The Bottleneck traveling salesman problem (bottleneck TSP) is a problem in discrete or combinatorial optimization. It is stated as follows: Find the Hamiltonian cycle in a weighted graph which minimizes the weight of the most weighty edge of the cycle.
The problem is known to be NP-hard. The decision problem version of this, "for a given length x, is there a Hamiltonian cycle in a graph g with no edge longer than x?", is NP-complete.
In an asymmetric bottleneck TSP, there are cases where the weight from node A to B is different from the weight from B to A (e. g. travel time between two cities with a traffic jam in one direction).
Euclidean bottleneck TSP, or planar bottleneck TSP, is the bottleneck TSP with the distance being the ordinary Euclidean distance. The problem still remains NP-hard, however many heuristics work better.
If the graph is a metric space then there is an efficient approximation algorithm that finds a Hamiltonian cycle with maximum edge weight being no more than twice the optimum.
Famous quotes containing the words traveling, salesman and/or problem:
“I see before me now a traveling army halting,
Below a fertile valley spread, with barns and the orchards of summer,
Behind, the terraced sides of a mountain, abrupt, in places rising high,”
—Walt Whitman (1819–1892)
“Nobody dast blame this man.... For a salesman, there is no rock bottom to the life. He don’t put a bolt to a nut, he don’t tell you the law or give you medicine. He’s a man way out there in the blue, riding on a smile and a shoeshine. And when they start not smiling back—that’s an earthquake. And then you get yourself a couple of spots on your hat, and you’re finished. Nobody dast blame this man. A salesman is got to dream, boy. It comes with the territory.”
—Arthur Miller (b. 1915)
“The problem is that we attempt to solve the simplest questions cleverly, thereby rendering them unusually complex. One should seek the simple solution.”
—Anton Pavlovich Chekhov (1860–1904)