Travelling salesman problem

If a salesman starts at point A, and if the distances between every pair of points are known, what is the shortest route which visits all points and returns to point A?

The 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 classified as NP-hard. Mathematical problems related to the traveling salesman problem were treated in the 1800s by the Irish mathematician Sir William Rowan Hamilton and by the British mathematician Thomas Penyngton Kirkman. A discussion of the early work of Hamilton and Kirkman can be found in Graph Theory 1736-1936. The general form of the TSP appears to have been first studied by mathematicians during the 1930s in Vienna and at Harvard, notably by Karl Menger. The problem was later undertaken by Hassler Whitney and Merrill Flood at Princeton. A detailed treatment of the connection between Menger and Whitney as well as the growth in the study of TSP can be found in Alexander Schrijver's 2005 paper "On the history of combinatorial optimization (till 1960)".

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?

Related Problems

1. 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.
2. Another 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 $O((n^{2})*(2^{n}))$. 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 better or exact 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 (Cook et al. 2006).

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 (Cook et al. 2006).

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 neighbour (NN) algorithm lets the salesman start from any one city and choose the nearest city not visited yet to be his next visit. This algorithm quickly yields an effectively short route. Rosenkrantz et al. [1977] showed that the NN algorithm has the approximation factor $\Theta (\log |V|)$. In 2D Euclidean TSP, NN algorithm result in a length about 1.26*(optimal length). Unfortunately, there exist some examples for which this algorithm gives a highly inefficient route. A bad result is due to the greedy nature of this algorithm.

Iterative improvement

• Pairwise exchange, or Lin-Kernighan 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.

Randomised improvement

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

• Random path change algorithms are currently the state-of-the-art search algorithms and work up to 100,000 cities. The concept is quite simple: Choose a random path, choose four nearby points, swap their ways to create a new random path, while in parallel decreasing the upper bound of the path length. If repeated until a certain number of trials of random path changes fail due to the upper bound, one has found a local minimum with high probability, and even further it's a global minimum with high probability (whereas high means that the rest probability decreases exponentially in the size of the problem - thus for 10,000 or more nodes, the chances of failure is negligible).

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

Example letting the inversion operator find a good solution

Suppose that the number of towns is = 60. For a random search process, this is like having a deck of cards numbered 1, 2, 3, ... 59, 60 where the number of permutations is of the same order of magnitude as the total number of atoms in the universe. If the hometown is not counted the number of possible tours becomes 60*59*58*...*4*3 (about 10⁸⁰, i. e. a 1 followed by 80 zeros).

Suppose that the salesman does not have a map showing the location of the towns, but only a deck of numbered cards, which he may permute, put in a card reader - like in the childhood of computers - and let the computer calculate the length of the tour. The probability to find the shortest tour by random permutation is about one in 10⁸⁰ so, it will never happen. So, should he give up?

No, by no means, evolution may be of great help to him; at least if it could be simulated on his computer. The natural evolution uses an inversion operator, which - in principle - is extremely well suited for finding good solutions to the problem. A part of the card deck (DNA) - chosen at random - is taken out, turned in opposite direction and put back in the deck again like in the figure below with 6 towns. The hometown (nr 1) is not counted.

If this inversion takes place where the tour happens to have a loop, then the loop is opened and the salesman is guaranteed a shorter tour. The probability that this will happen is greater than 1/(60*60) for any loop if we have 60 towns, so, in a population with one million card decks it might happen 1000000/3600 = 277 times that a loop will disappear.

This has been simulated with a population of 180 card decks, from which 60 decks are selected in every generation using MATLAB. The figure below shows a random tour at start

After about 1500 generations all loops have been removed and the length of the random tour at start has been reduced to 1/5 of the original tour. The human eye can see that some improvements can be made, but probably the random search has found a tour, which is not much longer than the shortest possible. See figure below.

