# Travelling Salesman Problem

Maybe Nemo

## Structure

1. Problem Statement
2. Problem Analysis
3. Basic Solutions
4. Applications
5. Heuristic & Other solutions

# What's a GRAPH?

## EdgeS

Ordered or Unordered pair of exactly 2 vertices.

## Vertices

Fundamental unit of a graph. Think of it as a point. Can be disconnected.

## GRAPH

Ordered Pair G =(V, E)
V = Set of vertices
E = Set of edges

## Directed

Edges are two way

Edge(a,b) != Edge(b,a)

## Undirected

Edges have no orientation.

Edge(a,b) =  Edge(b,a)

## Asymmetric

In the asymmetric TSP, paths may not exist in both directions or the distances might be different, forming a directed graph.

One-way streets, for eg.

## symmetric

The distance between two cities is the same in each opposite direction, forming an undirected graph.

This symmetry halves the number of possible solutions.

Most common type of TSP.

# Problem Statement

undirected weighted graph

city = graph's vertices

path = graph's edges

distance = edge's length

Minimize total distance after visiting all vertices.

Often, graph is complete as well

# Solve* TSP = GET \$1,000,000

## ComplexiTY

Year Vertices
1954 49 Cities (USA)
1971 64 Points
1975 80 Points
1977 120 Cities (Germany)
1987 318 Points
1987 532, 666, 1002, 2392
1992 3038 Points

# 24978 Cities

## 85,900 Points

Computer Chip Production

(2006)

# Applications

## Other Fields

• Data Mining
• Slewing
• X-ray Crystallography
• Engravings, Etchings
• Custom Chip Mfg.
• Cutting patterns
• Compression

# Ant Colony Optimization

1. Travel to all cities
2. Choose closer cities (visibility)
3. Probability of choosing ∝ pheromone
4. Deposit pheromone on path travelled
5. Iterate

## References

• By David Stanley from Nanaimo, Canada (Balloon Salesman  Uploaded by russavia) [CC-BY-2.0], via Wikimedia Commons
• TSP Cartoon, Courtesy Randal Munroe (xkcd.com)
• In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation, William J Cook

In Pursuit
of the
Travelling
Salesman

By Nemo

# Travelling Salesman Problem

A short talk on the Travelling Salesman Problem.

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