1. 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.

2. 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.

3. Chapter 2: Business Efficiency

4. • Given a collection of cities as
The well as the cost of travel
between each, we must find
Probl the cheapest way of visiting all
the cities and returning to the
em 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

5. Arranging school bus
routes Applicat
Delivery of meals to ions
homebound persons
Scheduling of a
machine to drill holes in
a circuit board
Genome sequencing
Scan chain optimization
in computer chips
Trip planning

6. For All Practical
Chapter 2: Business Efficiency Purposes
 Business Efficiency
 Visiting Vertices-Graph Theory Problem
Mathematical Literacy in
Today’s World, 8th ed.
 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
© 2009, W.H. Freeman and Company

7. Chapter 2: Business Efficiency
Business 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.
Applications:
 Salesman visiting
particular cities
 Delivering mail to
drop-off boxes
 Route taken by a
snowplow
 Pharmaceutical
representative
visiting doctors

8. Chapter 2: Business Efficiency
Hamiltonian Circuit
 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.
Starting at vertex A, the tour can be A different circuit visiting each vertex
written as ABDGIHFECA, or starting at once and only once would be
E, it would be EFHIGDBACE. CDBIGFEHAC (starting at vertex C).

9. Chapter 2: Business Efficiency
Hamiltonian Circuit vs. Euler Circuits
Hamiltonian vs. Euler Circuits Hamiltonian circuit –
 Similarities A tour (showed by
 Both forbid re-use. wiggly edges) that
 Hamiltonian do not reuse vertices. starts at a vertex of a
 Euler do not reuse edges. graph and visits each
vertex once and only
 Differences once, returning to
 Hamiltonian is a circuit of vertices. where it started.
 Euler is a circuit of edges.
 Euler graphs are easy to spot Euler circuit – A circuit
(connectedness and even valence). that traverses each
 Hamiltonian circuits are NOT as easy edge of a graph
to determine upon inspection. exactly once and starts
 Some certain family of graphs can be and stops at the same
known to have or not have Hamiltonian
circuits. point.

10. Chapter 2: Business Efficiency
Hamiltonian 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)
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.
Road mileage between four cities

11. Chapter 2: Business Efficiency
Hamiltonian Circuit
 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
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 Algorithm – A step-by-step description
minimum distance. of how to solve a problem.

12. Chapter 2: Business Efficiency
Hamiltonian Circuits
 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. Method of trees for vacation-planning problem

13. Chapter 2: Business Efficiency
Hamiltonian Circuits
 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. The three Hamiltonian circuits’ sums of the tours

14. Chapter 2: Business Efficiency
Hamiltonian Circuit
 Principle of Counting
Complete graph –
for Hamiltonian Circuits
 For a complete graph of n vertices, there are
A graph in
(n - 1)! possible routes. which every
 Half of these routes are repeats, the result is: pair of vertices
Possible unique Hamiltonian circuits are is joined by an
edge.
(n - 1)! / 2
 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.

15. Chapter 2: Business Efficiency
Traveling Salesman Problem
 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

16. Chapter 2: Business Efficiency
Traveling Salesman Problem — Nearest Neighbor
 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.
Hamiltonian Circuit: Hamiltonian Circuit:
A-B-C-E-D-A B-C-A-D-E-B
Nearest neighbor starting at vertex A Nearest neighbor starting at vertex B

17. Chapter 2: Business Efficiency
Traveling 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.
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.

18. Chapter 2: Business Efficiency
Minimum-Cost Spanning Trees
 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. Costs (in millions of dollars)
 The cost of redirecting the signal may be of installing Pictaphone
small compared to adding another link. service among five cities

19. Chapter 2: Business Efficiency
Minimum-Cost Spanning Trees
 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.

20. Chapter 2: Business Efficiency
Critical Path Analysis
 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.
An order-requirement digraph, tasks
A – E with task times in the circles
Critical Path is BE = 25 + 27 = 52 min.

21. Order-requirement digraph

22. Skill Objectives


23. 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.
.

24. 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.