ENGRI 1101 Engineering Applications of OR
Fall 2008 Handout 7.5
Study Guide for Prelim 1
1.
(a) What is a graph? How do you denote a graph?
(b) What is a node? What is an edge? What is an arc?
(c) What is a directed graph? What is an undirected graph? What is a bipartite graph?
(d) What is a path?
(e) What is a Hamiltonian path?
(f) What is a cycle?
(g) What is a tree?
(h) What is a spanning tree?
2.
(a) What is an algorithm?
(b) What is a heuristic?
3. Consider the Traveling Salesman problem (TSP):
(a) What is the input?
(b) How do we check if a cost matrix is symmetric or not? How about the triangular inequality?
(c) What is a feasible solution?
(d) What is an optimal solution?
(e) What is the mathematical formulation of TSP for a generic problem?
(f) Do you know how to solve a TSP?
(g) Do you use a directed or undirected graph in order to represent a symmetric TSP problem in
stance?
4. (Adapted from Winston, 531) A Hamiltonian path in a network is a closed path that passes exactly
once through each node in the network before returning to its starting point. Why is solving a TSP
equivalent to finding the shortest Hamiltonian path in a network? (HINT: Work on a 4city example.
Try adding a fifth dummy city and solving TSP on the new network. Which cities should this new city
connect to? What are the costs of these new connections?)
5. (Adapted from Winston, 531) There are 4 pins on a printed circuit. The distance between each pair
of pins (in inches) is given in Table 1. Suppose that we want to place three wires between the pins in
a way that connects all the wires and uses the minimum amount of wire. Also suppose that if more
than two wires touch a pin, a short circuit will incur. Set up a TSP (write down the mathematical
formulation explicitly!) that can be used to solve this problem. (HINT: Is this similar to one of the
previous exercises?)
6. Consider the Shortest Path (SP) problem:
(a) What is the input?
(b) What is a feasible solution?
1
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Table 1: Distance between each pair of pins, question 5
1
2
3
4
1
0
1
2
2
2
1
0
3
2.9
3
2
3
0
3
4
2
2.9
3
0
(c) What is an optimal solution?
(d) Do you know how to solve a SP problem? If so, name and describe the algorithm.
(e) What is the input and output of Dijkstra’s algorithm?
(f) We have assumed that the length of each arc is nonnegative for the SP problem, is this important?
Explain using a small example.
(g) Do you use a directed or undirected graph in order to represent a SP problem instance?
(h) If one of your friends claim that they have an optimal solution for a SP problem, can you verify
if he/she is correct or not? How?
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 '05
 TROTTER
 Operations Research, Flow network, Bipartite graph, Travelling salesman problem

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