Chapter 6 TRAVELLING SALESMAN PROBLEM


 Alicia Williams
 2 years ago
 Views:
Transcription
1 98 hapter 6 TRVLLING SLSMN PROLM 6.1 Introduction The Traveling Salesman Problem (TSP) is a problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pair wise distances, the task is to find a shortest possible tour that visits each city exactly once. The problem was first formulated as a mathematical problem by Menger (1930) and is one of the most intensively studied problems in optimization. However it was unnoticed till Menger (1994) published a book where he narrated the foundation of mathematical problem for the travelling salesman problem. It is used as a benchmark for many optimization methods. ven though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved. The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. If a slight modification is made in the problem, it appears as a subproblem in many areas, such as genome sequencing. In these applications, the concept city represents, for example, customers, soldering points, or N fragments, and the concept distance represents traveling times or cost, or a similarity measure between N fragments. In many applications, additional constraints such as limited resources or time windows make the problem considerably harder. In the theory of computational complexity, the decision version of TSP belongs to the class of NPcomplete problems. Thus, it is assumed that there is no efficient algorithm for solving TSP problems. In other words, it is likely that the worst case running time for any algorithm for TSP increases exponentially with the number of cities, so
2 99 even some instances with only hundreds of cities will take many PU years to solve exactly. The origins of the traveling salesman problem are unclear. handbook for traveling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, but contains no mathematical treatment. Hamilton (1800) and Kirkman (1800) expressed the concept of Mathematical problems related to the travelling salesman problem. The general form of the TSP appears to have been first studied by mathematicians notably by Menger (1930). Further Menger (1930) also defines the problem related with salesman ship based on bruteforce algorithm, and observes the nonoptimality of the nearest neighbor heuristic. However Whitney (1930) introduced the name travelling salesman problem. uring the period 1950 to 1960, the travelling salesman problem started getting popularity in scientific circle is especially in urope and the US. Many researchers like antzig, Fulkerson and Johnson (1954) at the RN orporation in Santa Monica expressed the problem as an integer linear program and developed the cutting plane method for its solution. With these new methods they solved an instance with 49 cities to optimality by constructing a tour and proving that no other tour could be shorter. In the following decades, the problem was studied by many researchers from mathematics, science, chemistry, physics, and other sciences. Karp (1972) showed that the Hamiltonian cycle problem was NPcomplete, which implies the NPhardness of TSP. This supplied a scientific explanation for the apparent computational difficulty of finding optimal tours. 6.2 pplication of New lternate Method of ssignment Problem in TSP We have so far already discussed the algorithm and examples for solving an assignment problem using a new alternate method in hapter 5. Now in this chapter we discuss how the new alternate method for solving an assignment
3 100 problem can be applied for Travelling Salesman Problem. For this we have considered an example related with travelling salesman problem and explain in detail how to find optimal solution using new alternate method of assignment problem. xample salesman has to visit five cities.,, and. The distances (in hundred kilometers) between the five cities are shown in Table 6.1. From ity Table 6.1 To city If the salesman starts from city and has to come back to city, which route should he select so that total distance traveled become minimum? Solution onsider the effective matrix. This is shown in Table 6.2. From ity Table 6.2 To city In this matrix first, we will take first row which is referred a city. We select that column (assignment) for which it contains minimum distance. For this example, incase of first row, column (assignment) has the minimum value. In the similar way, we select all the rows and find the minimum value for the respective columns. These are given in Table 6.3.
4 101 olumn 1(ity) Table 6.3 olum 2(ssignment) In this table, we observed that assignment occur only once with city. That is city is unique for city and hence we assign city to. This is shown in Table 6.4. Table However, for other job assignment occur more than once. Hence they are not unique. So how other job will be assigned further we discuss below. Next delete row and column. gain select minimum cost value for the remaining cities which is shown below Table 6.5. Table 6.5 olumn 1(ity) olum 2(ssignment) Since assignment occur with city and. Hence first we take the difference between the value of and next minimum value (here tie is happens). Here
5 102 maximum difference is 2 for and hence we assign to city. This is shown in Table 6.6. Table Next delete row and column. gain select minimum cost value for the remaining cities which is shown below Table 6.7. olumn 1(ity) Table 6.7 olum 2(ssignment), Since assignment occur with city, and. Hence we take the difference between the value of and next minimum value, here the maximum difference is 3 for Job. nd hence we assign to city. This is shown in Table 6.8. Table Next delete row and column. gain select minimum cost value for the remaining cities which is shown below Table 6.9. Table 6.9 olumn 1(ity) olumn 2(ssignment)
6 103 Here, we cannot assign to city. Therefore we only assign to city. Then obviously, we have no other choice rather to assign for ity. Finally, we can assign all the cities along with distance which is shown in Table Table 6.10 olumn 1(ity) olum 2(ssignment) istance Total 26 This solution is happened to be same as that of Hungarian method. Hence we can say that the minimum value is still 26 in both the methods. So this solution is optimal. However our method seems to be very simple, easy and takes very few steps in solving the method. 6.3 onclusion In this chapter, we have applied new alternate method of an assignment problem for solving Travelling salesman problem where it is shown that this method also gives optimal solution. Moreover the optimal solution obtained using this method is same as that of optimal solution obtained by Hungarian method. So we conclude that the Hungarian method and our method give same optimal solution. However the technique for solving Travelling Salesman problem using our method is more simple and easy as it takes few steps for the optimal solution.
