Travelling Salesman Problem

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TRAVELLING SALESMAN PROBLEM Abhay Rana 09118001

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STRUCTURE 1. Problem Statement 2. Problem Analysis 3. Basic Solutions 4. Applications 5. Heuristic & Other solutions 6. Links & References

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PROBLEM STATEMENT GIVEN A LIST OF CITIES AND THE DISTANCES BETWEEN EACH PAIR OF CITIES, WHAT IS THE SHORTEST POSSIBLE ROUTE THAT VISITS EACH CITY EXACTLY ONCE AND RETURNS TO THE ORIGIN CITY?

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MODEL IT AS A GRAPH

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WHAT'S A GRAPH? REPRESENTATION OF A SET OF OBJECTS WHERE SOME PAIRS OF OBJECTS ARE CONNECTED BY LINKS

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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

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DIRECTED Edges are two way Edge(a,b) != Edge(b,a) UNDIRECTED Edges have no orientation. Edge(a,b) = Edge(b,a)

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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.

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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

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SOLVE* TSP = GET $1,000,000 PROBLEM COMPLEXITY

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COMPLEXITY

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

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1998 - 13,509 CITIES IN USA

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SWEDEN (2004) 24978 CITIES

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85,900 POINTS Computer Chip Production (2006)

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APPLICATIONS

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NAVIGATION

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MAPPING GENOMES

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PRINTED CIRCUIT BOARDS

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

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SOLUTIONS

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BRUTE FORCE SEARCH

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NEAREST NEIGHBOUR

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PARTIAL TOUR METHOD

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ANT COLONY OPTIMIZATION

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

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REFERENCES By David Stanley from Nanaimo, Canada (Balloon Salesman Uploaded by russavia) [ ], 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 CC-BY-2.0