lecture 1: symmetric traveling salesman problem...lecture 3: traveling salesman problem symmetric...
TRANSCRIPT
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Lecture 1: Symmetric Traveling
Salesman Problem
Ola Svensson
ADFOCS 2015
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WHO AM I AND WHAT WILL
I TALK ABOUT?
Who?
Exercises
APPROX
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About Me
• Ola Svensson
• http://theory.epfl.ch/osven
• Assistant Prof at EPFL
• Research on algorithms
• Happy to get feedback
and answer questions
Ola in action
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Outline
LECTURE 1: Traveling Salesman Problem
LECTURE 2: Traveling Salesman Problem
LECTURE 3: Traveling Salesman Problem
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Outline
LECTURE 1: Traveling Salesman Problem
LECTURE 2: Traveling Salesman Problem
LECTURE 3: Traveling Salesman Problem
Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems
Asymmetric TSP, Cycle Cover Algorithm, Thin trees
Continuation of asymmetric TSP, Local-Connectivity Algorithm, Open Problems
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The Irresistible Traveling Salesman Problem
What is the cheapest way
to visit these cities?
![Page 7: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/7.jpg)
The Irresistible Traveling Salesman Problem
What is the cheapest way
to visit these cities?
![Page 8: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/8.jpg)
The Irresistible Traveling Salesman Problem
What is the cheapest way
to visit these cities?
Traveling Salesman Problem
![Page 9: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/9.jpg)
The Irresistible Traveling Salesman Problem
![Page 10: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/10.jpg)
The Irresistible Traveling Salesman Problem
33 city contest that Proctor and Gamble ran in 1962
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Rich History
• Variants studied in mathematics by Hamilton and Kirkman already in the
1800’s
• Benchmark problem in computer science from the “beginning”
• Today, probably the most studied NP-hard optimization problem
• Intractable: (current) exact algorithms require exponential time
Major open problem what efficient computation can accomplish
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HOW TO EVALUATE AN
ALGORITHM
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Solving intractable problems
• Heuristics
• good for “typical” instances
• bad instances do not happen too often
1
4
16
64
256
1024
4096
16384
50's 70's 80's 90's 00's
Dantzig, Fulkerson, and Johnson solve a 49-
city instance to optimality
Applegate, Bixby, Chvatal, Cook,
and Helsgaun solve a
24978-city instance
!Sweden has only 9
million inhabitants
≈ 360 persons/city
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Solving intractable problems
• Approximation Algorithms
• Perhaps we can efficiently find a reasonably good solution in polytime?
Approximation Ratio:
worst case over all instances
• α=1 is an exact polynomial time algorithm
• α=1.01 then algorithm finds a solution with at most 1% higher cost
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MOTIVATION OF TODAY’S
LECTURE
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Approximation algorithms for symmetric TSP
What is the best possible algorithm?
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1970’s
Christofides: 1.5-approximation algorithm for metric
distances
Held & Karp: Heuristic for calculating lower bound on a
tour
Coincides with the value of a linear
program known as Held-Karp or Subtour
Elimination Relaxation
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1980’s
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1990’s
Arora & Mitchell independently:
PTAS for Euclidian TSP
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1990’s
Arora & Mitchell independently:
PTAS for Euclidian TSP
Arora et al. and Grigni et al.
PTAS for planar TSP
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2000’s
Papadimitriou & Vempala:
NP-hard to approximate metric TSP within 220/219
Simplified and slightly improved by Lampis’12
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Today
Major open problem to understand
the approximability of TSP
• NP-hard to approximate metric TSP within 220/219
• Christofides’ 1.5-approximation algorithm still best
• Held-Karp relaxation conjectured to give 4/3-approximation
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TODAY’S Lecture
• The first approximation algorithm for TSP
• Christofides’ Algorithm
• Recent techniques that improve upon Christofides’ algorithm for
important special cases
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OUR FIRST APPROXIMATION
ALGORITHM
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The Traveling Salesman Problem
INPUT: 𝑛 cities with pairwise distances that satisfy the triangle inequality
OUTPUT: a tour of minimum total distance that visits each city once
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The Traveling Salesman Problem
INPUT: 𝑛 cities with pairwise distances that satisfy the triangle inequality
OUTPUT: a tour of minimum total distance that visits each city once
𝑥 ≤ 𝑦 + 𝑧
𝑦𝑧
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The Traveling Salesman Problem
INPUT: 𝑛 cities with pairwise distances that satisfy the triangle inequality
OUTPUT: a tour of minimum total distance that visits each city once
![Page 28: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/28.jpg)
How to analyze an approximation algorithm?
