Traveling Salesman Problem

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Traveling Salesman Problem

• DongChul Kim
• HwangRyol 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 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.

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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 of
  MST.
• MST lt Euler Cycle 2 MST lt 2.0 TSP

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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 lt Euler Cycle MWMMST lt 1.5 TSP
• (MWM ½ MST)

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

• Minimum-weighted Matching example

MWM ½ 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

• 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
Scheme)

• The status of Euclidean TSP remained open.
• PTAS Polynomial time algorithm, for each c gt 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 Cost Perfect Matching

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Demonstration