Travelling salesman problem



The Travelling Salesman Problem (TSP) is a problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once.

The problem was first formulated as a mathematical problem in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved.

The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as genome sequencing. In these applications, the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA fragments. In many applications, additional constraints such as limited resources or time windows make the problem considerably harder.

In the theory of computational complexity, the decision version of TSP belongs to the class of NP-complete problems. Thus, it is assumed that there is no efficient algorithm for solving TSPs. In other words, it is likely that the worst case running time for any algorithm for TSP increases exponentially with the number of cities, so even some instances with only hundreds of cities will take many CPU years to solve exactly.

History
The origins of the travelling salesman problem are unclear. A handbook for travelling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, but contains no mathematical treatment.



Mathematical problems related to the travelling salesman problem were treated in the 1800s by the Irish mathematician W. R. Hamilton and by the British mathematician Thomas Kirkman. Hamilton’s Icosian Game was a recreational puzzle based on finding a Hamiltonian cycle. 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, who defines the problem, considers the obvious brute-force algorithm, and observes the non-optimality of the nearest neighbour heuristic:

Hassler Whitney at Princeton University introduced the name travelling salesman problem soon after.

In the 1950s and 1960s, the problem became increasingly popular in scientific circles in Europe and the USA. Notable contributions were made by George Dantzig, Delbert Ray Fulkerson and Selmer M. Johnson at the RAND Corporation in Santa Monica, who expressed  the problem as an integer linear program and developed the cutting plane method for its solution. With these new methods they solved an instance with 49 cities to optimality by constructing a tour and proving that no other tour could be shorter. In the following decades, the problem was studied by many researchers from mathematics, computer science,chemistry, physics, and other sciences.

Richard M. Karp showed in 1972 that the Hamiltonian cycle problem was NP-complete, which implies the NP-hardness of TSP. This supplied a scientific explanation for the apparent computational difficulty of finding optimal tours.

Great progress was made in the late 1970s and 1980, when Grötschel, Padberg, Rinaldi and other managed to exactly solve instances with up to 2392 cities, using cutting planes and branch-and-bound.

In the 1990s, Applegate, Bixby, Chvátal, and Cook developed the program Concorde that has been used in many recent record solutions. Gerhard Reinelt published the TSPLIB in 1991, a collection of benchmark instances of varying difficulty, which has been used by many research groups for comparing results. In 2005, Cook and others computed an optimal tour through a 33,810-city instance given by  a microchip layout problem, currently the largest solved TSPLIB instance. For many other instances with millions of cities, solutions can be found that are provably within 1% of optimal tour.

As a graph problem


The TSP can be modelled as a graph, such that cities are the graph’s vertices, paths are the graph’s edges, and a path's distance is the edge's length. A TSP tour becomes a Hamiltonian cycle, and the optimal TSP tour is the shortest Hamiltonian cycle. Often, the model is a complete graph (i.e. an edge connects each pair of vertices). If no path exists between two cities, adding an arbitrarily long edge will complete the graph without affecting the optimal tour.

Asymmetric and symmetric
In the symmetric TSP, the distance between two cities is the same in each direction, forming an undirected graph. This symmetry halves the number of possible solutions. In the asymmetric TSP, paths may not exist in both directions or the distances might be different, forming a directed graph. Traffic collisions, one-way streets, and airfares for cities with different departure and arrival fees are examples of how this symmetry could break down.

With metric distances
In the metric TSP the intercity distances satisfy the triangle inequality. This can be understood as “no shortcuts”, in the sense that the direct connection from A to B is never longer than the detour via C.


 * $$c_{ij} \le c_{ik} + c_{kj}$$

These edge lengths define a metric on the set of vertices. When the cities are viewed as points in the plane, many natural distance functions are metrics.


 * In the Euclidian TSP the distance between two cities is the Euclidean distance between the corresponding points.


 * In the Rectilinear TSP the distance between two cities is the sum of the differences of their x- and y-coordinates. This metric is often called the Manhattan distance or city-block metric.


 * In the maximum metric, the distance between two points is the maximum of the differences of their x- and y-coordinates.

The last two metrics appear for example in routing a machine that drills a given set of holes in a printed circuit. The Manhattan metric corresponds to a machine that adjusts first one co-ordinate, and then the other, so the time to move to a new point is the sum of both movements. The maximum metric corresponds to a machine that adjusts both co-ordinates simultaneously, so the time to move to a new point is the slower of the two movements.

Non-metric distances
Distance measures that do not satisfy the triangle inequality arise in many routing problems. For example, one mode of transportation, such as travel by airplane, may be faster, even though it covers a longer distance.

