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DP IB Maths: AI HL

Revision Notes

Home / IB / Maths: AI HL / DP / Revision Notes / 3. Geometry & Trigonometry / 3.10 Graph Theory / 3.10.6 Bounds for Travelling Salesman Problem


3.10.6 Bounds for Travelling Salesman Problem


This revision note discusses more complex situations for the travelling salesman problem and you may wish to refer to the revision note 3.10.5 Travelling Salesman Problem.

Table of Least Distances

In some real-life contexts a graph may not be complete nor satisfy the triangle inequality, for example, when looking at a rail network, not every stop will be connected to every other stop and it may be quicker to travel from stop A to stop B via stop C rather than to travel from A to B directly. Thus, the problem is considered to be a practical travelling salesman problem.

Finding the table of least distances (or weights) can convert a practical travelling salesman problem into a classical travelling salesman problem that can then be analysed.

What is a table of least distances?

  • A table of least distances shows the shortest distance between any two vertices in a graph
    • In some cases, the direct route between two vertices may not be the shortest
  • By finding the table of least distances, a graph can be converted into a complete graph that satisfies the triangle inequality
  • STEP 1
    Fill in the information for vertices that are adjacent in the graph (at this stage check if the direct connections are actually the shortest route)
  • STEP 2
    Complete the rest of the table by finding the shortest route that can be travelled between each pair of vertices that are not adjacent

Exam Tip

  • Remember that the table of least values has a line of symmetry along the leading diagonal for an undirected graph, so complete one half carefully first, then map over to the second half

Worked Example

The graph G below contains six vertices representing villages and the roads that connect them. The weighting of the edges represents the time, in minutes, that it takes to walk along a particular road between two villages.

3-10-6-ib-ai-hl-bounds-for-travelling-salesman-problem-we-1

a)
Explain why G is not complete graph.

3-10-6-ib-ai-hl-bounds-for-travelling-salesman-problem-we-1a-solution

b)
Complete the table of least weights below.

  A B C D E F
A            
B            
C            
D            
E            
F            

3-10-6-ib-ai-hl-bounds-for-travelling-salesman-problem-we-1bi-solution

3-10-6-ib-ai-hl-bounds-for-travelling-salesman-problem-we-1bii-solution

Nearest Neighbour Algorithm

As the number of vertices in a graph increases, so does the number of possible Hamiltonian cycles and it can become impractical to solve. The nearest neighbour algorithm can be used to find the upper bound for the minimum weight Hamiltonian cycle.

What is the nearest neighbour algorithm?

  • For a complete graph with at least 3 vertices, performing the nearest neighbour algorithm will generate a low (but not necessarily least) weight Hamiltonian cycle
  • This low weight cycle can be considered the upper bound
  • The best upper bound is the upper bound with the smallest value
  • The nearest neighbour algorithm can only be used on a graph that is complete and satisfies the triangle inequality so the table of least distances should be found first

What are the steps of the nearest neighbour algorithm?

  • STEP 1
    Choose a starting vertex
  • STEP 2
    Follow the edge of least weight from the current vertex to an adjacent unvisited vertex (if there is more than one edge of least weight pick one at random)
  • STEP 3
    Repeat STEP 2 until all vertices have been visited
  • STEP 4
    Add the final edge to return to the starting vertex

Exam Tip

  • If asked to write down the route for the lower bound, don’t forget that some of the entries in the table of lowest distances may not be direct routes between vertices!

Worked Example

The table below contains six vertices representing villages and the roads that connect them. The weighting of the edges represents the time in minutes that it takes to walk along a particular road between two villages.

  A B C D E F
A - 14 7 11 13 21
B 14 - 13 18 13 9
C 7 13 - 5 6 14
D 11 18 5 - 10 18
E 13 13 6 10 - 8
F 21 9 14 18 8 -

Starting at village A, use the nearest neighbour algorithm to find the upper bound of the time it would take to visit each village and return to village A.

yTG4~P0v_3-10-6-ib-ai-hl-bounds-for-travelling-salesman-problem-we-2i-solution

3-10-6-ib-ai-hl-bounds-for-travelling-salesman-problem-we-2ii-solution

Deleted Vertex Algorithm

The deleted vertex algorithm can be used to find the lower bound for the minimum weight Hamiltonian cycle.

What is the deleted vertex algorithm?

  • The deleted vertex algorithm can only be used on a graph that is complete and satisfies the triangle inequality so the table of least distances should be found first
  • Deleting different vertices may give different results, the best lower bound is the lower bound with the highest value
  • If you have found a cycle the same length as the lower bound then you have found the shortest route for the travelling salesman problem
  • If the lower bound and the upper bound are the same weight then you have found the shortest route for the travelling salesman problem

What are the steps of the deleted vertex algorithm?

