Chapter 1. Skill Objectives

1. Determine by observation if a graph is connected. 2. Identify vertices and edges of a given graph. 3. Construct the graph of a given street network. 4. Determine by observation the valence of each vertex of

a graph. 5. Define an Euler circuit. 6. List the two conditions for the existence of an Euler

circuit. 7. Determine whether a graph contains an Euler circuit.

Chapter 1. Skill Objectives (cont’d)

8. If a graph contains an Euler circuit, list one such circuit by identifying the order in which the vertices are used by the circuit, or by identifying the order in which the edges are to be used.

9. If a graph does not contain an Euler circuit, add a minimum number of edges to “eulerize” the graph.

11. Identify management science problems whose solutions involve Euler circuits.

Chapter 2: Business Efficiency

The Problem

• Given a collection of cities as well as the cost of travel between each, we must find the cheapest way of visiting all the cities and returning to the starting point

• Studying the TSP has led to improved solutions in other math areas

• One of the most intensely studied problems

• No effective solution is known– $1,000,000 prize from the Clay

Mathematics Institute

Applications

Arranging school bus routes

Delivery of meals to homebound persons

Scheduling of a machine to drill holes in a circuit board

Genome sequencing

Scan chain optimization in computer chips

Trip planning

Chapter 2: Business Efficiency

Business Efficiency Visiting Vertices-Graph Theory Problem

Hamiltonian Circuits Vacation Planning Problem

Minimum Cost-Hamiltonian Circuit Method of Trees

Fundamental Principle of Counting

Traveling Salesman Problem

Helping Traveling Salesmen Nearest Neighbor and Sorted Edges Algorithms Minimum-Cost Spanning Trees

Kruskal’s Algorithm

Critical-Path Analysis

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Mathematical Literacy in Today’s World, 8th

ed.

For All Practical Purposes

© 2009, W.H. Freeman and Company

Chapter 2: Business EfficiencyBusiness Efficiency

Visiting Vertices In some graph theory problems, it is only necessary to visit

specific locations (using the travel routes, or streets available). Problem: Find an efficient route along distinct edges of a graph

that visits each vertex only once in a simple circuit.

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Applications: Salesman visiting

particular cities Delivering mail to

drop-off boxes Route taken by a

snowplow Pharmaceutical

representative visiting doctors

Hamiltonian Circuit A tour that starts and ends at the same vertex (circuit definition). Visits each vertex once. (Vertices cannot be reused or revisited.) Circuits can start at any location. Use wiggly edges to show the circuit.

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Starting at vertex A, the tour can be written as ABDGIHFECA, or starting at E, it would be EFHIGDBACE.

Chapter 2: Business EfficiencyHamiltonian Circuit

A different circuit visiting each vertex once and only once would be CDBIGFEHAC (starting at vertex C).

Hamiltonian vs. Euler Circuits Similarities

Both forbid re-use. Hamiltonian do not reuse vertices. Euler do not reuse edges.

Differences Hamiltonian is a circuit of vertices. Euler is a circuit of edges. Euler graphs are easy to spot

(connectedness and even valence). Hamiltonian circuits are NOT as easy

to determine upon inspection. Some certain family of graphs can be

known to have or not have Hamiltonian circuits.

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Chapter 2: Business EfficiencyHamiltonian Circuit vs. Euler Circuits

Hamiltonian circuit – A tour (showed by wiggly edges) that starts at a vertex of a graph and visits each vertex once and only once, returning to where it started.

Euler circuit – A circuit that traverses each edge of a graph exactly once and starts and stops at the same point.

Chapter 2: Business EfficiencyHamiltonian Circuits

Vacation–Planning Problem Hamiltonian circuit concept is used to find the best route that

minimizes the total distance traveled to visit friends in different cities. (assume less mileage less gas minimizes costs)

Road mileage between four cities10

Hamiltonian circuit with weighted edges Edges of the graph are given

weights, or in this case mileage or distance between cities.

As you travel from vertex to vertex, add the numbers (mileage in this case).

Each Hamiltonian circuit will produce a particular sum.

Chapter 2: Business EfficiencyHamiltonian Circuit

Algorithm – A step-by-step description of how to solve a problem.

Minimum-Cost Hamiltonian Circuit A Hamiltonian circuit with the lowest

possible sum of the weights of its edges.

Algorithm (step-by-step process) for Solving This Problem

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1. Generate all possible Hamiltonian tours (starting with Chicago).

2. Add up the distances on the edges of each tour.

3. Choose the tour of minimum distance.

Chapter 2: Business EfficiencyHamiltonian Circuits

Method of trees for vacation-planning problem12

Method of Trees For the first step of the algorithm, a systematic approach is

needed to generate all possible Hamiltonian tours (disregard distances during this step).

This method begins by selecting a starting vertex, say Chicago, and making a tree-diagram showing the next possible locations.

At each stage down, there will be one less choice (3, 2, then 1 choice).

In this example, the method of trees generated six different paths, all starting and ending with Chicago.

However, only three are unique circuits.

Chapter 2: Business EfficiencyHamiltonian Circuits

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The three Hamiltonian circuits’ sums of the tours

Minimum-Cost Hamiltonian Circuit: Vacation-Planning Example

1. Method of tree used to find all tours (for four cities: three unique paths). On the graph, the unique paths are drawn with wiggly lines.

2. Add up distance on edges of each unique tour. The smallest sum would give us the minimal distance, which is the minimum cost.

