Presentation on theme: "Hamiltonian Circuit & The Traveling Salesman Problem Drew Nash 26 March 2014."— Presentation transcript:
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?
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
Examples Use a greedy algorithm to solve TSP?
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
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 (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.