The traveling salesman problem (TSP), also known as the traveling salesperson problem, is a problem in discrete or combinatorial optimization. It is a prominent illustration of a class of problems in computational complexity theory which are hard to solve.

Table of contents

1 Problem statement 2 Computational complexity 3 Algorithms 3.1 Exact algorithms 3.2 Heuristics 3.2.1 Constructive heuristics 3.2.2 Iterative improvement 3.2.3 Randomized improvement 3.3 Special cases 3.3.4 Restricted locations 3.3.5 TSP with triangle inequality 4 References 5 Related articles 6 External Links

Problem statement

An equivalent formulation in terms of graph theory is: Find the shortest Hamiltonian cycle in a weighted graph.

It can be shown that the requirement of returning to the starting city does not change the computational complexity of the problem.

A related problem is the Bottleneck traveling salesman problem (bottleneck TSP): Find the Hamiltonian cycle in a weighted graph with the minimal length of the longest edge.

The problem is of considerable practical importance, apart from evident transportation and logistics areas. A classical example is in printed circuit manufacturing -- scheduling of a route of the drill machine to drill holes in a PCB. In robotic machining or drilling applications, the "cities" are parts to machine or holes (of different sizes) to drill, and the "cost of travel" includes time for retooling the robot (single machine job sequencing problem).

Computational complexity

The most direct solution would be to try all the combinations and see which one is cheapest, but given that the number of combinations is N! (the factorial of the number of cities), this solution rapidly becomes impractical.

The problem has been shown to be NP-hard, and the decision version of it ("given the costs and a number x, decide whether there is a roundtrip route cheaper than x") is NP-complete.

The bottleneck TSP is also NP-hard.

Algorithms

The traditional lines of attack for the NP-hard problems are the following:

• Devising algorithms for finding exact solutions (they will work reasonably fast only for relatively small problem sizes)
• Devising "suboptimal" or heuristic algorithms, i.e., algorithms that deliver seemingly or provably good solutions, but which could not proved to be optimal.
• Finding special cases for the problem ("subproblems") for which either exact or better heuristics are posible.

Exact algorithms

• Various branch and bound algorithms, which can be used to process TSPs containing 40-60 cities.
• Progressive improvement algorithms which use techniques reminiscent of linear programming works well up to 120-200 cities.

Heuristics

Various approximation algorithms, which "quickly" yield "good" solutions with "high" probability, have been devised. Modern methods can find solutions for extremely large problems (millions of cities) within a reasonable time which are provably 2-3% away from the optimal solution.

Several categories of heuristics are recognized.

Constructive heuristics

• The nearest neighbour algorithm, which is normally fairly close to the optimal route, and doesn't take too long to execute. Unfortunately it is can be provably reliable only for special cases of the TSP. In general case, however there exists an example for which the nearest neighbour algorithm gives the worst possible route.

• Optimised Markov chain algorithms which utilise local searching heuristical sub-algorithms can find a route extremely close to the optimal route for 700-800 cities.

Special cases

Restricted locations

Euclidean TSP, or planar TSP, is the TSP with the distance being the ordinary Euclidean distance. The problem still remains NP-hard, however many heuristics work better. It turns out that the instrumental property in this case is the triangle inequality.

The length of the minimum spanning tree of the network is a natural lower bound for the length of the optimal route. In the TSP with triangle inequality case it is possible to prove upper bounds in terms of the minimum spanning tree and design an algorithm that has a provable upper bound on the length of the route. The first published (and the simplest) example follows.

• Step 1: Construct the minimal spanning tree.
• Step 2: Duplicate all its edges. This gives us an Eulerian graph.
• Step 3: Find an Eulerian cycle in it. Clearly, its length is twice the length of the tree.
• Step 4: Convert the Eulerian cycle into the Hamiltonian one in the following way: walk along the Eulerian cycle, and each time you are about to come into an already visited vertex, skip it and try to go to the next one (along the Eulerian cycle).

Better implementations of this heuristic are known, as well as other heuristics with better worst case estimates.

References

• Tabu search
• Assignment problem