• Recall that we measure its performance by
Approximation Ratio:
worst case over all instances
• But calculating the cost of the optimal solution is NP-hard…
SOLUTION: Compare with a lower bound on OPT!
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What is a Lower bound on OPT?
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What is a Lower bound on OPT?
The weight of a minimum spanning tree is at most OPT:
𝒘 𝑴𝑺𝑻 ≤ 𝑶𝑷𝑻
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What is a Lower bound on OPT?
PROOF:
The weight of a minimum spanning tree is at most OPT:
𝒘 𝑴𝑺𝑻 ≤ 𝑶𝑷𝑻
![Page 32: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/32.jpg)
What is a Lower bound on OPT?
PROOF:
The weight of a minimum spanning tree is at most OPT:
𝒘 𝑴𝑺𝑻 ≤ 𝑶𝑷𝑻
• Take an optimal tour of cost OPT
• Drop an edge to obtain a tree T
• Distances/weights are non-negative
so 𝑤 𝑇 ≤ 𝑂𝑃𝑇
• Hence, 𝑤 𝑀𝑆𝑇 ≤ 𝑂𝑃𝑇
![Page 33: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/33.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
G
E
H
C
F G
![Page 34: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/34.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
G
E
H
C
F G
![Page 35: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/35.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
![Page 36: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/36.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
![Page 37: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/37.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
Eulerian tour: a walk that traverses each edge exactly once
![Page 38: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/38.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A
Eulerian tour: a walk that traverses each edge exactly once
![Page 39: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/39.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B
Eulerian tour: a walk that traverses each edge exactly once
![Page 40: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/40.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D
Eulerian tour: a walk that traverses each edge exactly once
![Page 41: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/41.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H
Eulerian tour: a walk that traverses each edge exactly once
![Page 42: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/42.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D
Eulerian tour: a walk that traverses each edge exactly once
![Page 43: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/43.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I
Eulerian tour: a walk that traverses each edge exactly once
![Page 44: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/44.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D
Eulerian tour: a walk that traverses each edge exactly once
![Page 45: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/45.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B
Eulerian tour: a walk that traverses each edge exactly once
![Page 46: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/46.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E
Eulerian tour: a walk that traverses each edge exactly once
![Page 47: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/47.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B
Eulerian tour: a walk that traverses each edge exactly once
![Page 48: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/48.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F
Eulerian tour: a walk that traverses each edge exactly once
![Page 49: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/49.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B
Eulerian tour: a walk that traverses each edge exactly once
![Page 50: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/50.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A
Eulerian tour: a walk that traverses each edge exactly once
![Page 51: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/51.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C
Eulerian tour: a walk that traverses each edge exactly once
![Page 52: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/52.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C,G
Eulerian tour: a walk that traverses each edge exactly once
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Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C,G,C
Eulerian tour: a walk that traverses each edge exactly once
![Page 54: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/54.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C,G,C,A
Eulerian tour: a walk that traverses each edge exactly once
![Page 55: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/55.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C,G,C,A
Eulerian tour: a walk that traverses each edge exactly once
Short cut to visit each vertex exactly once. By triangle inequality this doesn’t increase cost
![Page 56: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/56.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C,G,C,A
Eulerian tour: a walk that traverses each edge exactly once
Short cut to visit each vertex exactly once. By triangle inequality this doesn’t increase cost
![Page 57: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/57.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C,G,C,A
