The Bottleneck traveling salesman problem (bottleneck TSP) is a problem in
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
* Discrete group, ...
or
combinatorial optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combina ...
. The problem is to find the
Hamiltonian cycle
In the mathematics, mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path (graph theory), path in an undirected or directed graph that visits each vertex (graph theory), vertex exactly once. A Hamiltonian cycle (or ...
(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
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assumi ...
. The
decision problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
version of this, "for a given length is there a Hamiltonian cycle in a graph with no edge longer than ?", is
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''.
Somewhat more precisely, a problem is NP-complete when:
# It is a decision problem, meaning that for any ...
. 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 .
Alternatively, the problem can be solved by performing a
binary search
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the m ...
or
sequential search for the smallest such that the subgraph of edges of weight at most has a Hamiltonian cycle. This method leads to solutions whose running time is only a logarithmic factor larger than the time to find a Hamiltonian cycle.
Variations
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).
The Euclidean bottleneck TSP, or planar bottleneck TSP, is the bottleneck TSP with the distance being the ordinary
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
. The problem still remains NP-hard. However, many heuristics work better for it than for other distance functions.
The maximum scatter traveling salesman problem is another variation of the traveling salesman problem in which the goal is to find a Hamiltonian cycle that maximizes the minimum edge length rather than minimizing the maximum length. Its applications include the analysis of medical images, and the scheduling of metalworking steps in aircraft manufacture to avoid heat buildup from steps that are nearby in both time and space. It can be translated into an instance of the bottleneck TSP problem by negating all edge lengths (or, to keep the results positive, subtracting them all from a large enough constant). However, although this transformation preserves the optimal solution, it does not preserve the quality of approximations to that solution.
Metric approximation algorithm
If the graph is a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
then there is an efficient
approximation algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned sol ...
that finds a Hamiltonian cycle with maximum edge weight being no more than twice the optimum.
This result follows by
Fleischner's theorem, that the
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
of a
2-vertex-connected graph always contains a Hamiltonian cycle. It is easy to find a threshold value , the smallest value such that the edges of weight form a 2-connected graph. Then provides a valid lower bound on the bottleneck TSP weight, for the bottleneck TSP is itself a 2-connected graph and necessarily contains an edge of weight at least . However, the square of the subgraph of edges of weight at most is Hamiltonian. By the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of Degeneracy (mathematics)#T ...
for metric spaces, its Hamiltonian cycle has edges of weight at most .
[.][.]
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.
Without the assumption that the input is a metric space, no finite approximation ratio is possible.
See also
*
Travelling salesman problem
In the Computational complexity theory, theory of computational complexity, the travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible ...
References
{{reflist
Combinatorial optimization
Graph algorithms
Hamiltonian paths and cycles
NP-complete problems