In a special case when all towns are equidistantly placed along a circle, the optimal solution is found when all loops have been removed. This means that this simple random search is able to find one optimal tour out of as many as 10⁸⁰. See also Goldberg, 1989.

Special cases

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.

1. Construct the minimum spanning tree.
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.
3. Find a Eulerian cycle in it. Clearly, its length is twice the length of the tree.
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.

The 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 + 1/c) times the optimal for geometric instances of TSP (Arora); 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 optimisation using street-level routing (asymmetric due to one-way streets, slip-roads and motorways).

Solving by conversion to Symmetric TSP

Solving an asymmetric TSP graph can be somewhat complex. The following is a 3x3 matrix containing all possible path weights between the nodes A, B and C. One option is to turn an asymmetric matrix of size N into a symmetric matrix of size 2N, doubling the complexity.

Asymmetric Path Weights

	A	B	C
A		1	2
B	6		3
C	5	4

To double the size, each of the nodes in the graph is duplicated, creating a second ghost node. Using duplicate points with very low weights, such as -∞, provides a cheap route "linking" back to the real node and allowing symmetric evaluation to continue. The original 3x3 matrix shown above is visible in the bottom left and the inverse of the original in the top-right. Both copies of the matrix have had their diagonals replaced by the low-cost hop paths, represented by -∞.

Symmetric Path Weights

	A	B	C	A'	B'	C'
A				-∞	6	5
B				1	-∞	4
C				2	3	-∞
A'	-∞	1	2
B'	6	-∞	3
C'	5	4	-∞

The original 3x3 matrix would produce two Hamiltonian cycles (a path that visits every node once), namely A-B-C-A [score 12] and A-C-B-A [score 9]. Evaluating the 6x6 symmetric version of the same problem now produces many paths, including A-A'-B-B'-C-C'-A, A-B'-C-A'-A, A-A'-B-C'-A [all score 9-∞].

The important thing about each new sequence is that there will be an alternation between dashed (A',B',C') and un-dashed nodes (A,B,C) and that the link to "jump" between any related pair (A-A') is effectively free. A version of the algorithm could use any weight for the A-A' path, as long as that weight is lower than all other path weights present in the graph. As the path weight to "jump" must effectively be "free", the value zero (0) could be used to represent this cost— if zero is not being used for another purpose already (such as designating invalid paths). In the two examples above, non-existent paths between nodes are shown as a blank square.

Human performance on TSP

The TSP, in particular the Euclidean variant of the problem, has attracted the attention of researchers in cognitive psychology. It is observed that humans are able to produce good quality solutions quickly. The first issue of the Journal of Problem Solving is devoted to the topic of human performance on TSP.

TSP path length

Many quick algorithms yield approximate TSP solution for large city number. To have an idea of the precision of an approximation, one should measure the resulted path length and compare it to the exact path length. To find out the exact path length, there are 3 approaches:

1. find a lower bound of it,
2. find an upper bound of it with CPU time T, do extrapolation on T to infinity so result in a reasonable guess of the exact value, or
3. solve the exact value without solving the city sequence.

Lower bound

Consider N points are randomly distributed in one unit square, with N>>1. A simple lower bound of the shortest path length is ${\frac {1}{2}}{\sqrt {N}}$, obtained by considering each point connects to its nearest neighbor which is $\left.{\frac {1}{2}}\right/{\sqrt {N}}$ distance away on average.

Another lower bound is $\left({{\frac {1}{2}}+{\frac {3}{4}}}\right){\frac {\sqrt {N}}{2}}$, obtained by considering each point j connects to j's nearest neighbor, and j's second nearest neighbor connects to j. Since j's nearest neighbor is (1/2)/sqrt(N) distance away; j's second nearest neighbor is (3/4)/sqrt(N) distance away on average.