In its definition, the TSP does not allow cities to be visited twice, but many applications do not need this constraint. In such cases, a symmetric, non-metric instance can be reduced to a metric one. This replaces the original graph with a complete graph in which the inter-city distance $$c_{ij}$$ is replaced by the shortest path between $$i$$ and $$j$$ in the original graph.

Related problems

 * 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.


 * The requirement of returning to the starting city does not change the computational complexity of the problem, see Hamiltonian path problem.


 * Another related problem is the bottleneck travelling salesman problem (bottleneck TSP): Find a Hamiltonian cycle in a weighted graph with the minimal weight of the weightiest 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).


 * The generalized travelling salesman problem deals with "states" that have (one or more) "cities" and the salesman has to visit exactly one "city" from each "state". Also known as the "travelling politician problem". One application is encountered in ordering a solution to the cutting stock problem in order to minimise knife changes. Another is concerned with drilling in semiconductor manufacturing, see e.g. . Surprisingly, Behzad and Modarres demonstrated that the generalised travelling salesman problem can be transformed into a standard travelling salesman problem with the same number of cities, but a modified distance matrix.

Computing a solution
The traditional lines of attacking 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.

Computational complexity
The problem has been shown to be NP-hard (more precisely, it is complete for the complexity class FPNP; see function problem), and the decision problem version ("given the costs and a number x, decide whether there is a round-trip route cheaper than x") is NP-complete. The bottleneck travelling 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 there is an optimal tour that visits each city only once (otherwise, by the triangle inequality, a shortcut that skips a repeated visit would not increase the tour length).

Complexity of approximation
In the general case, finding a shortest travelling salesman tour is NPO-complete. If the distance measure is a metric and symmetric, the problem becomes APX-complete and Christofides’s algorithm approximates it within 3/2.

If the distances are restricted to 1 and 2 (but still are a metric) the approximation ratio becomes 7/6. In the asymmetric, metric case, only logarithmic performance guarantees are known, the best current algorithm achieves performance ratio 0.814 log n; it is an open question if a constant factor approximation exists.

The corresponding maximization problem of finding the longest travelling salesman tour is approximable within 63/38. If the distance function is symmetric, the longest tour can be approximated within 4/3 by a deterministic algorithm and within $$(33+\epsilon)/25$$ by a randomised algorithm.

Exact algorithms
The most direct solution would be to try all permutations (ordered combinations) and see which one is cheapest (using brute force search). The running time for this approach lies within a polynomial factor of $$O(n!)$$, the factorial of the number of cities, so this solution becomes impractical even for only 20 cities. One of the earliest applications of dynamic programming is an algorithm that solves the problem in time $$O(n^2 2^n)$$

The dynamic programming solution requires exponential space. Using inclusion–exclusion, the problem can be solved in time within a polynomial factor of $$2^n$$ and polynomial space.

Improving these time bounds seems to be difficult. For example, it is an open problem if there exists an exact algorithm for TSP that runs in time $$O(1.9999^n)$$

Other approaches include:


 * 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 200 cities.

An exact solution for 15,112 German towns 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 travelling salesman problem of visiting all 24,978 towns in Sweden was solved: a tour of length approximately 72,500 kilometers was found and it was proven that no shorter tour exists.
 * Implementations of branch-and-bound and problem-specific cut generation; this is the method of choice for solving large instances. This approach holds the current record, solving an instance with 85,900 cities, see

In March 2005, the travelling salesman problem of visiting all 33,810 points in a circuit board was solved using Concorde TSP Solver: 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). In April 2006 an instance with 85,900 points was solved using Concorde TSP Solver, taking over 136 CPU years, see.

Heuristic and approximation algorithms
Various heuristics and approximation algorithms, which quickly yield good solutions 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 (or so-called greedy algorithm) lets the salesman choose the nearest unvisited city as his next move. This algorithm quickly yields an effectively short route. For N cities randomly distributed on a plane, the algorithm averagely yields length = 1.25 * exact_shortest_length.

However, there exist many specially arranged city distributions which make the NN algorithm give the worst route (Gutin, Yeo, and Zverovich, 2002). This is true for both asymmetric and symmetric TSPs (Gutin and Yeo, 2007).

Rosenkrantz et al. [1977] showed that the NN algorithm has the approximation factor $$\Theta(\log |V|)$$ for instances satisfying the triangle inequality.

Recently a new constructive heuristic, Match Twice and Stitch (MTS) (Kahng, Reda 2004 ), has been proposed. MTS has been shown to empirically outperform all existing tour construction heuristics. MTS performs two sequential matchings, where the second matching is executed after deleting all the edges of the first matching, to yield a set of cycles. The cycles are then stitched to produce the final tour.

