The Bottleneck traveling salesman problem (bottleneck TSP) is a problem in discrete or combinatorial optimization. The problem is to find the Hamiltonian cycle (visiting each node exactly once) in a weighted graph which minimizes the weight of the highest-weight edge of the cycle. It was first formulated by with some additional constraints, and in its full generality by .
Complexity
The problem is known to be NP-hard. The decision problem version of this, "for a given length is there a Hamiltonian cycle in a graph with no edge longer than ?", is NP-complete. NP-completeness follows immediately by a reduction from the problem of finding a Hamiltonian cycle.
Algorithms
Another reduction, from the bottleneck TSP to the usual TSP (where the goal is to minimize the sum of edge lengths), allows any algorithm for the usual TSP to also be used to solve the bottleneck TSP.
If the edge weights of the bottleneck TSP are replaced by any other numbers that have the same relative order, then the bottleneck solution remains unchanged.
If, in addition, each number in the sequence exceeds the sum of all smaller numbers, then the bottleneck solution will also equal the usual TSP solution.
For instance, such a result may be attained by resetting each weight to where is the number of vertices in the graph and is the rank of the original weight of the edge in the sorted sequence of weights. For instance, following this transformation, the Held–Karp algorithm could be used to solve the bottleneck TSP in time .
This approximation ratio is best possible. For, any unweighted graph can be transformed into a metric space by setting its edge weights to and setting the distance between all nonadjacent pairs of vertices to . An approximation with ratio better than in this metric space could be used to determine whether the original graph contains a Hamiltonian cycle, an NP-complete problem.