ieor266 © 2008 1 classification of mcnf problems
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IEOR266 © 2008
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Classification of MCNF problems
IEOR266 © 2008
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The Bridges of Koenigsberg: Euler 1736
“Graph Theory” began in 1736 Leonard Euler– Visited Koenigsberg– People wondered whether it is possible to take
a walk, end up where you started from, and cross each bridge in Koenigsberg exactly once
– Generally it was believed to be impossible
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The Bridges of Koenigsberg: Euler 1736
A
D
C
B
1 2
4
3
7
65
Is it possible to start in A, cross over each bridge exactly once, and end up back in A?
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The Bridges of Koenigsberg: Euler 1736
A
D
C
B
1 2
4
3
7
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Conceptualization: Land masses are nodes
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The Bridges of Koenigsberg: Euler 1736
1 2
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3
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Conceptualization: Bridges are edges
A
C
D
B
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The Bridges of Koenigsberg: Euler 1736
1 2
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3
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Translation to graphs or networks: Is there a walk starting at A and ending at A and passing through each edge exactly once? Why isn’t there such a walk?
A
C
D
B
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Adding two bridges creates such a walk
A, 1, B, 5, D, 6, B, 4, C, 8, A, 3, C, 7, D, 9, B, 2, A
1 2
4
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A
C
D
B
8
9
Here is the walk.
Note: the number of edges incident to B is twice the number of times that B appears on the walk.
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Eulerian cycle: a closed walk that passes through each edge exactly once
Degree of a node = number of edges incident to the node
Necessary condition: each node has an even degree.
Why necessary? The degree of a node j is twice the number of times j appears on the walk (except for the initial and final node of the walk.)
This condition is also nearly sufficient. If every node has even degree, and if the graph is connected, then there is an eulerian cycle.
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Eulerian cycles
Eulerian cycles and extensions are used in practice
Mail Carrier routes:– visit each city block at least once– minimize travel time– other constraints in practice?
Trash pickup routes– visit each city block at least once– minimize travel time– other constraints in practice?
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Eulerian cycle: a closed walk that passes through each arc exactly once
Degree of a node = number of edges incident to the node
Necessary condition: each node has an even degree.
Why necessary? The degree of a node j is twice the number of times j appears on the walk (except for the initial and final node of the walk.)
Theorem. A graph has an eulerian cycle if and only if the graph is connected and every node has even degree.
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Eulerian path: a walk that is not closed and passes through each edge exactly once
Theorem. A graph has an eulerian path if and only if exactly two nodes have odd degree and the graph is connected.
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Exercise: Does the graph below have an Eulerian path or cycle? If so, find it.
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Eulerian cycles
Eulerian cycles and extensions are used in practice Mail Carrier routes:
– visit each city block at least once– minimize travel time– other constraints in practice?
Trash pickup routes– visit each city block at least once– minimize travel time– other constraints in practice?
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The Traveling Salesman Problem : find a tour that visit all cities and minimizes the total distance traveled.
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More on cycles
A hamiltonian cycle is a cycle that passes through each node of the graph exactly once.
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Hamilton’s Around the World Game
In 1857, the Irish mathematician, Sir William Rowan Hamilton invented a puzzle that he hoped would be very popular.
The objective was to make what we just called a hamiltonian cycle.
The game was not a commercial success, especially the 3D version.
But the mathematics of hamiltonian cycles is very popular today.
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Hamilton’s Around the World Game
We will see this problem again when we generalize it to be the traveling salesman problem.
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The knight’s tour problem
Can a knight visit all squares of a chessboard exactly once, starting at some square, and by making 63 legitimate moves?
The knight’s tour problem is a special case of the hamiltonian tour problem.
The answer is yes!