An Exact Algorithm for Steiner Tree Problem on Graphs
International Journal of Computers, Communications & Control Vol. I (2006), No. 1, pp. 4146 An Exact Algorithm for Steiner Tree Problem on Graphs Milan Stanojević, Mirko Vujošević Abstract: The paper
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 informationSolution of a LargeScale TravelingSalesman Problem
Chapter 1 Solution of a LargeScale TravelingSalesman Problem George B. Dantzig, Delbert R. Fulkerson, and Selmer M. Johnson Introduction by Vašek Chvátal and William Cook The birth of the cuttingplane
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 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 informationAssignment Problems. Guoming Tang
Assignment Problems Guoming Tang A Practical Problem Workers: A, B, C, D Jobs: P, Q, R, S. Cost matrix: Job P Job Q Job R Job S Worker A 1 2 3 4 Worker B 2 4 6 8 Worker C 3 6 9 12 Worker D 4 8 12 16 Given:
More informationThe TSP is a special case of VRP, which means VRP is NPhard.
49 6. ROUTING PROBLEMS 6.1. VEHICLE ROUTING PROBLEMS Vehicle Routing Problem, VRP: Customers i=1,...,n with demands of a product must be served using a fleet of vehicles for the deliveries. The vehicles,
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 informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
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 informationGreedy Heuristics with Regret, with Application to the Cheapest Insertion Algorithm for the TSP
Greedy Heuristics with Regret, with Application to the Cheapest Insertion Algorithm for the TSP Refael Hassin Ariel Keinan Abstract We considers greedy algorithms that allow partial regret. As an example
More informationHigh Performance Computing for Operation Research
High Performance Computing for Operation Research IEF  Paris Sud University claude.tadonki@upsud.fr INRIAAlchemy seminar, Thursday March 17 Research topics Fundamental Aspects of Algorithms and Complexity
More informationNPcomplete? NPhard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics
NPcomplete? NPhard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer
More informationResearch Paper Business Analytics. Applications for the Vehicle Routing Problem. Jelmer Blok
Research Paper Business Analytics Applications for the Vehicle Routing Problem Jelmer Blok Applications for the Vehicle Routing Problem Jelmer Blok Research Paper Vrije Universiteit Amsterdam Faculteit
More informationDesign and Analysis of ACO algorithms for edge matching problems
Design and Analysis of ACO algorithms for edge matching problems Carl Martin Dissing Söderlind Kgs. Lyngby 2010 DTU Informatics Department of Informatics and Mathematical Modelling Technical University
More informationAutomated SEO. A Market Brew White Paper
Automated SEO A Market Brew White Paper Abstract In this paper, we use the term Reach to suggest the forecasted traffic to a particular webpage or website. Reach is a traffic metric that describes an expected
More informationON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu
ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a
More informationMATHEMATICAL ENGINEERING TECHNICAL REPORTS. An Improved Approximation Algorithm for the Traveling Tournament Problem
MATHEMATICAL ENGINEERING TECHNICAL REPORTS An Improved Approximation Algorithm for the Traveling Tournament Problem Daisuke YAMAGUCHI, Shinji IMAHORI, Ryuhei MIYASHIRO, Tomomi MATSUI METR 2009 42 September
More information5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
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 informationChapter 15 Introduction to Linear Programming
Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 WeiTa Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of
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 informationSolving the Travelling Salesman Problem Using the Ant Colony Optimization
Ivan Brezina Jr. Zuzana Čičková Solving the Travelling Salesman Problem Using the Ant Colony Optimization Article Info:, Vol. 6 (2011), No. 4, pp. 010014 Received 12 July 2010 Accepted 23 September 2011
More informationLecture 6: Approximation via LP Rounding
Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other
More informationHow to speedup hard problem resolution using GLPK?
