Bottleneck traveling salesman problem
Encyclopedia
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.
Discrete optimization
Discrete optimization is a branch of optimization in applied mathematics and computer science.As opposed to continuous optimization, the variables used in the mathematical program are restricted to assume only discrete values, such as the integers.Two notable branches of discrete optimization...
or combinatorial optimization
Combinatorial optimization
In applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not feasible...
. It is stated as follows: Find the Hamiltonian cycle
Hamiltonian path
In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected graph that visits each vertex exactly once. A Hamiltonian cycle is a cycle in an undirected graph that visits each vertex exactly once and also returns to the starting vertex...
in a weighted graph
Glossary of graph theory
Graph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different meanings. Some authors use different words to mean the same thing. This page attempts to keep up with current usage....
which minimizes the weight of the most weighty edge
Glossary of graph theory
Graph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different meanings. Some authors use different words to mean the same thing. This page attempts to keep up with current usage....
of the cycle.
The problem is known to be NP-hard
NP-hard
NP-hard , in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP". A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H...
. The decision problem
Decision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a 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
NP-complete
In computational complexity theory, the complexity class NP-complete is a class of decision problems. A decision problem L is NP-complete if it is in the set of NP problems so that any given solution to the decision problem can be verified in polynomial time, and also in the set of NP-hard...
.
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
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...
. The problem still remains NP-hard, however many heuristics work better.
If the graph is a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
then there is an efficient approximation algorithm
Approximation algorithm
In computer science and operations research, approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP-hard problems; since it is unlikely that there can ever be efficient polynomial time exact...
that finds a Hamiltonian cycle with maximum edge weight being no more than twice the optimum.