# DongChul Kim_ HwangRyol Ryu - Traveling Salesman Problem

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```					Traveling Salesman Problem

DongChul Kim
HwangRyol Ryu
Introduction

   Research Goal

   What you will learn …
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?
History of TSP

   Irish mathematician Sir William Rowan
Hamilton and the British mathematician
Thomas Penyngton Kirkman

Hamilton’s Iconsian game
History of TSP (1)

   The general form of TSP appeared in 1930s
by 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, Robert
Bixby, Vasek Chvatal, William Cook, and
Keld Helsgaun in May 2004.
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.
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.
2.0 TSP Approximation Algorithm (2)

   2.0? TSP
   2.0 is TSP version number?
   Tour of length is at most twice the length of
MST.

MST < Euler Cycle = 2 * MST <= 2.0 TSP
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.
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 <= 1.5 TSP
(MWM = ½ MST)
1.5 TSP Approximation Algorithm (3)
(Known as Christofides Heuristics)

   Minimum-weighted Matching example

MWM = ½ MST
Matching Algorithm
   Smile Matching Algorithm

   Better matching

Matching Algorithm (2)

   Improved Smile Matching Algorithm

1. Choose the two nodes in the farthest
distance
Matching Algorithm (3)

   Improved Smile Matching Algorithm

2. Each end node is connected to the node in
the closest distance.
PTAS Algorithm
(Polynomial Time Approximation Scheme)
   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.
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.
Other approximation schemes

   Minimum Steiner Tree
   K-TSP and K-MST
   Min Cost Perfect Matching
Demonstration

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