How to speedup hard problem resolution using GLPK? Onfroy B. & Cohen N. September 27, 2010 Contents 1 Introduction 2 1.1 What is GLPK?.......................................... 2 1.2 GLPK, the best one?.......................................
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 informationInteger Programming Approach to Printed Circuit Board Assembly Time Optimization
Integer Programming Approach to Printed Circuit Board Assembly Time Optimization Ratnesh Kumar Haomin Li Department of Electrical Engineering University of Kentucky Lexington, KY 405060046 Abstract A
More informationThe Hungarian Method for the Assignment Problem
Chapter 2 The Hungarian Method for the Assignment Problem Harold W. Kuhn Introduction by Harold W. Kuhn This paper has always been one of my favorite children, combining as it does elements of the duality
More informationPlanar Tree Transformation: Results and Counterexample
Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem
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 informationLecture 3. Mathematical Induction
Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion
More informationEvaluation of Complexity of Some Programming Languages on the Travelling Salesman Problem
International Journal of Applied Science and Technology Vol. 3 No. 8; December 2013 Evaluation of Complexity of Some Programming Languages on the Travelling Salesman Problem D. R. Aremu O. A. Gbadamosi
More informationDynamic Programming.S1 Sequencing Problems
Dynamic Programming.S1 Sequencing Problems Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard Many operational problems in manufacturing, service and distribution require the sequencing
More informationHeuristics and Local Search. Version 2 c 2009, 2001 José Fernando Oliveira, Maria Antónia Carravilla FEUP
Heuristics and Local Search Version c 009, 001 José Fernando Oliveira, Maria Antónia Carravilla FEUP Approximate methods to solve combinatorial optimization problems Heuristics Aim to efficiently generate
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 informationA greedy algorithm for the DNA sequencing by hybridization with positive and negative errors and information about repetitions
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 59, No. 1, 2011 DOI: 10.2478/v1017501100150 Varia A greedy algorithm for the DNA sequencing by hybridization with positive and negative
More information6. Mixed Integer Linear Programming
6. Mixed Integer Linear Programming Javier Larrosa Albert Oliveras Enric RodríguezCarbonell Problem Solving and Constraint Programming (RPAR) Session 6 p.1/40 Mixed Integer Linear Programming A mixed
More informationDiscrete (and Continuous) Optimization Solutions of Exercises 1 WI4 131
Discrete (and Continuous) Optimization Solutions of Exercises 1 WI4 131 Kees Roos Technische Universiteit Delft Faculteit Informatietechnologie en Systemen Afdeling Informatie, Systemen en Algoritmiek
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationINTEGER PROGRAMMING. Integer Programming. Prototype example. BIP model. BIP models
Integer Programming INTEGER PROGRAMMING In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is
More informationis your answer, but you are asked to give the answer (correct to 2 decimal places). then the second number is increased by 1.
DECIMAL PLACES Sometimes you are required to give a shorter answer than the one which you have worked out. Example 1 3.68472 is your answer, but you are asked to give the answer correct to 2 decimal places
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
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 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 informationOffline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com
More information1 Basic Definitions and Concepts in Graph Theory
CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered
More information1. Sorting (assuming sorting into ascending order) a) BUBBLE SORT
DECISION 1 Revision Notes 1. Sorting (assuming sorting into ascending order) a) BUBBLE SORT Make sure you show comparisons clearly and label each pass First Pass 8 4 3 6 1 4 8 3 6 1 4 3 8 6 1 4 3 6 8 1
More informationAIMA Chapter 3: Solving problems by searching
AIMA Chapter 3: Solving problems by searching Ana Huaman January 13, 2011 1. Define in your own words the following terms: state, state space, search tree, search node, goal, action, transition model and
More informationConstruction Heuristics
Outline DM811 HEURISTICS AND LOCAL SEARCH ALGORITHMS FOR COMBINATORIAL OPTIMZATION Lecture 5 Construction Heuristics Marco Chiarandini 1. Complete Search Methods 2. Incomplete Search Methods 3. Other Tree
More informationVehicle Routing and Scheduling. Martin Savelsbergh The Logistics Institute Georgia Institute of Technology
Vehicle Routing and Scheduling Martin Savelsbergh The Logistics Institute Georgia Institute of Technology Vehicle Routing and Scheduling Part I: Basic Models and Algorithms Introduction Freight routing
More informationComputational complexity theory