Eulerian tour: a walk that traverses each edge exactly once
Short cut to visit each vertex exactly once. By triangle inequality this doesn’t increase cost
![Page 58: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/58.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C,G,C,A
Eulerian tour: a walk that traverses each edge exactly once
Short cut to visit each vertex exactly once. By triangle inequality this doesn’t increase cost
![Page 59: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/59.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C,G,C,A
Eulerian tour: a walk that traverses each edge exactly once
Short cut to visit each vertex exactly once. By triangle inequality this doesn’t increase cost
![Page 60: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/60.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C,G,C,A
Eulerian tour: a walk that traverses each edge exactly once
Short cut to visit each vertex exactly once. By triangle inequality this doesn’t increase cost
![Page 61: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/61.jpg)
Double Tree AlgorithmFind minimum spanning tree T
Duplicate T
Return Eulerian tour plus shortcutting
A
B
D
H
E
I
C
F G
A,B,D,H,D,I,D,B,E,B,F,B,A,C,G,C,A
Eulerian tour: a walk that traverses each edge exactly once
Short cut to visit each vertex exactly once. By triangle inequality this doesn’t increase cost
Cost of tour is at most
2 ⋅ 𝑤 𝑇 ≤ 2 ⋅ 𝑂𝑃𝑇
Hence, Double Tree Algorithm is a
2-approximation algorithm for TSP
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CHRISTOFIDES ALGORITHM
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The recipe
• Hence, to solve TSP it is sufficient to find a cheap connected subgraph so
that each vertex has even degree
• Double spanning tree algorithm does this by simply duplicating a spanning
tree
• Christofides algorithm will also ensure connectivity by taking a MST but
then be more clever in the correction of the parity of vertices
A graph has an Eulerian walk iff each vertex has even degree
Euler’1736
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Christofides AlgorithmFind minimum spanning tree T
Find …
Return T + M (with shortcutting)
A
B
D
G
E
H
C
F G
![Page 65: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/65.jpg)
Christofides AlgorithmFind minimum spanning tree T
Find …
Return T + M (with shortcutting)
A
B
D
G
E
H
C
F G
![Page 66: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/66.jpg)
Christofides AlgorithmFind minimum spanning tree T
Find …
Return T + M (with shortcutting)
A
B
D
G
E
H
C
F G
Hint: A graph has always an even
number of vertices (exercise)
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Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
A
B
D
G
E
H
C
F G
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Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
A
B
D
G
E
H
C
F G
![Page 69: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/69.jpg)
Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
A
B
D
G
E
H
C
F G
![Page 70: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/70.jpg)
Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
A
B
D
G
E
H
C
F G
![Page 71: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/71.jpg)
Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
A
B
D
G
E
H
C
F G
Cost of tour is at most
𝑤 𝑇 + 𝑤 𝑀 ≤ 𝑤 𝑂𝑃𝑇 + 𝑤(𝑀)
How can we bound 𝒘 𝑴 ?
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Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
A
B
D
G
E
H
C
F G
Cost of tour is at most
𝑤 𝑇 + 𝑤 𝑀 ≤ 𝑤 𝑂𝑃𝑇 + 𝑤(𝑀)
How can we bound 𝒘 𝑴 ?
![Page 73: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/73.jpg)
Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
We have 𝒘 𝑴 ≤𝑶𝑷𝑻
𝟐
PROOF:
• Take an optimal tour of cost OPT
![Page 74: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/74.jpg)
Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
We have 𝒘 𝑴 ≤𝑶𝑷𝑻
𝟐
PROOF:
• Take an optimal tour of cost OPT
• Consider vertices that are odd in T
![Page 75: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/75.jpg)
Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
We have 𝒘 𝑴 ≤𝑶𝑷𝑻
𝟐
PROOF:
• Take an optimal tour of cost OPT
• Consider vertices that are odd in T
• Shortcut to obtain tour on odd-degree
vertices of no larger cost
![Page 76: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/76.jpg)
Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
We have 𝒘 𝑴 ≤𝑶𝑷𝑻
𝟐
PROOF:
• Take an optimal tour of cost OPT
• Consider vertices that are odd in T
• Shortcut to obtain tour on odd-degree
vertices of no larger cost
• These edges partition into two
matchings 𝑀1and 𝑀2 such that
𝑤 𝑀1 + 𝑤 𝑀2 ≤ 𝑂𝑃𝑇
![Page 77: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/77.jpg)
Christofides AlgorithmFind minimum spanning tree T
Find min matching M of odd degree vertices
Return T + M (with shortcutting)
We have 𝒘 𝑴 ≤𝑶𝑷𝑻
𝟐
PROOF:
• Take an optimal tour of cost OPT
• Consider vertices that are odd in T
• Shortcut to obtain tour on odd-degree
vertices of no larger cost
• These edges partition into two
matchings 𝑀1and 𝑀2 such that
𝑤 𝑀1 + 𝑤 𝑀2 ≤ 𝑂𝑃𝑇
• Hence the minimum has weight ≤𝑂𝑃𝑇
2
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Summary sofar
• Saw our first approximation algorithm
• We were a little bit more clever to obtain Christofides algorithm
• This is the best known in spite of a lot of research!
