Travelling Salesman Problem

A small talk on TSP. Made using Slides.com. Also available at http://slides.com/captn3m0/travelling-salesman/

Transcript

1. TRAVELLING SALESMAN PROBLEM Abhay Rana 09118001
2. STRUCTURE 1. Problem Statement 2. Problem Analysis 3. Basic Solutions 4. Applications 5. Heuristic & Other solutions 6. Links & References
3. 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?
4. MODEL IT AS A GRAPH
5. WHAT'S A GRAPH? REPRESENTATION OF A SET OF OBJECTS WHERE SOME PAIRS OF OBJECTS ARE CONNECTED BY LINKS
6. 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
7. DIRECTED Edges are two way Edge(a,b) != Edge(b,a) UNDIRECTED Edges have no orientation. Edge(a,b) = Edge(b,a)
8. 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.
9. 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
10. SOLVE* TSP = GET $1,000,000 PROBLEM COMPLEXITY
11. COMPLEXITY
12. 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
13. 1998 - 13,509 CITIES IN USA
14. SWEDEN (2004) 24978 CITIES
15. 85,900 POINTS Computer Chip Production (2006)
16. APPLICATIONS
17. NAVIGATION
18. MAPPING GENOMES
19. PRINTED CIRCUIT BOARDS
20. OTHER FIELDS Data Mining Slewing X-ray Crystallography Engravings, Etchings Custom Chip Mfg. Cutting patterns Compression
21. SOLUTIONS
22. BRUTE FORCE SEARCH
23. NEAREST NEIGHBOUR
24. PARTIAL TOUR METHOD
25. ANT COLONY OPTIMIZATION
26. 1. Travel to all cities 2. Choose closer cities (visibility) 3. Probability of choosing ∝ pheromone 4. Deposit pheromone on path travelled 5. Iterate
27. 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
28. WILLIAM J COOK In Pursuit of the Travelling Salesman
29. THANK YOU TRAVELLING SALESMAN MOVIE .COM