## Traveling Salesman Problem

The traveling salesman problem (TSP) asks the question, "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city and returns to the origin city?".

## This project

- The goal of this site is to be an educational resource to help visualize, learn, and develop different algorithms for the traveling salesman problem in a way that's easily accessible
- As you apply different algorithms, the current best path is saved and used as input to whatever you run next. (e.g. shortest path first -> branch and bound). The order in which you apply different algorithms to the problem is sometimes referred to the meta-heuristic strategy.

## Heuristic algorithms

Heuristic algorithms attempt to find a good approximation of the optimal path within a more reasonable amount of time.

Construction - Build a path (e.g. shortest path)

- Shortest Path

## Arbitrary Insertion

Furthest insertion.

- Nearest Insertion
- Convex Hull Insertion*
- Simulated Annealing*

Improvement - Attempt to take an existing constructed path and improve on it

- 2-Opt Inversion
- 2-Opt Reciprcal Exchange*

## Exhaustive algorithms

Exhaustive algorithms will always find the best possible solution by evaluating every possible path. These algorithms are typically significantly more expensive then the heuristic algorithms discussed next. The exhaustive algorithms implemented so far include:

- Random Paths

## Depth First Search (Brute Force)

- Branch and Bound (Cost)
- Branch and Bound (Cost, Intersections)*

## Dependencies

These are the main tools used to build this site:

- web workers
- material-ui

## Contributing

Pull requests are always welcome! Also, feel free to raise any ideas, suggestions, or bugs as an issue.

This is an exhaustive, brute-force algorithm. It is guaranteed to find the best possible path, however depending on the number of points in the traveling salesman problem it is likely impractical. For example,

- With 10 points there are 181,400 paths to evaluate.
- With 11 points, there are 1,814,000.
- With 12 points there are 19,960,000.
- With 20 points there are 60,820,000,000,000,000, give or take.
- With 25 points there are 310,200,000,000,000,000,000,000, give or take.

This is factorial growth, and it quickly makes the TSP impractical to brute force. That is why heuristics exist to give a good approximation of the best path, but it is very difficult to determine without a doubt what the best path is for a reasonably sized traveling salesman problem.

This is a recursive, depth-first-search algorithm, as follows:

- From the starting point
- For all other points not visited
- If there are no points left return the current cost/path
- Else, go to every remaining point and
- Mark that point as visited
- " recurse " through those paths (go back to 1. )

## Implementation

Nearest neighbor.

This is a heuristic, greedy algorithm also known as nearest neighbor. It continually chooses the best looking option from the current state.

- sort the remaining available points based on cost (distance)
- Choose the closest point and go there
- Chosen point is no longer an "available point"
- Continue this way until there are no available points, and then return to the start.

## Two-Opt inversion

This algorithm is also known as 2-opt, 2-opt mutation, and cross-aversion. The general goal is to find places where the path crosses over itself, and then "undo" that crossing. It repeats until there are no crossings. A characteristic of this algorithm is that afterwards the path is guaranteed to have no crossings.

- While a better path has not been found.
- For each pair of points:
- Reverse the path between the selected points.
- If the new path is cheaper (shorter), keep it and continue searching. Remember that we found a better path.
- If not, revert the path and continue searching.

This is an impractical, albeit exhaustive algorithm. It is here only for demonstration purposes, but will not find a reasonable path for traveling salesman problems above 7 or 8 points.

I consider it exhaustive because if it runs for infinity, eventually it will encounter every possible path.

- From the starting path
- Randomly shuffle the path
- If it's better, keep it
- If not, ditch it and keep going

This is a heuristic construction algorithm. It select a random point, and then figures out where the best place to put it will be.

- First, go to the closest point
- Choose a random point to go to
- Find the cheapest place to add it in the path
- Continue from #3 until there are no available points, and then return to the start.

## Two-Opt Reciprocal Exchange

This algorithm is similar to the 2-opt mutation or inversion algorithm, although generally will find a less optimal path. However, the computational cost of calculating new solutions is less intensive.

The big difference with 2-opt mutation is not reversing the path between the 2 points. This algorithm is not always going to find a path that doesn't cross itself.

It could be worthwhile to try this algorithm prior to 2-opt inversion because of the cheaper cost of calculation, but probably not.

