traveling salesman

(classic problem)

Definition: Find a path through a weighted graph that starts and ends at the same vertex, includes every other vertex exactly once, and minimizes the total cost of edges.

Also known as TSP.

See also bottleneck traveling salesman, Hamiltonian cycle, optimization problem, Christofides algorithm, similar problems: all pairs shortest path, minimum spanning tree, vehicle routing problem.

Note: Less formally, find a path for a salesman to visit every listed city at the lowest total cost.

The above described path is always a Hamiltonian cycle, or tour, however a Hamiltonian cycle need not be optimal. The problem is an optimization problem, that is, to find the shortest tour. The corresponding decision problem asks if there is a tour with a cost less than some given amount. See [CLR90, page 969].

If the triangle inequality does not hold, that is dik > dij + djk for some i, j, k, there is no possible polynomial time algorithm that guarantees near-optimal result (unless P=NP).

If the triangle inequality holds, you can quickly get a near-optimal solution by finding the minimum spanning tree. Convert the tree to a path by traversing the tree, say by depth-first search, but go directly to the next unvisited node, rather than repeating nodes. Christofides algorithm starts with a minimum spanning tree, but is more careful about converting the tree to a path with results closer to optimal.

Author: PEB


Steven Skiena's Stony Brook Algorithm Repository (C, Fortran, Pascal, Mathematica, and C++). Animation and implementation (Icon).

More information

links to challenges, advances, etc..

Go to the Dictionary of Algorithms and Data Structures home page.

If you have suggestions, corrections, or comments, please get in touch with Paul Black.

Entry modified 26 July 2021.
HTML page formatted Mon Jul 26 11:58:06 2021.

Cite this as:
Paul E. Black, "traveling salesman", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed. 26 July 2021. (accessed TODAY) Available from: