lecture 9: chinese postman problemlecture 9: chinese postman problem • problem: find a minimum...

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 and Even Points", Chinese Math, (1962), 273-277.

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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.

Page 2: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Observation

Page 3: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Basic Concept

Page 4: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

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

Page 5: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Questions• Q1: How to find the Euler path for an Eulerian graph?

• Q2: How to make a graph Eulerian?

Page 6: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

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.

Page 7: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Chinese Postman’s Algorithm

Page 8: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Chinese Postman’s Algorithm

Page 9: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Example

• Remaining Work:Nonbipartite (Max) Weighted Matching

Page 10: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Nonbipartite Weighted Matching

Page 11: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Example 1

Page 12: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Dual Problem

Page 13: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Example 1

Page 14: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Example 2 - Homework

Page 15: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Optimality Conditions

Page 16: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

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).

Page 17: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Initial Solution

Page 18: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Observations• Observation 1: When

Null matching is optimal!

• Observation 2: In general, (8.2) is violated at some nodes with a positive

potential.

Page 19: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

General Step

Page 20: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

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.

Page 21: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Possible Outcomes• Case 2: If no augmentation can be found (Hungarian),

then we adjust node/blossom potentials such that

Page 22: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Effects

Page 23: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Example

Page 24: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

A Critical Issue

Page 25: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Restrictions

Page 26: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Restrictions

Page 27: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Restrictions

Page 28: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Selection Rule

Page 29: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Results

Page 30: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

SUMMARY OF MAX WEIGHTED MATCHING ALGORITHM

Page 31: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

SUMMARY OF MAX WEIGHTED MATCHING ALGORITHM

Page 32: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Implementation

Page 33: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Implementation

Page 34: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Implementation

Page 35: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Implementation

Page 36: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Implementation

Page 37: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Implementation

Page 38: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Example 1

Page 39: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Iterations (1)• Iteration 1 Iteration 2

Page 40: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Iterations (2)• Iteration 3 Iteration 4

Page 41: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Iterations (3)• Iteration 5 Iteration 6

Page 42: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Iterations (4)• Iteration 7 Iteration 8

Page 43: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Iterations (5)• Iteration 9 Iteration 10

Page 44: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Example 2

Page 45: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Iterations (1)• Iteration 1 Iteration 2

Page 46: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Iterations (2)• Iteration 3 Iteration 4

Page 47: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Iterations (3)• Iteration 5 Iteration 6

Page 48: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Iterations (4)• Iteration 7 Iteration 8

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Iterations (5)• Iteration 9 Iteration 10

Page 50: Lecture 9: Chinese Postman ProblemLecture 9: Chinese Postman Problem • Problem: Find a minimum length closed path (from and back to the post office), with repeated arcs as necessary,

Iterations (6)• Iteration 11 Iteration 12