Travelling salesman problem

The travelling salesman problemThe more common spelling in American English is traveling salesman problem. (TSP) 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.

Problem statement

Given a number of cities and the costs of traveling from any city to any other city, what is the cheapest round-trip route that visits each city exactly once and then returns to the starting city?

An equivalent formulation in terms of graph theory is: Given a complete weighted graph (where the vertices would represent the cities, the edges would represent the roads, and the weights would be the cost or distance of that road), find a Hamiltonian cycle with the least weight.

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 a 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 classic 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 permutations (ordered combinations) and see which one is cheapest (using brute force search), but given that the number of permutations is n! (the factorial of the number of cities, n), this solution rapidly becomes impractical.

Using the techniques of dynamic programming, one can solve the problem exactly in time $O(2^n)$ . Although this is exponential, it is still much better than $O(n!)$ . See Big O notation.

NP-hardness

The problem has been shown to be NP-hard (more precisely, it is complete for the complexity class FP^NP; see the function problem article), and the decision problem version ("given the costs and a number x, decide whether there is a roundtrip route cheaper than x") is NP-complete.

The bottleneck traveling salesman problem is also NP-hard.

The problem remains NP-hard even for the case when the cities are in the plane with Euclidean distances, as well as in a number of other restrictive cases.

Removing the condition of visiting each city "only once" does not remove the NP-hardness, since it is easily seen that in the planar case an optimal tour visits cities only once (otherwise, by the triangle inequality, a shortcut that skips a repeated visit would decrease the tour length).

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 either seemingly or probably good solutions, but which could not be proved to be optimal.
Finding special cases for the problem ("subproblems") for which either exact or better heuristics are possible.

For benchmarking of TSP algorithms, TSPLIB a library of sample instances of the TSP and related problems is maintained, see the TSPLIB external reference. Many of them are lists of actual cities and layouts of actual printed circuits.

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 for up to 120-200 cities.
Recent implementations of branch-and-bound and cut based on linear programming works very well for up to 5,000 cities, and this approach has been used to solve instances with up to 33,810 cities.

An exact solution for 15,112 German cities from TSPLIB was found in 2001 using the cutting-plane method proposed by George Dantzig, Ray Fulkerson, and Selmer Johnson in 1954, based on linear programming. The computations were performed on a network of 110 processors located at Rice University and Princeton University (see the Princeton external link). The total computation time was equivalent to 22.6 years on a single 500 MHz Alpha processor. In May 2004, the traveling salesman problem of visiting all 24,978 cities in Sweden was solved: a tour of length approximately 72,500 kilometers was found and it was proven that no shorter tour exists.

In March 2005, the traveling salesman problem of visiting all 33,810 points in a circuit board was solved using CONCORDE: a tour of length 66,048,945 units was found and it was proven that no shorter tour exists, the computation took approximately 15.7 CPU years.

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 with a high probability just 2-3% away from the optimal solution.

Several categories of heuristics are recognized.

Constructive heuristics

The nearest neighbor algorithm, which is normally fairly close to the optimal route, and does not take too long to execute. Rosenkrantz et al. ^* showed that the nearest neighbour algorithm has approximation factor $\Theta(\log |V|)$ . Unfortunately, it is provably reliable only for special cases of the TSP, namely those where the triangle inequality is satisfied. In the general case, there exists an example for which the nearest neighbor algorithm gives the worst possible route. In practice, this heuristic provides solutions which are in average 10 to 15 percent above the optimal. An example of the nearest neighbor algorithm finding the worst case path would be to travel along a line, starting at 0, to alternating negative and positive powers of 2 minus 1, therefore, from 0,1,-3,7,-15,31,-63... $(2^n -1)(-1)^n$ .

Iterative improvement

Pairwise exchange, or Kernighan-Lin heuristics.

k-opt heuristic: Take a given tour and delete k mutually disjoint edges. Reassemble the remaining fragments into a tour, leaving no disjoint subtours (that is, don't connect a fragment's endpoints together). This in effect simplifies the TSP under consideration into a much simpler problem. Each fragment endpoint can be connected to 2k − 2 other possibilities: of 2k total fragment endpoints available, the two endpoints of the fragment under consideration are disallowed. Such a constrained 2k-city TSP can then be solved with brute force methods to find the least-cost recombination of the original fragments.

Randomized improvement

Optimized Markov chain algorithms which utilize local searching heuristic sub-algorithms can find a route extremely close to the optimal route for 700-800 cities.

TSP is a touchstone for many general heuristics devised for combinatorial optimization: genetic algorithms, simulated annealing, Tabu search, neural nets, ant system.

Special cases

Restricted locations

A trivial special case is when all cities are located on the perimeter of a convex polygon.
A good exercise in combinatorial algorithms is to solve the TSP for a set of cities located along two concentric circles.

Triangle inequality and the Christofides algorithm

A very natural restriction of the TSP is the triangle inequality. That is, for any 3 cities A, B and C, the distance between A and C must be at most the distance from A to B plus the distance from B to C. Most natural instances of TSP satisfy this constraint.

In this case, there is a constant-factor approximation algorithm (due to Christofides, 1975) which always finds a tour of length at most 1.5 times the shortest tour. In the next paragraphs, we explain a weaker (but simpler) algorithm which finds a tour of length at most twice the shortest tour.

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. That is, wherever there is an edge from u to v, add a second edge from u to v. This gives us an Eulerian graph.
Step 3: Find a 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).

