Bottleneck traveling salesman problem

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.[1]

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.[1]

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.[2]

See also

References

  1. ^ a b Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5.  A2.3: ND24, pg.212.
  2. ^ R. Garey Parker and Ronald L. Rardin (1984). Guaranteed performance heuristics for the bottleneck traveling salesman problem. Operations Research Letters.  2(6):269–272

3. G. Carpaneto, S. Martello and P. Toth (1984). An algorithm for the bottleneck travelling salesman problem. Operations Research 32(2):380–389.