Iterative improvement

 * Pairwise exchange, or Lin–Kernighan heuristics.


 * The pairwise exchange or '2-opt' technique involves iteratively removing two edges and replacing these with two different edges that reconnect the fragments created by edge removal into a new and shorter tour. This is a special case of the k-opt method. Note that the label 'Lin–Kernighan' is an often heard misnomer for 2-opt. Lin–Kernighan is actually a more general method.


 * 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 &minus; 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. The k-opt technique is a special case of the V-opt or variable-opt technique. The most popular of the k-opt methods are 3-opt, and these were introduced by Shen Lin of Bell Labs in 1965. There is a special case of 3-opt where the edges are not disjoint (two of the edges are adjacent to one another). In practice, it is often possible to achieve substantial improvement over 2-opt without the combinatorial cost of the general 3-opt by restricting the 3-changes to this special subset where two of the removed edges are adjacent. This so-called two-and-a-half-opt typically falls roughly midway between 2-opt and 3-opt, both in terms of the quality of tours achieved and the time required to achieve those tours.


 * V-opt heuristic


 * The variable-opt method is related to, and a generalization of the k-opt method. Whereas the k-opt methods remove a fixed number (k) of edges from the original tour, the variable-opt methods do not fix the size of the edge set to remove. Instead they grow the set as the search process continues. The best known method in this family is the Lin–Kernighan method (mentioned above as a misnomer for 2-opt). Shen Lin and Brian Kernighan first published their method in 1972, and it was the most reliable heuristic for solving travelling salesman problems for nearly two decades. More advanced variable-opt methods were developed at Bell Labs in the late 1980s by David Johnson and his research team. These methods (sometimes called Lin–Kernighan–Johnson) build on the Lin–Kernighan method, adding ideas from tabu search and evolutionary computing. The basic Lin–Kernighan technique gives results that are guaranteed to be at least 3-opt. The Lin–Kernighan–Johnson methods compute a Lin–Kernighan tour, and then perturb the tour by what has been described as a mutation that removes at least four edges and reconnecting the tour in a different way, then v-opting the new tour. The mutation is often enough to move the tour from the local minimum identified by Lin–Kernighan. V-opt methods are widely considered the most powerful heuristics for the problem, and are able to address special cases, such as the Hamilton Cycle Problem and other non-metric TSPs that other heuristics fail on. For many years Lin–Kernighan–Johnson had identified optimal solutions for all TSPs where an optimal solution was known and had identified the best known solutions for all other TSPs on which the method had been tried.

Randomised improvement

 * Optimised Markov chain algorithms which use 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 further it is a global minimum with high probability (where 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 optimization such as genetic algorithms, simulated annealing, Tabu search, ant colony optimization, and the cross entropy method.

Example: Using inversion to find an approximate solution
While David Goldberg wrote about this method in terms of DNA, it is repeated here in terms of a travelling salesman. This method shortens the path by opening loops at random. After all the loops are opened, one final loop remains that defines a perimeter. While a perimeter is not always the optimal path, it is usually very close. In a special case when all towns are equidistantly placed along a circle, the perimeter is the optimal solution.

Suppose a salesman has a deck of 60 cards, each with the location of a different town. He wants to find the optimal path by arranging the cards, calculating the length, and comparing results over and over. If the hometown is not counted, then the number of possible permutations becomes 60*59*58*...*4*3 or 4.2×1081. This is more than the estimated number of atoms in the observable universe! With this method, he likely won't find the optimal path or even get close before dying of old age.

Using inversion instead, the salesman starts with the shuffled cards laid out in front of him. First, he calculates the path length. Then he randomly picks four contiguous cards and inverts their order (i.e. A B C D -> D C B A). After recalculating the path length, he keeps the new order if it is shorter as in the figure below with 6 towns (note: the hometown, point 1, is not counted).


 * [[Image:Rundtur.GIF]]

For 60 towns, a loop is randomly opened more than once in every 602 attempts. If the salesman has a friend with an identical card deck, then they can double their chances of opening a loop. They both lay out the cards in the same order, then randomly select different sets of four cards to invert. Naturally, more friends working in parallel will continue to improve the odds. The figure below shows a randomly generated path connecting 60 points.


 * [[Image:Slumptur.GIF]]

Each iteration has 180 parallel inversion operations (i.e. friends), 60 of which are combined to make a new path. After about 1500 iterations, all loops have been opened, reducing the tour to 1/5 of the original length. See figure below.


 * [[Image:Kortad-rundtur.GIF]]

Out of 4.2×1081 possible permutations, the inversion method found an approximately optimal path with 180*1500 = 2.7×105 random inversion operations.

