Top Banner
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.
68

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,

Jul 11, 2020

Download

Documents

dariahiddleston
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: 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,

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

Page 49: 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 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

Page 51: 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 (7)• Iteration 13

Page 52: 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,

An Weighted Matching Algorithm

Page 53: 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,

Rescanning Labels

Page 54: 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,

An Weighted Matching Algorithm

Page 55: 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,

An Weighted Matching Algorithm

Page 56: 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,

An Weighted Matching Algorithm

Page 57: 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,

An Weighted Matching Algorithm

Page 58: 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,

An Weighted Matching Algorithm

Page 59: 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,

An Weighted Matching Algorithm

Page 60: 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,

An Weighted Matching Algorithm

Page 61: 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,

An Weighted Matching Algorithm

Page 62: 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,

An Weighted Matching Algorithm

Page 63: 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,

References

Page 64: 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,

References

Page 65: 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,

References

Page 66: 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,

References

Page 67: 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,

References

Page 68: 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,

References


Related Documents
Postman Technopoly

Postman Technopoly

Category: Documents
A Cycle-Trade Heuristic for the Weighted k-Chinese Postman ...liu.diva-portal.org/smash/get/diva2:1275190/FULLTEXT01.pdf · Postman Problem Anton Hölscher Supervisor : Kaj Holmberg

A Cycle-Trade Heuristic for the Weighted k-Chinese Postman.....

Category: Documents
The Random Chinese Postman Tour is - arXiv.org e-Print … · 2016-08-03 · The Random Chinese Postman Tour is 4=3-approximate for network search Thomas Lidbetter Department of Mathematics,

The Random Chinese Postman Tour is - arXiv.org e-Print...

Category: Documents
PROBLEME DU POSTIER CHINOIS « CHINESE POSTMAN PROBLEM » ( CCP)

PROBLEME DU POSTIER CHINOIS « CHINESE POSTMAN...

Category: Documents
Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has

Fundamentals of Informatics Introduction to Algorithms and.....

Category: Documents
Spring 2015 Mathematics in Management Science Review for Exam 1 Chinese Postman Problem Traveling Salesman Problem Network Problem Critical Path Analysis.

Spring 2015 Mathematics in Management Science Review for...

Category: Documents
8. The Postman Problems The Chinese postman problem (CPP)cc.etsii.ull.es/.../postman_problems.pdf8. The Postman Problems The Chinese postman problem (CPP) A postal carrier must pick

8. The Postman Problems The Chinese postman problem...

Category: Documents
The Problem of Continuity in Chinese Businesses in ... · The Problem of Continuity in Chinese Businesses in Southeast Asia Kunio YOSHIHARA* Chinese businesses and Chinese business-I

The Problem of Continuity in Chinese Businesses in ... ·.....

Category: Documents
A tabu search algorithm for the min max k-chinese postman problem

A tabu search algorithm for the min max k-chinese postman...

Category: Software
Today: Reading: Exercisesadean/MC3021309/ScreenSlides/MC302_131031_S.pdfThursday, 10/31/13, Slide #2 The Icosian Game ... Chinese Postman Problem

Today: Reading:...

Category: Documents
CHAPTER 3 Chinese postman problem - Mark Greenaway Projects/Projects/Topology and... · 46 Chinese postman problem Graph 3 When the order of all the vertices is even, the graph is

CHAPTER 3 Chinese postman problem - Mark Greenaway...

Category: Documents
13: ' # '9& *#1 & 2 · 2018. 4. 13. · problem to search the shortest path or route, modelled as a famous Chinese Postman Problem or Traveling Salesman Problem (TSP ) . The computer

13: ' # '9& *#1 & 2 · 2018. 4. 13. · problem to search.....

Category: Documents