Transcript
Page 1: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

Lecture 1: Symmetric Traveling

Salesman Problem

Ola Svensson

ADFOCS 2015

Page 2: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

WHO AM I AND WHAT WILL

I TALK ABOUT?

Who?

Exercises

APPROX

Page 3: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

About Me

• Ola Svensson

[email protected]

• http://theory.epfl.ch/osven

• Assistant Prof at EPFL

• Research on algorithms

• Happy to get feedback

and answer questions

Ola in action

Page 4: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

Outline

LECTURE 1: Traveling Salesman Problem

LECTURE 2: Traveling Salesman Problem

LECTURE 3: Traveling Salesman Problem

Page 5: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 6: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

The Irresistible Traveling Salesman Problem

What is the cheapest way

to visit these cities?

Page 7: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

The Irresistible Traveling Salesman Problem

What is the cheapest way

to visit these cities?

Page 8: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

The Irresistible Traveling Salesman Problem

What is the cheapest way

to visit these cities?

Traveling Salesman Problem

Page 9: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

The Irresistible Traveling Salesman Problem

Page 10: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

The Irresistible Traveling Salesman Problem

33 city contest that Proctor and Gamble ran in 1962

Page 11: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 12: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

HOW TO EVALUATE AN

ALGORITHM

Page 13: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 14: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 15: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

MOTIVATION OF TODAY’S

LECTURE

Page 16: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

Approximation algorithms for symmetric TSP

What is the best possible algorithm?

Page 17: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 18: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

1980’s

Page 19: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

1990’s

Arora & Mitchell independently:

PTAS for Euclidian TSP

Page 20: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

1990’s

Arora & Mitchell independently:

PTAS for Euclidian TSP

Arora et al. and Grigni et al.

PTAS for planar TSP

Page 21: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

2000’s

Papadimitriou & Vempala:

NP-hard to approximate metric TSP within 220/219

Simplified and slightly improved by Lampis’12

Page 22: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 23: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

TODAY’S Lecture

• The first approximation algorithm for TSP

• Christofides’ Algorithm

• Recent techniques that improve upon Christofides’ algorithm for

important special cases

Page 24: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

OUR FIRST APPROXIMATION

ALGORITHM

Page 25: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 26: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 27: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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!

Page 29: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

What is a Lower bound on OPT?

Page 30: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

What is a Lower bound on OPT?

The weight of a minimum spanning tree is at most OPT:

𝒘 𝑴𝑺𝑻 ≤ 𝑶𝑷𝑻

Page 31: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 53: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 62: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

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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 66: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 72: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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)

We have 𝒘 𝑴 ≤𝑶𝑷𝑻

𝟐

PROOF:

• Take an optimal tour of cost OPT

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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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 78: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

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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

Page 84: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

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Relating 2-VC and Tours

Input

Sample perfect matching M so that

each edge is take with prob. 1/3

Return E + M

Page 90: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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?

Page 91: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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?

Page 92: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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∗

Page 94: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 95: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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∗

Page 96: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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∗

Page 97: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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∗

Page 98: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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|

Page 99: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 105: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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 106: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 110: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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

Page 120: Lecture 1: Symmetric Traveling Salesman Problemresources.mpi-inf.mpg.de/conferences/adfocs-15/material/Ola-Lect1.pdfOutline LECTURE 1: Traveling Salesman Problem LECTURE 2: Traveling

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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