  • STEP 1
    Choose a vertex and delete it along with all edges that are connected to it
  • STEP 2
    Find the minimum spanning tree for the remaining graph (see revision note 3.10.3 Minimum Spanning Trees)
  • STEP 3
    Add the two shortest edges that were deleted from the original graph to the weight of the minimum spanning tree

Exam Tip

  • Be careful when using a weighted adjacency table not to get confused between using Prim’s algorithm and the nearest neighbour algorithm.
    • Remember that Prim’s is used to find a minimum spanning tree, so vertices can be connected to several other vertices and hence can have more than one value in a column circled
    • When using the table for the nearest neighbour algorithm, vertices cannot be revisited so only one value will be circled in each column

Worked Example

The table below contains six vertices representing villages and the roads that connect them. The weighting of the edges represents the time in minutes that it takes to walk along a particular road between two villages.

  A B C D E F
A - 14 7 11 13 21
B 14 - 13 18 13 9
C 7 13 - 5 6 14
D 11 18 5 - 10 8
E 13 13 6 10 - 8
F 21 9 14 18 8 -

a)
By deleting vertex A and using Prim’s algorithm, find a lower bound for the time taken to start at village A, visit each of the other villages and return to village A

3-10-6-ib-ai-hl-bounds-for-travelling-salesman-problem-we-3a-solution

b)
Show that by deleting vertex B instead, a higher lower bound can be found.

3-10-6-ib-ai-hl-bounds-for-travelling-salesman-problem-we-3b-solution

 