- Swap the points in the path. That is, go to point B before point A, continue along the same path, and go to point A where point B was.

## Branch and Bound on Cost

This is a recursive algorithm, similar to depth first search, that is guaranteed to find the optimal solution.

The candidate solution space is generated by systematically traversing possible paths, and discarding large subsets of fruitless candidates by comparing the current solution to an upper and lower bound. In this case, the upper bound is the best path found so far.

While evaluating paths, if at any point the current solution is already more expensive (longer) than the best complete path discovered, there is no point continuing.

For example, imagine:

- A -> B -> C -> D -> E -> A was already found with a cost of 100.
- We are evaluating A -> C -> E, which has a cost of 110. There is no point evaluating the remaining solutions.

Instead of continuing to evaluate all of the child solutions from here, we can go down a different path, eliminating candidates not worth evaluating:

- A -> C -> E -> D -> B -> A
- A -> C -> E -> B -> D -> A

Implementation is very similar to depth first search, with the exception that we cut paths that are already longer than the current best.

This is a heuristic construction algorithm. It selects the closest point to the path, and then figures out where the best place to put it will be.

- Choose the point that is nearest to the current path

## Branch and Bound (Cost, Intersections)

This is the same as branch and bound on cost, with an additional heuristic added to further minimize the search space.

While traversing paths, if at any point the path intersects (crosses over) itself, than backtrack and try the next way. It's been proven that an optimal path will never contain crossings.

Implementation is almost identical to branch and bound on cost only, with the added heuristic below:

This is a heuristic construction algorithm. It selects the furthest point from the path, and then figures out where the best place to put it will be.

- Choose the point that is furthest from any of the points on the path

## Convex Hull

This is a heuristic construction algorithm. It starts by building the convex hull , and adding interior points from there. This implmentation uses another heuristic for insertion based on the ratio of the cost of adding the new point to the overall length of the segment, however any insertion algorithm could be applied after building the hull.

There are a number of algorithms to determine the convex hull. This implementation uses the gift wrapping algorithm .

In essence, the steps are:

- Determine the leftmost point
- Continually add the most counterclockwise point until the convex hull is formed
- For each remaining point p, find the segment i => j in the hull that minimizes cost(i -> p) + cost(p -> j) - cost(i -> j)
- Of those, choose p that minimizes cost(i -> p -> j) / cost(i -> j)
- Add p to the path between i and j
- Repeat from #3 until there are no remaining points

## Simulated Annealing

Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem.

For problems where finding an approximate global optimum is more important than finding a precise local optimum in a fixed amount of time, simulated annealing may be preferable to exact algorithms

## Visualize algorithms for the traveling salesman problem. Use the controls below to plot points, choose an algorithm, and control execution. (Hint: try a construction alogorithm followed by an improvement algorithm)

- ENGINEERING
- What's the point?
- life cycles
- engineering
- data visualization
- biogeochemistry
- social science
- computer science
- epidemiology

## 11 Animated Algorithms for the Traveling Salesman Problem

For the visual learners, here’s an animated collection of some well-known heuristics and algorithms in action.

## TSP Algorithms and heuristics

Although we haven’t been able to quickly find optimal solutions to NP problems like the Traveling Salesman Problem, "good-enough" solutions to NP problems can be quickly found [1] .

For the visual learners, here’s an animated collection of some well-known heuristics and algorithms in action. Researchers often use these methods as sub-routines for their own algorithms and heuristics. This is not an exhaustive list.

For ease of visual comparison we use Dantzig49 as the common TSP problem, in Euclidean space. Dantzig49 has 49 cities — one city in each contiguous US State, plus Washington DC.

## 1: Greedy Algorithm

A greedy algorithm is a general term for algorithms that try to add the lowest cost possible in each iteration, even if they result in sub-optimal combinations.

In this example, all possible edges are sorted by distance, shortest to longest. Then the shortest edge that will neither create a vertex with more than 2 edges, nor a cycle with less than the total number of cities is added. This is repeated until we have a cycle containing all of the cities.

Although all the heuristics here cannot guarantee an optimal solution, greedy algorithms are known to be especially sub-optimal for the TSP.

