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log n Approximation Ratio for the Asymmetric Traveling Salesman Path Problem Chandra Chekuri Martin Pal Presented by Instructor Rahmtin Rotabi Prof ID: 526756

factor path proof atspp path factor atspp proof cycle augmentation lemma traveling salesman density dominates atsp problem metric cont

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Slide1
Slide2

An

O(log

n) Approximation Ratio for theAsymmetric Traveling Salesman PathProblemChandra Chekuri Martin

P´al

Presented by: Instructor:

Rahmtin Rotabi Prof. Zarrabi-Zadeh

2Slide3

Introduction

ATSPP:

Asymmetric Traveling Salesman Path ProblemGiven Info

Objective

Find optimum

-

path in

NP Hard

 

3Slide4

Past works

Metric-TSP

ChristofidesATSP

factor

Best known factor:

Metric-TSPP

best known factor

 

4Slide5

Past works (cont’d)

ATSPP- our problem

approximation

Proved by Lam and Newman

 

5Slide6

ATSP (Tour)

-factor for ATSPP

-factor for ATSP

Two algorithms for ATSP

Reducing vertices by cycle cover

Factor

Proof is straight forward

Min-Density Cycle Algorithm

Factor Proof is just like “set cover” 6Slide7

ATSPP- Our work

denotes the set of all

paths

denotes cycle not containing s and t

Density?

 

7Slide8

Density lemma

Assumption

:let

be the min-density path of non-trivial path in

Objective:

We can either find the min-density path

Or a cycle in

with a lower densityIdea of proof:Binary searchBellman-ford 8Slide9

Augmentation lemma

Definitions

:DominationExtensionSuccessorAssumptions:

Let

in

such that

dominates

Objective:

There is a path that dominates

, extends

 

9Slide10

Augmentation lemma proof

Define

Mark some members of

with an algorithm

Name them

Obtain

P

3 from

P1Replace

of

by the sub-path

 

10Slide11

Augmentation lemma proof(cont’d)

The path extends

The path dominates

Straight-forward with following in-equalities

(I1) For

we have

(I2) For

we have

(I3) For

we have

Corollary:

Replace

with

 

11Slide12

Algorithm

Start with only one edge

Use proxies Until we have a spanning pathUse path or cycle augmentationIt will finish after at most

iterations

Implemented naively:

 

12Slide13

Claims and proof

In every iteration, if

is the augmenting path or cycle in that iteration,

Use augmentation path lemma

Algorithm factor is

.

Step is from k1 to k2 vertices

Path step

Cycle step

 

13Slide14

Path-constrained ATSPP

Start from

Instead of

Same analysis

Best integrality gap for ATSPP is 2

Best integrality gap for ATSP is

LP:

 

14Slide15

Any Question?

15Slide16

References

An

O(logn) Approximation Ratio for theAsymmetric Traveling Salesman Path Problem, THEORY OF COMPUTING, Volume 3 (2007), pp. 197–209.

Traveling salesman path problem, Mathematical Journal, Volume 113, Issue 1,

pp

39-5916Slide17

Thank you for your time

17

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