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Approximation & Estimation
          • 1.1.3 GDC: Solving Equations
          • 1.2 Exponentials & Logs
            • 1.2.1 Exponents
              • 1.2.2 Logarithms
              • 1.3 Sequences & Series
                • 1.3.1 Language of Sequences & Series
                  • 1.3.2 Arithmetic Sequences & Series
                    • 1.3.3 Geometric Sequences & Series
                      • 1.3.4 Applications of Sequences & Series
                      • 1.4 Financial Applications
                        • 1.4.1 Compound Interest & Depreciation
                          • 1.4.2 Amortisation & Annuities
                          • 1.5 Complex Numbers
                            • 1.5.1 Intro to Complex Numbers
                              • 1.5.2 Modulus & Argument
                                • 1.5.3 Introduction to Argand Diagrams
                                • 1.6 Further Complex Numbers
                                  • 1.6.1 Geometry of Complex Numbers
                                    • 1.6.2 Forms of Complex Numbers
                                      • 1.6.3 Applications of Complex Numbers
                                      • 1.7 Matrices
                                        • 1.7.1 Introduction to Matrices
                                          • 1.7.2 Operations with Matrices
                                            • 1.7.3 Determinants & Inverses
                                              • 1.7.4 Solving Systems of Linear Equations with Matrices
                                              • 1.8 Eigenvalues & Eigenvectors
                                                • 1.8.1 Eigenvalues & Eigenvectors
                                                  • 1.8.2 Applications of Matrices
                                                • 2. Functions
                                                  • 2.1 Linear Functions & Graphs
                                                    • 2.1.1 Equations of a Straight Line
                                                    • 2.2 Further Functions & Graphs
                                                      • 2.2.1 Functions
                                                        • 2.2.2 Graphing Functions
                                                          • 2.2.3 Properties of Graphs
                                                          • 2.3 Modelling with Functions
                                                            • 2.3.1 Linear Models
                                                              • 2.3.2 Quadratic & Cubic Models
                                                                • 2.3.3 Exponential Models
                                                                  • 2.3.4 Direct & Inverse Variation
                                                                    • 2.3.5 Sinusoidal Models
                                                                      • 2.3.6 Strategy for Modelling Functions
                                                                      • 2.4 Functions Toolkit
                                                                        • 2.4.1 Composite & Inverse Functions
                                                                        • 2.5 Transformations of Graphs
                                                                          • 2.5.1 Translations of Graphs
                                                                            • 2.5.2 Reflections of Graphs
                                                                              • 2.5.3 Stretches of Graphs
                                                                                • 2.5.4 Composite Transformations of Graphs
                                                                                • 2.6 Further Modelling with Functions
                                                                                  • 2.6.1 Properties of Further Graphs
                                                                                    • 2.6.2 Natural Logarithmic Models
                                                                                      • 2.6.3 Logistic Models
                                                                                        • 2.6.4 Piecewise Models
                                                                                      • 3. Geometry & Trigonometry
                                                                                        • 3.1 Geometry Toolkit
                                                                                          • 3.1.1 Coordinate Geometry
                                                                                            • 3.1.2 Radian Measure
                                                                                              • 3.1.3 Arcs & Sectors
                                                                                              • 3.2 Geometry of 3D Shapes
                                                                                                • 3.2.1 3D Coordinate Geometry
                                                                                                  • 3.2.2 Volume & Surface Area
                                                                                                  • 3.3 Trigonometry
                                                                                                    • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                      • 3.3.2 Non Right-Angled Trigonometry
                                                                                                        • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                        • 3.4 Further Trigonometry
                                                                                                          • 3.4.1 The Unit Circle
                                                                                                            • 3.4.2 Simple Identities
                                                                                                              • 3.4.3 Solving Trigonometric Equations
                                                                                                              • 3.5 Voronoi Diagrams
                                                                                                                • 3.5.1 Voronoi Diagrams
                                                                                                                  • 3.5.2 Toxic Waste Dump Problem
                                                                                                                  • 3.6 Matrix Transformations
                                                                                                                    • 3.6.1 Matrix Transformations
                                                                                                                      • 3.6.2 Determinant of a Transformation Matrix
                                                                                                                      • 3.7 Vector Properties
                                                                                                                        • 3.7.1 Introduction to Vectors
                                                                                                                          • 3.7.2 Position & Displacement Vectors
                                                                                                                            • 3.7.3 Magnitude of a Vector
                                                                                                                              • 3.7.4 The Scalar Product
                                                                                                                                • 3.7.5 The Vector Product
                                                                                                                                  • 3.7.6 Components of Vectors
                                                                                                                                    • 3.7.7 Geometric Proof with Vectors
                                                                                                                                    • 3.8 Vector Equations of Lines
                                                                                                                                      • 3.8.1 Vector Equations of Lines
                                                                                                                                        • 3.8.2 Shortest Distances with Lines
                                                                                                                                        • 3.9 Modelling with Vectors
                                                                                                                                          • 3.9.1 Kinematics with Vectors
                                                                                                                                            • 3.9.2 Constant & Variable Velocity
                                                                                                                                            • 3.10 Graph Theory
                                                                                                                                              • 3.10.1 Introduction to Graph Theory
                                                                                                                                                • 3.10.2 Walks & Adjacency Matrices
                                                                                                                                                  • 3.10.3 Minimum Spanning Trees
                                                                                                                                                    • 3.10.4 Chinese Postman Problem
                                                                                                                                                      • 3.10.5 Travelling Salesman Problem
                                                                                                                                                        • 3.10.6 Bounds for Travelling Salesman Problem
                                                                                                                                                      • 4. Statistics & Probability
                                                                                                                                                        • 4.1 Statistics Toolkit
                                                                                                                                                          • 4.1.1 Sampling
                                                                                                                                                            • 4.1.2 Data Collection
                                                                                                                                                              • 4.1.3 Statistical Measures
                                                                                                                                                                • 4.1.4 Frequency Tables
                                                                                                                                                                  • 4.1.5 Linear Transformations of Data
                                                                                                                                                                    • 4.1.6 Outliers
                                                                                                                                                                      • 4.1.7 Univariate Data
                                                                                                                                                                        • 4.1.8 Interpreting Data
                                                                                                                                                                        • 4.2 Correlation & Regression
                                                                                                                                                                          • 4.2.1 Bivariate Data
                                                                                                                                                                            • 4.2.2 Correlation Coefficients
                                                                                                                                                                              • 4.2.3 Linear Regression
                                                                                                                                                                              • 4.3 Further Correlation & Regression
                                                                                                                                                                                • 4.3.1 Non-linear Regression
                                                                                                                                                                                  • 4.3.2 Logarithmic Scales
                                                                                                                                                                                    • 4.3.3 Linearising using Logarithms
                                                                                                                                                                                    • 4.4 Probability
                                                                                                                                                                                      • 4.4.1 Probability & Types of Events
                                                                                                                                                                                        • 4.4.2 Conditional Probability
                                                                                                                                                                                          • 4.4.3 Sample Space Diagrams
                                                                                                                                                                                          • 4.5 Probability Distributions
                                                                                                                                                                                            • 4.5.1 Discrete Probability Distributions
