lecture 9: chinese postman problemlecture 9: chinese postman problem • problem: find a minimum...
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Lecture 9: Chinese Postman Problem• Problem: Find a minimum length closed path (from and
back to the post office), with repeated arcs as necessary, which contains every arc of a given undirected network.
Mei-Ko Kwan, “Graph Programming Using Odd andEven Points", Chinese Math, (1962), 273-277.
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Observation
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Basic Concept
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Main Theorem• Theorem 9.1 (Chapter 1)A graph G is Eulerian if and only if G is connected andeach node of G has an even degree.
Proof: Homework
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Questions• Q1: How to find the Euler path for an Eulerian graph?
• Q2: How to make a graph Eulerian?
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Main Results• Theorem: There are an even number of nodes with odd
degrees in a connected graph.
• Theorem:(Chinese Postman's Problem)Pairing the odd degree nodes with the shortest distancein between results in an Eulerian graph that providessolution to the Chinese Postman's Problem.
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Chinese Postman’s Algorithm
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Chinese Postman’s Algorithm
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Example
• Remaining Work:Nonbipartite (Max) Weighted Matching
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Nonbipartite Weighted Matching
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Example 1
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Dual Problem
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Example 1
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Example 2 - Homework
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Optimality Conditions
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Solution Strategy• (A) Keep primal feasibility (a matching) and dual
feasibility (a non-negative node/blossom potential).
• (B) Maintain valid (8.1) and (8.3).
• (C) Reduce the violation of (8.2).
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Initial Solution
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Observations• Observation 1: When
Null matching is optimal!
• Observation 2: In general, (8.2) is violated at some nodes with a positive
potential.
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General Step
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Possible Outcomes• Case 1: If an augmenting path is found, then we get a
new valid matching and reduce the violation of (8.2).
• The new matching takes care of the maximum matching in a blossom automatically for (8.3). Also because we only work on a sub-graph in which
(8.1) is maintained.
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Possible Outcomes• Case 2: If no augmentation can be found (Hungarian),
then we adjust node/blossom potentials such that
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Effects
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Example
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A Critical Issue
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Restrictions
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Restrictions
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Restrictions
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Selection Rule
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Results
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SUMMARY OF MAX WEIGHTED MATCHING ALGORITHM
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SUMMARY OF MAX WEIGHTED MATCHING ALGORITHM
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Implementation
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Implementation
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Implementation
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Implementation
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Implementation
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Implementation
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Example 1
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Iterations (1)• Iteration 1 Iteration 2
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Iterations (2)• Iteration 3 Iteration 4
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Iterations (3)• Iteration 5 Iteration 6
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Iterations (4)• Iteration 7 Iteration 8
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Iterations (5)• Iteration 9 Iteration 10
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Example 2
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Iterations (1)• Iteration 1 Iteration 2
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Iterations (2)• Iteration 3 Iteration 4
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Iterations (3)• Iteration 5 Iteration 6
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Iterations (4)• Iteration 7 Iteration 8
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Iterations (5)• Iteration 9 Iteration 10
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Iterations (6)• Iteration 11 Iteration 12
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Iterations (7)• Iteration 13
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An Weighted Matching Algorithm
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Rescanning Labels
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An Weighted Matching Algorithm
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An Weighted Matching Algorithm
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An Weighted Matching Algorithm
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An Weighted Matching Algorithm
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An Weighted Matching Algorithm
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An Weighted Matching Algorithm
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An Weighted Matching Algorithm
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An Weighted Matching Algorithm
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An Weighted Matching Algorithm
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References
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References
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References
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References
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References
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References