DongChul Kim, HwangRyol Ryu _ TSP Slides

8/6/2019 DongChul Kim, HwangRyol Ryu _ TSP Slides

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Traveling Salesman Problem

DongChul KimHwangRyol Ryu


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Introduction

Research Goal

What you will learn


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What Is TSP?

Shortest Hamiltonian cycle (i.e. tour).

Grow exponentially

Current Definition of TSP:Given a number of cities and the costs of

traveling from one to the other, what is the

cheapest round trip route that visits each city

and then returns to the starting city?


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History of TSP

Irish mathematician Sir William Rowan

Hamilton and the British mathematician

Thomas Penyngton Kirkman

Hamiltons Iconsian game


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History of TSP (1)

The general form of TSP appeared in 1930sby Karl Menger in Vienna and Havard.

A breakthrough by George Dantzig, Ray

Fulkerson, and Selmer Johnson in1954. 49 - 120 550 - 2,392 - 7,397 19,509 cities

From year 1954 to year 2001.

24,098 cities by David Applegate, RobertBixby, Vasek Chvatal, William Cook, andKeld Helsgaun in May 2004.


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Branch & Lower Bound

An algorithmic technique to find the optimal

solution by keeping the best solution found so

far.

Standard to measure performance of TSP

heuristics.


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2.0 TSP Approximation Algorithm

Double Minimum Spanning Tree

Return a tour of length at most twice the shortest

tour.

Algorithm:

1. Construct the minimal spanning tree

2. Duplicate all its edges. This gives us an Euler

cycle.

3. Traverse the cycle, but do not visit any node

more than once, taking shortcuts when it passes a

visited node.


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2.0 TSP Approximation Algorithm (2)

2.0? TSP

2.0 is TSP version number?

Tour of length is at most twice the length ofMST.

MST < Euler Cycle = 2 * MST


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1.5 TSP Approximation Algorithm(Known as Christofides Heuristics)

Professor Nicos Christofides extended the 2.0 TSP

and published that the worst-case ratio of the

extended algorithm was 3/2.

Algorithm:1. Compute MST graph T.

2. Compute a minimum-weighted matching graph M.

3. Combine T and M as edge set and Compute an

Euler Cycle.

4. Traverse each vertex taking shortcuts to avoid

visited nodes.


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1.5 TSP Approximation Algorithm (2)(Known as Christofides Heuristics)

What is a Minimum-weighted Matching?

It creates a MWM on a set of the nodes

having an odd degree.

Why odd degree?

Property of Euler Cycle

Why 1.5 TSP?

MST < Euler Cycle = MWM+MST


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1.5 TSP Approximation Algorithm (3)(Known as Christofides Heuristics)

Minimum-weighted Matching example

MWM = MST


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Matching Algorithm

Smile Matching Algorithm Bad matching

Better matching

Fixed Bad matching problem.


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Matching Algorithm (2)

Improved Smile Matching Algorithm

1. Choose the two nodes in the farthest

distance


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Matching Algorithm (3)

Improved Smile Matching Algorithm

2. Each end node is connected to the node in

the closest distance.


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PTAS Algorithm

(Polynomial Time Approximation Sch

eme) The status of Euclidean TSP remained open.

PTAS = Polynomial time algorithm, for each

c > 1, can approximate the problem within a

factor 1 + 1/c.


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PTAS Algorithm (2)

The central idea of the PTAS is that the plane

can be recursively partitioned and by using a

dynamic programming on Quadtree, it finds

an optimal tour.


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Other approximation schemes

Minimum Steiner Tree

K-TSP and K-MST

Min CostP

erfect Matching


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Demonstration

DongChul Kim, HwangRyol Ryu _ TSP Slides

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