It is easy to prove that the last step works. Moreover, thanks to the triangle inequality, each skipping at Step 4 is in fact a shortcut, i.e., the length of the cycle does not increase. Hence it gives us a TSP tour no more than twice as long as the optimal one.

Christofides algorithm follows a similar outline but combines the minimum spanning tree with a solution of another problem, minimum-weight perfect matching. This gives a TSP tour which is at most 1.5 times the optimal. It is a long-standing (since 1975) open problem to improve 1.5 to a smaller constant. It is known, however, that there is no polynomial time algorithm that finds a tour of length at most 1/219 more than optimal, unless P = NP (Papadimitriou and Vempala, 2000). In the case of the bounded metrics, it is known, that there is no polynomial time algorithm that constructs a tour of length at most 1/388 more than optimal, unless P = NP (Engebretsen and Karpinski, 2001). The best known polynomial time approximation algorithm for the TSP problem with distances one and two finds a tour of length at most 1/7 more than optimal (Berman and Karpinski, 2006).

The Christofides algorithm was one of the first approximation algorithms, and was in part responsible for drawing attention to approximation algorithms as a practical approach to intractable problems. As a matter of fact, the term "algorithm" was not commonly extended to approximation algorithms until later. At the time of publication, the Christofides algorithm was referred to as the Christofides heuristic.

Euclidean TSP

Euclidean TSP, or planar TSP, is the TSP with the distance being the ordinary Euclidean distance. Although the problem still remains NP-hard, it is known that there exists a subexponential time algorithm for it. Moreover, many heuristics work better.

Euclidean TSP is a particular case of TSP with triangle inequality, since distances in plane obey triangle inequality. However, it seems to be easier than general TSP with triangle inequality. For example, the minimum spanning tree of the graph associated with an instance of Euclidean TSP is a Euclidean minimum spanning tree, and so can be computed in expected O(n log n) time for n points (considerably less than the number of edges). This enables the simple 2-approximation algorithm for TSP with triangle inequality above to operate more quickly.

In general, for any c > 0, there is a polynomial-time algorithm that finds a tour of length at most (1 + c) times the optimal for geometric instances of TSP (Arora; Mitchell; Rao and Smith); this is called a polynomial-time approximation scheme. This result is an important theoretical algorithm but is not likely to be practical. Instead, heuristics with weaker guarantees are often used, but they also perform better on instances of Euclidean TSP than on general instances.

Asymmetric TSP

In most cases, the distance between two nodes in the TSP network is the same in both directions. The case where the distance from A to B is not equal to the distance from B to A is called asymmetric TSP. A practical application of an asymmetric TSP is route optimization using street-level routing (asymmetric due to one-way streets, slip-roads and motorways).

Notes

References

G. B. Dantzig, R. Fulkerson, and S. M. Johnson, Solution of a large-scale traveling salesman problem, Operations Research 2 (1954), 393-410.
S. Arora. Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and other Geometric Problems. Journal of ACM, 45(1998), 753-782.
P. Berman, M. Karpinski, 8/7-Approximation Algorithm for (1,2)-TSP, Proc. 17th ACM-SIAM SODA (2006), 641-648.
N. Christofides, Worst-case analysis of a new heuristic for the travelling salesman problem, Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, 1976.
L. Engebretsen, M. Karpinski, Approximation hardness of TSP with bounded metrics, Proceedings of 28th ICALP (2001), LNCS 2076, Springer 2001, 201-212.
J. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems, SIAM Journal on Computing, 28(1999), 1298-1309.
S. Rao, W. Smith. Approximating geometrical graphs via 'spanners' and 'banyans'. Proc. 30th Annual ACM Symposium on Theory of Computing, 1998, pp 540-550.
C. H. Papadimitriou, S. Vempala: On the approximability of the traveling salesman problem (extended abstract). Proceedings of STOC'2000, 126-133.
D. S. Johnson and L. A. McGeoch, The Traveling Salesman Problem: A Case Study in Local Optimization, Local Search in Combinatorial Optimization, E. H. L. Aarts and J.K. Lenstra (ed), John Wiley and Sons Ltd, 1997, pp 215-310.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 35.2: The traveling-salesman problem, pp.1027–1033.
A2.3: ND22–24, pp.211–212.

External links

TSPLIB
TSP page
Private web page with 2 algorithms & demo program to download
Example of finding approximate solution of TSP problem using a genetic algorithm
A Java implementation of a TSP-solution using JGAP (Java Genetic Algorithms Package). The technique used is a Genetic Algorithm.
Solution of the Traveling Salesman Problem using a Kohonen Map
Solution of the Traveling Salesman Problem using a Kohonen Map (in 3-dimensions!)
Simulated annealing method for solving this problem, written in the functional programming language Objective Caml
The Travelling Salesman Problem Discusses the use of simulated annealing to solve this problem and allows download of a DOS/Windows command-line program demonstrating problem solution.
Most TSP loop families grow polynomially Private web page shows that a method exists for obtaining a set of optimal "traveling salesman" routes that is a member of a family that grows no faster than about 2ⁿIn2. However, even for large n, the method yields a polynomial rate for most sets of fields of nodes.
VisualBots - Freeware multi-agent simulator in Microsoft Excel. Sample programs include genetic algorithm, ACO, and simulated annealing solutions to TSP.

Graph theory | NP-complete problems

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