Ant colony optimization
Artificial intelligence researcher Marco Dorigo described in 1997 a method of heuristically generating "good solutions" to the TSP using a simulation of an ant colony called ACS (Ant Colony System). It uses some of the same ideas used by real ants to find short paths between food sources and their nest, an emergent behaviour resulting from each ant's preference to follow trail pheromones deposited by other ants.

ACS sends out a large number of virtual ant agents to explore many possible routes on the map. Each ant probabilistically chooses the next city to visit based on a heuristic combining the distance to the city and the amount of virtual pheromone deposited on the edge to the city. The ants explore, depositing pheromone on each edge that they cross, until they have all completed a tour. At this point the ant which completed the shortest tour deposits virtual pheromone along its complete tour route (global trail updating). The amount of pheromone deposited is inversely proportional to the tour length; the shorter the tour, the more it deposits.



Metric TSP
A very natural restriction of the TSP is to require that the distances between cities form a metric, i.e., they satisfy 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 that 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.


 * Construct the minimum spanning tree.


 * 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.


 * Find a Eulerian cycle in it. Clearly, its length is twice the length of the tree.


 * 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. 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; the Christofides algorithm was initially referred to as the Christofides heuristic.

In the special case that distances between cities are all either one or two (and thus the triangle inequality is necessarily satisfied), there is a polynomial-time approximation algorithm that finds a tour of length at most 8/7 times the optimal tour length. However, it is a long-standing (since 1975) open problem to improve the Christofides approximation factor of 1.5 for general metric TSP to a smaller constant. It is known that, unless P = NP, there is no polynomial-time algorithm that finds a tour of length at most 220/219=1.00456… times the optimal tour's length. In the case of 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.

Euclidean TSP
Euclidean TSP, or planar TSP, is the TSP with the distance being the ordinary Euclidean distance.

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 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 in O(n (log n)^O(c)) time; this is called a polynomial-time approximation scheme In practice, heuristics with weaker guarantees continue to be used.

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.


 * {| class="wikitable"


 * - align="center"
 * + Asymmetric Path Weights

! !! A !! B !! C
 * - align="center"

! A
 * || 1 || 2
 * - align="center"

! B
 * 6 || || 3
 * - align="center"

! 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 -∞.


 * {| class="wikitable"


 * - align="center" class="wikitable"
 * + Symmetric Path Weights

! !! A !! B !! C !! A' !! B' !! C'
 * - align="center"

! A
 * || || || -∞ || 6 || 5
 * - align="center"

! B
 * || || || 1 || -∞ || 4
 * - align="center"

! C
 * || || || 2 || 3 || -∞
 * - align="center"

! A'
 * -∞ || 1 || 2 || || ||
 * - align="center"

! B'
 * 6 || -∞ || 3 || || ||
 * - align="center"

! 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 9] and A-C-B-A [score 12]. 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.

Benchmarks
For benchmarking of TSP algorithms, TSPLIB is 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.

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 for random pointset in a square
Consider N points randomly distributed in a 1 x 1 square with N>>1.

It is known that, for N points in a unit square, the TSP always has length at most proportional to the square root of N, and that random pointset will have length at least proportional to the square root. However, the constants of proportionality, both for worst-case pointset and for random pointset, are unknown yet.

Lower bound
A lower bound of the shortest tour length is $$\frac{1}{2} \sqrt{N}$$, obtained by assuming i be a point in the tour sequence and i has its nearest neighbor as its next in the tour.

A better lower bound is $$\left({\frac{1}{4} + \frac{3}{8}}\right)\sqrt{N}$$, obtained by assuming i's next is i's nearest, and i's previous is i's second nearest.

Divide tour sequence into 2 parts as before_i and after_i with each part contains N/2 points. Delete before_i points from the square to build a diluted environment. Now the best next point that i can find in after_i part is $$\sqrt{\frac{1}{2N}}$$ away. This leads to a lower bound $$\sqrt{\frac{N}{2}}$$.

$$0.7080\sqrt{N}+0.522$$, where 0.522 comes from the points near square boundary which have fewer neighbors.
 * David S. Johnson obtained a lower bound by computer experiment:

$$0.7078\sqrt{N}+0.551$$
 * Christine L. Valenzuela and Antonia J. Jones obtained another lower bound by computer experiment:

Upper bound
By applying simulated annealing method on samples of N=40000, computer shows an upper bound

$$\left(\sqrt{\frac{N}{2}}+0.72\right) \cdot 1.015 $$, where 0.72 comes from boundary effect.

Because the actual solution is only the shortest path, for the purposes of programmatic search another upper bound is the length of any previously discovered approximation.