• David S. Johnson had got a lower bound by experiment

0.708*sqrt(N)+0.522

• Christine L. Valenzuela and Antonia J. Jones have also explored a better lower bound by experiment

0.708*sqrt(N)+0.551

Marco Dorigo

• Marco Dorigo (1999) announced that he and his colleagues in the Free University of Brussels had hit upon a way of reaching "near optimal" solutions to the TSP. Solving the problem in the same manner a colony of ants would. Sending out an army of virtual salesmen to explore all possible routes on the map. when a salesmen successfully completes a journey to all cities , it traces a path back to the starting city, depositing a small amount of virtual pheromone. The total amount of pheromone is finite, such that it is spread more thinly along the longer routes, more heavily along the shorter ones. With thousands of ants exploring the map some sections of shorter routes quickly accumulate thick layers of pheromone, less efficient routes have almost no pheromone at all. (Taken from Johnson, S., Emergence, 2001, Penguin Books)

Miscellanea

• The board game Elfenland resembles the traveling salesman problem.

References

• E. L. Lawler and Jan Karel Lenstra and A. H. G. Rinnooy Khan and D. B. Shmoys (1985). The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley & Sons. ISBN 0-471-90413-9.
• G. Gutin and A. P. Punnen (2006). The Traveling Salesman Problem and Its Variations. Springer. ISBN 0-387-44459-9.
• G. B. Dantzig, R. Fulkerson, and S. M. Johnson, Solution of a large-scale traveling salesman problem, Operations Research 2 (1954), pp. 393-410.
• S. Arora. "Polynomial Time Approximation Schemes for Euclidean Traveling Salesman and other Geometric Problems". Journal of ACM, 45 (1998), pp. 753-782.
• P. Berman, M. Karpinski, "8/7-Approximation Algorithm for (1,2)-TSP", Proc. 17th ACM-SIAM SODA (2006), pp. 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, pp. 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), pp. 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 and Santosh Vempala, "On the approximability of the traveling salesman problem", Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000.
• Daniel J. Rosenkrantz and Richard E. Stearns and Phlip M. Lewis II (1977). "An Analysis of Several Heuristics for the Traveling Salesman Problem". SIAM J. Comput. 6 (5): 563–581.
• D. S. Johnson and L. A. McGeoch, The Traveling Salesman Problem: A Case Study in Local Optimization, Local Search in Combinatorial Optimisation, 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 0-262-03293-7. Section 35.2: The traveling-salesman problem, pp. 1027–1033.
• Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5. A2.3: ND22–24, pp.211–212.
• MacGregor, J. N., & Ormerod, T. (1996). Human performance on the traveling salesman problem. Perception & Psychophysics, 58(4), pp. 527–539.
• Vickers, D., Butavicius, M., Lee, M., & Medvedev, A. (2001). Human performance on visually presented traveling salesman problems. Psychological Research, 65, pp. 34–45.
• William Cook, Daniel Espinoza, Marcos Goycoole (2006). Computing with domino-parity inequalities for the TSP. INFORMS Journal on Computing. Accepted.
• Goldberg, D. E. Genetic Algorithms in Search, Optimization & Machine Learning. Addison-Wesley, New York, 1989.

See also

• Tabu search
• Assignment problem
• route inspection problem
• Greedy algorithm
• Six degrees of separation

External links

• Solution of the Travelling Salesman Problem using a nearest vertex add Algorithm
• Demo applet of a genetic algorithm solving TSP and VRPTW online
• TSPLIB
• lalena.com: TSP Solution in C#.NET (downloadable or run as applet) using Genetic Algorithms
• 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 Travelling Salesman Problem using a Kohonen Map
• Kohonen Neural Network applied to the Traveling Salesman Problem (using three dimensions).
• Most TSP loop families grow polynomially Private web page shows that a method exists for obtaining a set of optimal "travelling 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.
• Work-in-progress attempted proof on private web page of polynomial calculation time on the 2-D undirected graph.
• Links to TSP source codes
• http://www.delphiforfun.org/Programs/traveling_salesman.htm The Traveling Salesman Problem (TSP) requires that we find the shortest path visiting each of a given set of cities and returning to the starting point.