
V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer
More informationThe Classes P and NP. mohamed@elwakil.net
Intractable Problems The Classes P and NP Mohamed M. El Wakil mohamed@elwakil.net 1 Agenda 1. What is a problem? 2. Decidable or not? 3. The P class 4. The NP Class 5. TheNP Complete class 2 What is a
More informationGRAPH THEORY and APPLICATIONS. Trees
GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.
More informationKeywords: Travelling Salesman Problem, Map Reduce, Genetic Algorithm. I. INTRODUCTION
ISSN: 23217782 (Online) Impact Factor: 6.047 Volume 4, Issue 6, June 2016 International Journal of Advance Research in Computer Science and Management Studies Research Article / Survey Paper / Case Study
More informationIntroduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24
Introduction to Algorithms Review information for Prelim 1 CS 4820, Spring 2010 Distributed Wednesday, February 24 The final exam will cover seven topics. 1. greedy algorithms 2. divideandconquer algorithms
More informationUsing the Simplex Method in Mixed Integer Linear Programming
Integer Using the Simplex Method in Mixed Integer UTFSM Nancy, 17 december 2015 Using the Simplex Method in Mixed Integer Outline Mathematical Programming Integer 1 Mathematical Programming Optimisation
More informationMinimum Spanning Trees
Minimum Spanning Trees Algorithms and 18.304 Presentation Outline 1 Graph Terminology Minimum Spanning Trees 2 3 Outline Graph Terminology Minimum Spanning Trees 1 Graph Terminology Minimum Spanning Trees
More information5.1 Bipartite Matching
CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the FordFulkerson
More informationWhere the Really Hard Problems Are
Where the Really Hard Problems Are Peter Cheeseman Bob Kanefsky William M. Taylor RIACS* Sterling Software Sterling Software Artificial Intelligence Research Branch NASA Ames Research Center, Mail Stop
More informationP versus NP, and More
1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify
More informationData Structures and Algorithms Written Examination
Data Structures and Algorithms Written Examination 22 February 2013 FIRST NAME STUDENT NUMBER LAST NAME SIGNATURE Instructions for students: Write First Name, Last Name, Student Number and Signature where
More informationPlanarity Planarity
Planarity 8.1 71 Planarity Up until now, graphs have been completely abstract. In Topological Graph Theory, it matters how the graphs are drawn. Do the edges cross? Are there knots in the graph structure?
More informationSocial Media Mining. Network Measures
Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the likeminded users
More information! X is a set of strings. ! Instance: string s. ! Algorithm A solves problem X: A(s) = yes iff s! X.
Decision Problems 8.2 Definition of NP Decision problem. X is a set of strings. Instance: string s. Algorithm A solves problem X: A(s) = yes iff s X. Polynomial time. Algorithm A runs in polytime if for
More informationDynamic Programming. Applies when the following Principle of Optimality
Dynamic Programming Applies when the following Principle of Optimality holds: In an optimal sequence of decisions or choices, each subsequence must be optimal. Translation: There s a recursive solution.
More informationChapter 8 Independence
Chapter 8 Independence Section 8.1 Vertex Independence and Coverings Next, we consider a problem that strikes close to home for us all, final exams. At the end of each term, students are required to take
More informationCS311H. Prof: Peter Stone. Department of Computer Science The University of Texas at Austin
CS311H Prof: Department of Computer Science The University of Texas at Austin Good Morning, Colleagues Good Morning, Colleagues Are there any questions? Logistics Class survey Logistics Class survey Homework
More informationAn Introduction to APGL
An Introduction to APGL Charanpal Dhanjal February 2012 Abstract Another Python Graph Library (APGL) is a graph library written using pure Python, NumPy and SciPy. Users new to the library can gain an
More informationIMO Training 2010 Russianstyle Problems Alexander Remorov
Solutions: Combinatorial Geometry 1. No. Call a lattice point even if the sum of its coordinates is even, and call it odd otherwise. Call one of the legs first, and other one second. Then every time a
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationGraph. Consider a graph, G in Fig Then the vertex V and edge E can be represented as:
Graph A graph G consist of 1. Set of vertices V (called nodes), (V = {v1, v2, v3, v4...}) and 2. Set of edges E (i.e., E {e1, e2, e3...cm} A graph can be represents as G = (V, E), where V is a finite and
More informationTHE PROBLEM WORMS (1) WORMS (2) THE PROBLEM OF WORM PROPAGATION/PREVENTION THE MINIMUM VERTEX COVER PROBLEM
1 THE PROBLEM OF WORM PROPAGATION/PREVENTION I.E. THE MINIMUM VERTEX COVER PROBLEM Prof. Tiziana Calamoneri Network Algorithms A.y. 2014/15 2 THE PROBLEM WORMS (1)! A computer worm is a standalone malware
More informationChapter 6: Graph Theory
Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.