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Today
Major open problem to understand
the approximability of TSP
• NP-hard to approximate metric TSP within 220/219
• Christofides’ 1.5-approximation algorithm still best
• Held-Karp relaxation conjectured to give 4/3-approximation
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4
Cost = #edges
Graph-TSP
Given an unweighted undirected graph G=(V,E)
Find shortest tour where
d(u,v) = shortest path between u and v Find spanning Eulerian multigraph with minimum #edges
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4/3
1.5
2000
Major open problem to understand
the approximability of TSP
• NP-hard to approximate metric TSP within 𝟐𝟐𝟎
𝟐𝟏𝟗≈ 𝟏. 𝟎𝟎𝟓
• Christofides’ 1.5-approximation algorithm still best
• Held-Karp relaxation conjectured to give 4/3-
approximation
2005 2010
Gamarnik, Lewenstein & Sviridenko’05:
1.487-approximation for cubic 3-edge connected graphs
4/3 – approximation for cubic graphs
7/5 – approximation for subcubic graphs
Boyd, Sitters, van der Star & Stougie’10:
Oveis Gharan, Saberi & Singh:
𝟏. 𝟓 − 𝝐 -approximation algorithm for graph-TSP
1.5-ε
1,461
Progress on TSP vs mobiles
![Page 82: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/82.jpg)
4/3
1.5 1.5-ε
1,461
2000 2005 2010
Mömke & S.:
1.461-approximation algorithm for graph-TSP
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4/3
1.5 1.5-ε
1,461
2000 2005 2010
Mucha:
1.444-approximation algorithm for graph-TSP
1.444
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4/3
1.5 1.5-ε
1,461
2000 2005 2010
Sebö & Vygen:
1.4-approximation algorithm for graph-TSP
1.444
1.4
![Page 85: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/85.jpg)
4/3
1.5 1.5-ε
1,461
2000 2005 2010
Sebö & Vygen:
1.4-approximation algorithm for graph-TSP
1.444
1.4
?
![Page 86: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/86.jpg)
Oveis Gharan, Saberi, Singh
Sampling spanning trees
Mömke & S.
Removable pairings
Sebö & Vygen
Ear decompositions
Further results and future directions
Christofides Algorithm’76Frederickson & Ja’Ja’82
Monma, Munson & Pulleyblank’90
![Page 87: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/87.jpg)
Approximating TSP by removable pairings
• Different more “graph theoretic approach”
• Promising applications (apart from 1.4 approximation for graph-TSP)
• Among other things settled a conjecture by Boyd et al.
Subcubic 2-edge connected graphs have a tour of length 4n/3-2/3
We will illustrate the techniques by proving above statement for cubic graphs
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Relating 2-VC and Tours
A 2-VC (cubic) graph G=(V,E) has a tour of length at most 4/3|E|
Frederickson & Ja’Ja’82 and Monma, Munson & Pulleyblank’90
![Page 89: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/89.jpg)
Relating 2-VC and Tours
Input
Sample perfect matching M so that
each edge is take with prob. 1/3
Return E + M
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Relating 2-VC and ToursSample perfect matching M so that
each edge is take with prob. 1/3
Return E + M
Input
Berge-Fulkerson Conjecture:
Any cubic 2-edge connected graph has 6 matchings so that each
edge appears in exactly two of them?
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Relating 2-VC and ToursSample perfect matching M so that
each edge is take with prob. 1/3
Return E + M
Input
How to sample M in general?
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Relating 2-VC and ToursSample perfect matching M so that
each edge is take with prob. 1/3
Return E + M
Input
How to sample M in general?
Edmond’s perfect matching polytope
𝑒∈𝛿(𝑣) 𝑥𝑒 = 1 ∀𝑣 ∈ 𝑉
𝑒∈𝛿(𝑆) 𝑥𝑒 ≥ 1 ∀𝑆 ⊆ 𝑉, 𝑆 odd
𝑥 ≥ 0
![Page 93: Lecture 1: Symmetric Traveling Salesman Problem...LECTURE 3: Traveling Salesman Problem Symmetric TSP, Christofides’ Algorithm, Removable Edges, Open Problems Asymmetric TSP, Cycle](https://reader034.vdocuments.us/reader034/viewer/2022042116/5e937af12cf19022df741418/html5/thumbnails/93.jpg)
Relating 2-VC and ToursSample perfect matching M so that
each edge is take with prob. 1/3
Return E + M
Input
How to sample M in general?
