Hamiltonian Circuit & The Traveling Salesman Problem Drew Nash 26 March 2014

Definitions  You are a traveling salesperson…  On assignment… death-of-a-salesman-interactive.html?_r=0

Definitions  Traveling Salesman Problem – If given a list of cities and potential paths between them, with each path having a certain “cost” (Applegate) or distance, which route that starts at an origin city, visits every other city exactly once, and terminates at the origin city, would yield the lowest cost or shortest distance?

Definitions

 Hamiltonian Path – A path containing all vertices in a graph G  Hamiltonian Circuit – A circuit containing all vertices in a graph G  Hamiltonian Graph – A graph containing a Hamiltonian circuit

Definitions  Back to Traveling Salesman Problem.. – An answer to the TSP is referred to as a tour or circuit – If the TSP is graphed, the resulting tour will be a Hamiltonian circuit of that graph – In other words, find the Hamiltonian circuit in a weighted graph with the least weight

History  One of the most well-known researchers of this problem was Merrill Flood of Princeton University.  He was working on school bus routing in the 1930s.  The term Traveling Salesman Problem became popular in the mid-1950s. history/faculty/merrill-m-flood

History  It is thought that the first written reference to the TSP term was in Julia Robinson’s 1949 report, “On the Hamiltonian game (a traveling salesman problem).”  Some consider the Knight’s Tour problem to be the predecessor of the TSP.

History  Hamiltonian circuits are named after Sir William Rowan Hamilton  Hamilton is known for his work with the dodecahedron because he showed that no matter which five of the twenty points you start with, you can complete the tour along the edges by visiting the remaining 15 points and terminating adjacent to the origin

History  1962 Proctor and Gamble Contest

History

 Timeline: – cities – cities – cities – cities – cities – cities – cities – cities – 19871,002 cities – 19872,392 cities

History  Concorde Timeline: – 19923,038 cities – 19934,461 cities – 19947,397 cities – ,509 cities – ,112 cities – ,978 cities – ,810 cities – ,900 cities

Examples  Use a greedy algorithm to solve TSP?

Examples

 Use heuristics: – In 2011, Rego, et al. published a paper stating that many TSP heuristics have a decent chance of yielding a solution with less than 5% of the selected edges being incorrect – However, the guarantee is only that the heuristics will generate a path that is at most 50% longer than the optimal solution

Examples  Lin-Kernighan Heuristic: – In 1973, a heuristic was developed to solve the TSP if the cost is based on Euclidean distance – The method relies on swapping pairs of paths between cities in order to find a shorter tour

Examples

West

Examples

West

Examples

Applications  Route-planning – School bus routes (how it is though TSP got started) – Post Office routes – Airline routes – Wire route to various nodes – (How is cost determined?)

Applications

Applications (sort of)  White blood cells are known to be able to accurately solve small TSPs  Psychologists have observed the abilities of adults, children, and animals to solve a given TSP

Open Problems  Further develop the TSPLIB program to be capable of realistically solving TSPs with more than 100,000 cities

Open Problems  Does P = NP? – This is one of the seven Millennium Prize Problems given by the Clay Mathematics Institute in  Further develop the TSPLIB program to be capable of realistically solving TSPs with more than 100,000 cities

References  Applegate, David L., et al. The Traveling Salesman Problem. Princeton: Princeton UP, Print.  Beineke, Lowell W., et al. Selected Topics in Graph Theory. London: Academic, Print.  Christopfides, Nicos. Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem. Pittsburgh: Carnegie-Mellon, Print.  Cormen, Thomas H., et al. Introduction to Algorithms. Cambridge: MIT, Print.  Gross, Jonathan L., et al. Handbook of Graph Theory. Boca Raton: CRC, Print.  Gutin, Gregory, et al. The Traveling Salesman Problem and Its Variations. Norwell: Kluwer, Print.

References  Karp, Richard M. (1972). "Reducibility Among Combinatorial Problems". In R. E. Miller and J. W. Thatcher (editors). Complexity of Computer Computations. New York: Plenum. pp. 85–103.  Lin, Shen; Kernighan, B. W. (1973). "An Effective Heuristic Algorithm for the Traveling-Salesman Problem". Operations Research 21 (2): 498–516.  Rego, César; Gamboa, Dorabela; Glover, Fred; Osterman, Colin (2011), "Traveling salesman problem heuristics: leading methods, implementations and latest advances", European Journal of Operational Research 211 (3): 427–441.  Lawler, E.L., et al. The Traveling Salesman Problem. London: Wiley, Print.  Tomescu, Ioan, et al. Problems in Combinatorics and Graph Theory. London: Wiley, Print.  West, Douglas B. Introduction to Graph Theory. Upper Saddle River: Prentice Hall, Print

Homework 1  Prove or Disprove: – The following graph has a Hamiltonian circuit.

Homework 2

Homework 3