3. Choose the tour of minimum distance. The smallest sum would give us the minimal distance, which is the minimum cost.

Chapter 2: Business EfficiencyHamiltonian Circuit

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Principle of Counting for Hamiltonian Circuits For a complete graph of n vertices, there are

(n - 1)! possible routes. Half of these routes are repeats, the result is:

Possible unique Hamiltonian circuits are

(n - 1)! / 2

Complete graph – A graph in which every pair of vertices is joined by an edge.

Fundamental Principle of Counting If there are a ways of choosing one thing,

b ways of choosing a second after the first is chosen,

c ways of choosing a third after the second is chosen…, and so on…,

and z ways of choosing the last item after the earlier choices,

then the total number of choice patterns is a × b × c × … × z.

Example: Jack has 9 shirts and 4 pairs of pants. He can wear 9 × 4 = 36 shirt-pant outfits.

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Traveling Salesman Problem (TSP) Difficult to solve Hamiltonian circuits when the number of vertices

in a complete graph increases (n becomes very large). This problem originated from a salesman determining his trip that

minimizes costs (less mileage) as he visits the cities in a sales territory, starting and ending the trip in the same city.

Many applications today: bus schedules, mail drop-offs, telephone booth coin pick-up routes, etc.

How can the TSP be solved? Computer program can find optimal route (not always

practical). Heuristic methods can be used to find a “fast” answer, but

does not guarantee that it is always the optimal answer. Nearest neighbor algorithm Sorted edges algorithm

Chapter 2: Business EfficiencyTraveling Salesman Problem

Chapter 2: Business EfficiencyTraveling Salesman Problem — Nearest Neighbor

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Nearest neighbor starting at vertex A

Nearest Neighbor Algorithm (to solve TSP) Starting from the “home” city (or vertex), first visit the nearest city

(one with the least mileage from “home”). As you travel from city to city, always choose the next city (vertex)

that can be reached quickest (i.e., nearest with the least miles), that has not already been visited.

When all other vertices have been visited, the tour returns home.

Nearest neighbor starting at vertex B

Hamiltonian Circuit: A-B-C-E-D-A

Hamiltonian Circuit: B-C-A-D-E-B

Chapter 2: Business EfficiencyTraveling Salesman Problem — Sorted Edges Sorted Edges Algorithm (to solve TSP)

Start by sorting, or arranging, the edges in order of increasing cost (sort smallest to largest mileage between cities).

At each stage, select that edge of least cost until all the edges are connected at the end while following these rules: If an edge is added that results in three edges meeting at a

vertex, eliminate the longest edge. Always include all vertices in the

finished circuit.

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Example using sorted edges Edges selected are DE at 400, BC at 500, AD at 550, and AB at 600 (AC and AE are not chosen because they result in three edges meeting at A). Lastly, CE at 750 is chosen to complete the circuit of 2800 miles.

Chapter 2: Business EfficiencyMinimum-Cost Spanning Trees

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Costs (in millions of dollars) of installing Pictaphone service among five cities

Minimum-Cost Spanning Trees Another graph theory optimization problem that links all the

vertices together, in order of increasing costs, to form a “tree.” The cost of the tree is the sum of the weights on the edges.

Example: What is the cost to construct a Pictaphone service (telephone service with video image of the callers) among

five cities? The diagram shows the cost to build the

connection from each vertex to all other vertices (connected graph).

Cities are linked in order of increasing costs to make the connection.

The cost of redirecting the signal may be small compared to adding another link.

Chapter 2: Business EfficiencyMinimum-Cost Spanning Trees

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Kruskal’s Algorithm — Developed by Joseph Kruskal (AT&T research). Goal of minimum-cost spanning tree: Create a tree that links all

the vertices together and achieves the lowest cost to create. Add links in order of cheapest cost according to the rules:

No circuit is created (no loops). If a circuit (or loop) is created by adding the next largest link, eliminate this

largest (most expensive link)—it is not needed. Every vertex must be included in the final tree.

Chapter 2: Business EfficiencyCritical Path Analysis

An order-requirement digraph, tasks A – E with task times in the circles

Critical Path is BE = 25 + 27 = 52 min.20

Critical Path Analysis Most often, scheduling jobs consists of complicated tasks that

cannot be done in a random order. Due to a pre-defined order of tasks, the entire job may not be

done any sooner than the longest path of dependent tasks. Order-Requirement Digraph

A directed graph (digraph) that shows which tasks precede other tasks among the collection of tasks making up a job.

Critical Path The longest path in an order-requirement digraph. The length is measured in terms of summing the task times of the

tasks making up the path.

Order-requirement digraph

Skill Objectives

1. Give the definition of a Hamiltonian circuit. 2. Explain the difference between an Euler circuit and a

Hamiltonian circuit. 3. Identify a given application as being an Euler circuit

problem or a Hamiltonian circuit problem. 4. Calculate n! for a given value of n. 5. Apply the formula to calculate the number of

Hamiltonian circuits in a graph with a given number of vertices.

6. Define algorithm.

Skill Objectives (cont’d)

7. Explain the term heuristic algorithm and list both an advantage and a disadvantage.

8. Discuss the difficulties inherent in the application of the brute force method for finding the minimum-cost Hamiltonian circuit.

9. Describe the steps in the nearest-neighbor algorithm. 10. Find an approximate solution to the traveling salesman

problem by applying the nearest-neighbor algorithm. 11. Describe the steps in the sorted-edges algorithm. 12. Find an approximate solution to the traveling salesman

problem by applying the sorted-edges algorithm.

.

Skill Objectives (cont’d)

13. Give the definition of a tree. 14. Given a graph with edge weights, determine a

minimum-cost spanning tree. 15. Identify the critical path in an order-requirement

digraph. 16. Find the earliest possible completion time for a

collection of tasks by finding the critical path in an order-requirement digraph.

17. Explain the difference between a graph and a directed graph.