More informationIntroduction to computer science
Introduction to computer science Michael A. Nielsen University of Queensland Goals: 1. Introduce the notion of the computational complexity of a problem, and define the major computational complexity classes.
More information136 CHAPTER 4. INDUCTION, GRAPHS AND TREES
136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics
More information(Vertex) Colorings. We can properly color W 6 with. colors and no fewer. Of interest: What is the fewest colors necessary to properly color G?
Vertex Coloring 2.1 33 (Vertex) Colorings Definition: A coloring of a graph G is a labeling of the vertices of G with colors. [Technically, it is a function f : V (G) {1, 2,...,l}.] Definition: A proper
More informationNetwork (Tree) Topology Inference Based on Prüfer Sequence
Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,
More informationRANDOMIZATION IN APPROXIMATION AND ONLINE ALGORITHMS. Manos Thanos Randomized Algorithms NTUA
RANDOMIZATION IN APPROXIMATION AND ONLINE ALGORITHMS 1 Manos Thanos Randomized Algorithms NTUA RANDOMIZED ROUNDING Linear Programming problems are problems for the optimization of a linear objective function,
More informationCMSC 451: Graph Properties, DFS, BFS, etc.
CMSC 451: Graph Properties, DFS, BFS, etc. Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Chapter 3 of Algorithm Design by Kleinberg & Tardos. Graphs
More informationHomework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS
Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C
More informationFigure 4.0 CPMS Architecture
CHAPTER4 SYSTEM DESIGN System architecture of our project can be explained by using following diagram. Figure 4.0 CPMS Architecture Basically our system is divided in to two parts 1]. Web Role 2]. Worker
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;
More informationCOT5405 Analysis of Algorithms Homework 3 Solutions
COT0 Analysis of Algorithms Homework 3 Solutions. Prove or give a counter example: (a) In the textbook, we have two routines for graph traversal  DFS(G) and BFS(G,s)  where G is a graph and s is any
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More information(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7
(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition
More informationNear Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NPComplete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
More informationConnections between decomposition trees of 3connected plane triangulations and hamiltonian properties
Introduction Hamiltonicity Other hamiltonian properties Connections between decomposition trees of 3connected plane triangulations and hamiltonian properties Gunnar Brinkmann Jasper Souffriau Nico Van
More informationTrees and Fundamental Circuits
Trees and Fundamental Circuits Tree A connected graph without any circuits. o must have at least one vertex. o definition implies that it must be a simple graph. o only finite trees are being considered
More informationGraph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis
Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.0, steen@cs.vu.nl Chapter 06: Network analysis Version: April 8, 04 / 3 Contents Chapter
More informationGeneralized Induced Factor Problems
Egerváry Research Group on Combinatorial Optimization Technical reports TR200207. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
More information1. Relevant standard graph theory
Color identical pairs in 4chromatic graphs Asbjørn Brændeland I argue that, given a 4chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4coloring
More informationWhy Study NP hardness. NP Hardness/Completeness Overview. P and NP. Scaling 9/3/13. Ron Parr CPS 570. NP hardness is not an AI topic
Why Study NP hardness NP Hardness/Completeness Overview Ron Parr CPS 570 NP hardness is not an AI topic It s important for all computer scienhsts Understanding it will deepen your understanding of AI
More informationSome Minesweeper Configurations
Some Minesweeper Configurations Richard Kaye School of Mathematics The University of Birmingham Birmingham B15 2TT RWKaye@bhamacuk http://webmatbhamacuk/rwkaye/ 31st May 2007 Contents 1 Introduction 2
More informationBicolored Shortest Paths in Graphs with Applications to Network Overlay Design
Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design Hongsik Choi and HyeongAh Choi Department of Electrical Engineering and Computer Science George Washington University Washington,
More informationPrinciple of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))
Outline We will cover (over the next few weeks) Mathematical Induction (or Weak Induction) Strong (Mathematical) Induction Constructive Induction Structural Induction Principle of (Weak) Mathematical Induction
More informationTransportation Polytopes: a Twenty year Update
Transportation Polytopes: a Twenty year Update Jesús Antonio De Loera University of California, Davis Based on various papers joint with R. Hemmecke, E.Kim, F. Liu, U. Rothblum, F. Santos, S. Onn, R. Yoshida,
More information