                                                                                                                                                                                              • 4.5.2 Expected Values
                                                                                                                                                                                              • 4.6 Random Variables
                                                                                                                                                                                                • 4.6.1 Linear Combinations of Random Variables
                                                                                                                                                                                                  • 4.6.2 Unbiased Estimates
                                                                                                                                                                                                  • 4.7 Binomial Distribution
                                                                                                                                                                                                    • 4.7.1 The Binomial Distribution
                                                                                                                                                                                                      • 4.7.2 Calculating Binomial Probabilities
                                                                                                                                                                                                      • 4.8 Normal Distribution
                                                                                                                                                                                                        • 4.8.1 The Normal Distribution
                                                                                                                                                                                                          • 4.8.2 Calculations with Normal Distribution
                                                                                                                                                                                                          • 4.9 Further Normal Distribution (inc Central Limit Theorem)
                                                                                                                                                                                                            • 4.9.1 Sample Mean Distribution
                                                                                                                                                                                                              • 4.9.2 Confidence Interval for the Mean
                                                                                                                                                                                                              • 4.10 Poisson Distribution
                                                                                                                                                                                                                • 4.10.1 Poisson Distribution
                                                                                                                                                                                                                  • 4.10.2 Calculating Poisson Probabilities
                                                                                                                                                                                                                  • 4.11 Hypothesis Testing
                                                                                                                                                                                                                    • 4.11.1 Hypothesis Testing
                                                                                                                                                                                                                      • 4.11.2 Chi-squared Test for Independence
                                                                                                                                                                                                                        • 4.11.3 Goodness of Fit Test
                                                                                                                                                                                                                        • 4.12 Further Hypothesis Testing
                                                                                                                                                                                                                          • 4.12.1 Hypothesis Testing for Mean (One Sample)
                                                                                                                                                                                                                            • 4.12.2 Hypothesis Testing for Mean (Two Sample)
                                                                                                                                                                                                                              • 4.12.3 Binomial Hypothesis Testing
                                                                                                                                                                                                                                • 4.12.4 Poisson Hypothesis Testing
                                                                                                                                                                                                                                  • 4.12.5 Hypothesis Testing for Correlation
                                                                                                                                                                                                                                    • 4.12.6 Type I & Type II Errors
                                                                                                                                                                                                                                    • 4.13 Transition Matrices & Markov Chains
                                                                                                                                                                                                                                      • 4.13.1 Markov Chains
                                                                                                                                                                                                                                        • 4.13.2 Transition Matrices
                                                                                                                                                                                                                                      • 5. Calculus
                                                                                                                                                                                                                                        • 5.1 Differentiation
                                                                                                                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                                                                                                                              • 5.1.3 Modelling with Differentiation
                                                                                                                                                                                                                                              • 5.2 Further Differentiation
                                                                                                                                                                                                                                                • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                  • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                    • 5.2.3 Related Rates of Change
                                                                                                                                                                                                                                                      • 5.2.4 Second Order Derivatives
                                                                                                                                                                                                                                                        • 5.2.5 Further Applications of Differentiation
                                                                                                                                                                                                                                                          • 5.2.6 Concavity & Points of Inflection
                                                                                                                                                                                                                                                          • 5.3 Integration
                                                                                                                                                                                                                                                            • 5.3.1 Trapezoid Rule: Numerical Integration
                                                                                                                                                                                                                                                              • 5.3.2 Introduction to Integration
                                                                                                                                                                                                                                                                • 5.3.3 Applications of Integration
                                                                                                                                                                                                                                                                • 5.4 Further Integration
                                                                                                                                                                                                                                                                  • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                    • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                      • 5.4.3 Further Applications of Integration
                                                                                                                                                                                                                                                                        • 5.4.4 Volumes of Revolution
                                                                                                                                                                                                                                                                        • 5.5 Kinematics
                                                                                                                                                                                                                                                                          • 5.5.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                            • 5.5.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                            • 5.6 Differential Equations
                                                                                                                                                                                                                                                                              • 5.6.1 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                • 5.6.2 Separation of Variables
                                                                                                                                                                                                                                                                                  • 5.6.3 Slope Fields
                                                                                                                                                                                                                                                                                    • 5.6.4 Approximate Solutions to Differential Equations
                                                                                                                                                                                                                                                                                    • 5.7 Further Differential Equations
                                                                                                                                                                                                                                                                                      • 5.7.1 Coupled Differential Equations
                                                                                                                                                                                                                                                                                        • 5.7.2 Second Order Differential Equations
                                                                                                                                                                                                                                                                                      Naomi Calverd

                                                                                                                                                                                                                                                                                      Author: Naomi

                                                                                                                                                                                                                                                                                      Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.


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