Edmond’s perfect matching polytope
𝑒∈𝛿(𝑣) 𝑥𝑒 = 1 ∀𝑣 ∈ 𝑉
𝑒∈𝛿(𝑆) 𝑥𝑒 ≥ 1 ∀𝑆 ⊆ 𝑉, 𝑆 odd
𝑥 ≥ 0Every extreme point perfect matching
𝑥∗
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Relating 2-VC and ToursSample perfect matching M so that
each edge is take with prob. 1/3
Return E + M
Input
How to sample M in general?
Edmond’s perfect matching polytope
𝑒∈𝛿(𝑣) 𝑥𝑒 = 1 ∀𝑣 ∈ 𝑉
𝑒∈𝛿(𝑆) 𝑥𝑒 ≥ 1 ∀𝑆 ⊆ 𝑉, 𝑆 odd
𝑥 ≥ 0Every extreme point perfect matching
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Relating 2-VC and ToursSample perfect matching M so that
each edge is take with prob. 1/3
Return E + M
Input
How to sample M in general?
Edmond’s perfect matching polytope
𝑒∈𝛿(𝑣) 𝑥𝑒 = 1 ∀𝑣 ∈ 𝑉
𝑒∈𝛿(𝑆) 𝑥𝑒 ≥ 1 ∀𝑆 ⊆ 𝑉, 𝑆 odd
𝑥 ≥ 0Every extreme point perfect matching
• For 2-connected cubic graphs 𝑥𝑒∗ = 1/3 for is a feasible solution (exercise)
𝑥∗
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Relating 2-VC and ToursSample perfect matching M so that
each edge is take with prob. 1/3
Return E + M
Input
How to sample M in general?
Edmond’s perfect matching polytope
𝑒∈𝛿(𝑣) 𝑥𝑒 = 1 ∀𝑣 ∈ 𝑉
𝑒∈𝛿(𝑆) 𝑥𝑒 ≥ 1 ∀𝑆 ⊆ 𝑉, 𝑆 odd
𝑥 ≥ 0Every extreme point perfect matching
• For 2-connected cubic graphs 𝑥𝑒∗ = 1/3 for is a feasible solution (exercise)
• By Edmond, 𝑥∗ = 𝜆𝑖𝑀(𝑖) can be written as a convex combination of perfect matchings
𝑥∗
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Relating 2-VC and ToursSample perfect matching M so that
each edge is take with prob. 1/3
Return E + M
Input
How to sample M in general?
Edmond’s perfect matching polytope
𝑒∈𝛿(𝑣) 𝑥𝑒 = 1 ∀𝑣 ∈ 𝑉
𝑒∈𝛿(𝑆) 𝑥𝑒 ≥ 1 ∀𝑆 ⊆ 𝑉, 𝑆 odd
𝑥 ≥ 0Every extreme point perfect matching
• For 2-connected cubic graphs 𝑥𝑒∗ = 1/3 for is a feasible solution (exercise)
• By Edmond, 𝑥∗ = 𝜆𝑖𝑀(𝑖) can be written as a convex combination of perfect matchings
• Sampling 𝑀(𝑖) with probability 𝜆𝑖 gives a distribution of perfect matchings where each
edge is taken with probability 1/3
𝑥∗
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Relating 2-VC and ToursSample perfect matching M so that
each edge is take with prob. 1/3
Return E + M
Input
2n = 4/3|E| edges
Output: E+M
2n = 4/3|E| edges
Output: E+M
2n = 4/3|E| edges
Output: E+M
=> A 2-VC (cubic) graph G=(V,E) has a tour of length at most 4/3|E|
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Instead of just adding edges from matching
remove some of them
Output: E+M
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Instead of just adding edges from matching
remove some of them
Output: E+M
Need to keep connectivity!