Computational complexity theory Goal: A general theory of the resources needed to solve computational problems What types of resources? Time What types of computational problems? decision problem Decision
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 informationMinimize subject to. x S R
Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such
More informationDeterministic Problems
Chapter 2 Deterministic Problems 1 In this chapter, we focus on deterministic control problems (with perfect information), i.e, there is no disturbance w k in the dynamics equation. For such problems there
More informationDiscrete Optimization Introduction & applications
Discrete Optimization 2013 1/21 Discrete Optimization Introduction & applications Bertrand Cornélusse ULg  Institut Montefiore 2013 Discrete Optimization 2013 2/21 Outline Introduction Some applications
More informationCMSC 858F: Algorithmic Lower Bounds Fall 2014 Puzzles and Reductions from 3Partition
CMSC 858F: Algorithmic Lower Bounds Fall 2014 Puzzles and Reductions from 3Partition Instructor: Mohammad T. Hajiaghayi Scribe: Thomas Pensyl September 9, 2014 1 Overview In this lecture, we first examine
More informationOptimal Scheduling for Dependent Details Processing Using MS Excel Solver
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 8, No 2 Sofia 2008 Optimal Scheduling for Dependent Details Processing Using MS Excel Solver Daniela Borissova Institute of
More informationAPPLYING MINIMUM TRAVEL COST APPROACH ON 17 NODES TRAVELLING SALESMAN PROBLEM
Geomatikai Közlemények, XIII/2, 2010 APPLYING MINIMUM TRAVEL COST APPROACH ON 17 NODES TRAVELLING SALESMAN PROBLEM Mohamed Eleiche and Bela Markus A minimális utazási költség alkalmazása a 17 csomópontos
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationNPCompleteness and Cook s Theorem
NPCompleteness and Cook s Theorem Lecture notes for COM3412 Logic and Computation 15th January 2002 1 NP decision problems The decision problem D L for a formal language L Σ is the computational task:
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 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 informationDepartment of Industrial Engineering
Department of Industrial Engineering Master of Engineering Program in Industrial Engineering (International Program) M.Eng. (Industrial Engineering) Plan A Option 2: Total credits required: minimum 39
More informationSUPPLEMENT TO CHAPTER 6 MINIMUM SPANNINGTREE PROBLEMS
S h 6 SUPPLMNT TO HPTR 6 MINIMUM SPNNINTR PROLMS hapter 6 focuses on network optimization problems. These are problems that can be described in terms of a complete network that has both nodes and links
More informationHybridization of One s Assignment and Stepping Stone Method
Mathematica Aeterna, Vol. 5, 2015, no. 1, 211216 Hybridization of One s Assignment and Stepping Stone Method Kirtiwant Ghadle Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad
More informationRow Echelon Form and Reduced Row Echelon Form
These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for inclass presentation
More informationIE 680 Special Topics in Production Systems: Networks, Routing and Logistics*
IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti
More information2 The Euclidean algorithm
2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In
More informationA MULTISTAGE HEURISTIC APPROACH TO RESOURCE ALLOCATION (AMHARA) FOR AUTOMATED SCHEDULING
エシアン ゾロナル オフ ソシルサエニセズ アンドヒオメニテズ ISSN: 21868492, ISSN: 21868484 Print A MULTISTAGE HEURISTIC APPROACH TO RESOURCE ALLOCATION (AMHARA) FOR AUTOMATED SCHEDULING Mehryar Nooriafshar * Faculty of Business,
More informationSpecial Session on Integrating Constraint Programming and Operations Research ISAIM 2016
Titles Special Session on Integrating Constraint Programming and Operations Research ISAIM 2016 1. GrammarBased Integer Programming Models and Methods for Employee Scheduling Problems 2. Detecting and
More informationCHAPTER 9. Integer Programming
CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral
More informationOn the Hardness of Topology Inference
On the Hardness of Topology Inference H. B. Acharya 1 and M. G. Gouda 2 1 The University of Texas at Austin, USA acharya@cs.utexas.edu 2 The National Science Foundation, USA mgouda@nsf.gov Abstract. Many
More informationBattleships Searching Algorithms
Activity 6 Battleships Searching Algorithms Summary Computers are often required to find information in large collections of data. They need to develop quick and efficient ways of doing this. This activity
More informationCracking the Sudoku: A Deterministic Approach
Cracking the Sudoku: A Deterministic Approach David Martin Erica Cross Matt Alexander Youngstown State University Center for Undergraduate Research in Mathematics Abstract The model begins with the formulation
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationCSE8393 Introduction to Bioinformatics Lecture 3: More problems, Global Alignment. DNA sequencing
SE8393 Introduction to Bioinformatics Lecture 3: More problems, Global lignment DN sequencing Recall that in biological experiments only relatively short segments of the DN can be investigated. To investigate
More informationCan linear programs solve NPhard problems?
Can linear programs solve NPhard problems? p. 1/9 Can linear programs solve NPhard problems? Ronald de Wolf Linear programs Can linear programs solve NPhard problems? p. 2/9 Can linear programs solve
More informationCharles Fleurent Director  Optimization algorithms
Software Tools for Transit Scheduling and Routing at GIRO Charles Fleurent Director  Optimization algorithms Objectives Provide an overview of software tools and optimization algorithms offered by GIRO
More informationInteger programming solution methods  introduction
Integer programming solution methods  introduction J E Beasley Capital budgeting There are four possible projects, which each run for 3 years and have the following characteristics. Capital requirements
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 informationOperations Research for Telecommunication Linear Programming and Network Routing
Operations Research for Telecommunication Linear Programming and Network Routing tks@imm.dtu.dk Informatics and Mathematical Modeling Technical University of Denmark 1 Operations Research (OR) OR: Mathematics
More informationSolving the Vehicle Routing Problem with Genetic Algorithm and Simulated Annealing
Solving the Vehicle Routing Problem with Genetic Algorithm and Simulated Annealing Master Thesis Computer Engineering Ákos Kovács Högskolan Dalarna, Sweden May 2008 DEGREE PROJECT in Computer Engineering
More informationBranchandPrice Approach to the Vehicle Routing Problem with Time Windows
TECHNISCHE UNIVERSITEIT EINDHOVEN BranchandPrice Approach to the Vehicle Routing Problem with Time Windows Lloyd A. Fasting May 2014 Supervisors: dr. M. Firat dr.ir. M.A.A. Boon J. van Twist MSc. Contents
More informationLoad Balancing of Telecommunication Networks based on Multiple Spanning Trees
Load Balancing of Telecommunication Networks based on Multiple Spanning Trees Dorabella Santos Amaro de Sousa Filipe Alvelos Instituto de Telecomunicações 3810193 Aveiro, Portugal dorabella@av.it.pt Instituto
More informationReductions & NPcompleteness as part of Foundations of Computer Science undergraduate course
Reductions & NPcompleteness as part of Foundations of Computer Science undergraduate course Alex Angelopoulos, NTUA January 22, 2015 Outline Alex Angelopoulos (NTUA) FoCS: Reductions & NPcompleteness
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 informationA Robust Formulation of the Uncertain Set Covering Problem
A Robust Formulation of the Uncertain Set Covering Problem Dirk Degel Pascal Lutter Chair of Management, especially Operations Research RuhrUniversity Bochum Universitaetsstrasse 150, 44801 Bochum, Germany
More informationSUPPLY CHAIN OPTIMIZATION MODELS IN THE AREA OF OPERATION
SUPPLY CHAIN OPTIMIZATION MODELS IN THE AREA OF OPERATION Tomáš DVOŘÁK, Martin VLKOVSKÝ Abstract: Main idea of this paper is to choose both suitable and applicable operations research methods for military
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 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 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 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 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 informationDiscrete Mathematics, Chapter 3: Algorithms
Discrete Mathematics, Chapter 3: Algorithms Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 3 1 / 28 Outline 1 Properties of Algorithms
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 informationOptimising Patient Transportation in Hospitals
Optimising Patient Transportation in Hospitals Thomas Hanne 1 Fraunhofer Institute for Industrial Mathematics (ITWM), FraunhoferPlatz 1, 67663 Kaiserslautern, Germany, hanne@itwm.fhg.de 1 Introduction
More informationA Note on Maximum Independent Sets in Rectangle Intersection Graphs
A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,
More informationmax cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x
Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study
More informationAn Effective Implementation of the LinKernighan Traveling Salesman Heuristic
An Effective Implementation of the LinKernighan Traveling Salesman Heuristic Keld Helsgaun Email: keld@ruc.dk Department of Computer Science Roskilde University DK4000 Roskilde, Denmark Abstract This
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
More information