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Removing EdgesTake a DFS spanning tree T
Same algorithm as before but
return
Input
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Removing EdgesTake a DFS spanning tree T
Same algorithm as before but
return
Input
5/3n = 10/9|E| edges
Output
5/3n = 10/9|E| edges
Output
4/3n = 8/10|E| edges
Output
=> A 2-VC (cubic) graph G=(V,E) has a tour of length at most 2/3(n-1) + 2/3|E|
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Increasing number of removable edges
• Use structure of perfect matching to increase the set of removable edges
• Define a “removable pairing”
• Pair of edges: only one edge in each pair can occur in a matching
• Graph obtained by removing at most one edge in each pair is connected
𝑅 contains all back edges
and paired tree edges
2𝑏 − 1 removable edges
where 𝑏 is number of
back edges
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Removable PairingsTake a DFS spanning tree T
Same algorithm as before but
return
Input
4/3n = 8/10|E| edges
Output
4/3n = 8/10|E| edges
Output
n = 2/3|E| edges
Output
=> A 2-VC subcubic graph G=(V,E) has a tour of length at most 4/3n-2/3
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Oveis Gharan, Saberi, Singh
Sampling spanning trees
Mömke & S.
Removable pairings
Sebö & Vygen
Ear decompositions
Further results and future directions
Christofides Algorithm’76Frederickson & Ja’Ja’82
Monma, Munson & Pulleyblank’90
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Instead of minimum spanning tree
sample one from the Held-Karp relaxationInspired by work on asymmetric traveling salesman problem (Asadpour et al’10)
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Sample Spanning TreeSolve Held-Karp relaxation
Sample spanning tree T from solution
Run Christofides starting from T
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Sample Spanning TreeSolve Held-Karp relaxation
Sample spanning tree T from solution
Run Christofides starting from T
Held-Karp Relaxation
A tour (after shortcutting) has |V| edges
No subtours
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Sample Spanning TreeSolve Held-Karp relaxation
Sample spanning tree T from solution
Run Christofides starting from T
Held-Karp Relaxation Spanning-Tree Polytope
=> Sample spanning tree whose expected cost equals value of Held-Karp
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Sample Spanning TreeSolve Held-Karp relaxation
Sample spanning tree T from solution
Run Christofides starting from T
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Sample Spanning TreeSolve Held-Karp relaxation
Sample spanning tree T from solution
Run Christofides starting from T
Cost of spanning tree
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Sample Spanning TreeSolve Held-Karp relaxation
Sample spanning tree T from solution
Run Christofides starting from T
Cost of spanning tree
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Sample Spanning TreeSolve Held-Karp relaxation
Sample spanning tree T from solution
Run Christofides starting from T
Cost of spanning tree
Cost of matching is at most
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Sample Spanning Tree
• Involved proof by Oveis Gharan et al
• maximum entropy distribution of spanning trees
• Structure of near-min cuts etc.
Solve Held-Karp relaxation
Sample spanning tree T from solution
Run Christofides starting from T
Cost of spanning tree
Cost of matching is at most
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Summary sofar
• Two very different approaches for improved algorithms for graph-TSP
• Many different concepts from graph theory
• Structure of perfect matchings, Ear decompositions … hopefully more
1,331,351,371,391,411,431,451,471,49
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Future results and future directions
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GENERAL METRICS
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No progress on TSP yet but …
A proof of the Boyd-Carr conjecture (Schalekamp, Williamson, van Zuylen’12)
• Tight analysis of cost of 2-matching vs Held-Karp relaxation
Improved algorithms for st-path TSP (An, Kleinberg, Shmoys’12, Sebö’13)
• Tight analysis of cost of 2-matching vs Held-Karp relaxation
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2-Matching Extreme Points
• A graph consisting of disjoint odd cycles connected a matching of paths
• Arbitrary distances on edges
• Held-Karp value = half the weight of cycles + total weight of paths
Can you always find a tour of cost at most 4/3 of Held-Karp?
2 2
3
2 3
1
1
1
2
2
2
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2-Matching Extreme Points
• A graph consisting of disjoint odd cycles connected a matching of paths
• Arbitrary distances on edges
• Held-Karp value = half the weight of cycles + total weight of paths
Can you always find a tour of cost at most 4/3 of Held-Karp?
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Node-Weighted Symmetric TSP
• distance of 𝑢, 𝑣 ∈ 𝐸 is w 𝑢 + 𝑤 𝑣
1
3
13
23
5
5
4
4
4
Can you do better than Christofides (1.5)?
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Summary
• Classic algorithms for TSP
• Our first approximation algorithm + Christofides
• Two new approaches
• Nice technique: Interpreting a fractional solution in an integral polytope as
a distribution
• Great open questions!
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