## 2: Nearest Neighbor

The nearest neighbor heuristic is another greedy algorithm, or what some may call naive. It starts at one city and connects with the closest unvisited city. It repeats until every city has been visited. It then returns to the starting city.

Karl Menger, who first defined the TSP, noted that nearest neighbor is a sub-optimal method:

"The rule that one first should go from the staring point to the closest point, then to the point closest to this, etc., in general does not yield the shortest route."

The time complexity of the nearest neighbor algorithm is O(n^2) . The number of computations required will not grow faster than n^2.

## 3: Nearest Insertion

Insertion algorithms add new points between existing points on a tour as it grows.

One implementation of Nearest Insertion begins with two cities. It then repeatedly finds the city not already in the tour that is closest to any city in the tour, and places it between whichever two cities would cause the resulting tour to be the shortest possible. It stops when no more insertions remain.

The nearest insertion algorithm is O(n^2)

## 4: Cheapest Insertion

Like Nearest Insertion, Cheapest Insertion also begins with two cities. It then finds the city not already in the tour that when placed between two connected cities in the subtour will result in the shortest possible tour. It inserts the city between the two connected cities, and repeats until there are no more insertions left.

The cheapest insertion algorithm is O(n^2 log2(n))

## 5: Random Insertion

Random Insertion also begins with two cities. It then randomly selects a city not already in the tour and inserts it between two cities in the tour. Rinse, wash, repeat.

Time complexity: O(n^2)

## 6: Farthest Insertion

Unlike the other insertions, Farthest Insertion begins with a city and connects it with the city that is furthest from it.

It then repeatedly finds the city not already in the tour that is furthest from any city in the tour, and places it between whichever two cities would cause the resulting tour to be the shortest possible.

## 7: Christofides Algorithm

Christofides algorithm is a heuristic with a 3/2 approximation guarantee. In the worst case the tour is no longer than 3/2 the length of the optimum tour.

Due to its speed and 3/2 approximation guarantee, Christofides algorithm is often used to construct an upper bound, as an initial tour which will be further optimized using tour improvement heuristics, or as an upper bound to help limit the search space for branch and cut techniques used in search of the optimal route.

For it to work, it requires distances between cities to be symmetric and obey the triangle inequality, which is what you'll find in a typical x,y coordinate plane (metric space). Published in 1976, it continues to hold the record for the best approximation ratio for metric space.

The algorithm is intricate [2] . Its time complexity is O(n^4)

A problem is called k-Optimal if we cannot improve the tour by switching k edges.

Each k-Opt iteration takes O(n^k) time.

2-Opt is a local search tour improvement algorithm proposed by Croes in 1958 [3] . It originates from the idea that tours with edges that cross over aren’t optimal. 2-opt will consider every possible 2-edge swap, swapping 2 edges when it results in an improved tour.

2-opt takes O(n^2) time per iteration.

3-opt is a generalization of 2-opt, where 3 edges are swapped at a time. When 3 edges are removed, there are 7 different ways of reconnecting them, so they're all considered.

The time complexity of 3-opt is O(n^3) for every 3-opt iteration.

## 10: Lin-Kernighan Heuristic

Lin-Kernighan is an optimized k-Opt tour-improvement heuristic. It takes a tour and tries to improve it.

By allowing some of the intermediate tours to be more costly than the initial tour, Lin-Kernighan can go well beyond the point where a simple 2-Opt would terminate [4] .

Implementations of the Lin-Kernighan heuristic such as Keld Helsgaun's LKH may use "walk" sequences of 2-Opt, 3-Opt, 4-Opt, 5-Opt, “kicks” to escape local minima, sensitivity analysis to direct and restrict the search, as well as other methods.

LKH has 2 versions; the original and LKH-2 released later. Although it's a heuristic and not an exact algorithm, it frequently produces optimal solutions. It has converged upon the optimum route of every tour with a known optimum length. At one point in time or another it has also set records for every problem with unknown optimums, such as the World TSP, which has 1,900,000 locations.

## 11: Chained Lin-Kernighan

Chained Lin-Kernighan is a tour improvement method built on top of the Lin-Kernighan heuristic:

## Lawrence Weru

Larry is a TEDx speaker, Harvard Medical School Dean's Scholar, Florida State University "Notable Nole," and has served as an invited speaker at Harvard, FSU, and USF. Larry's contributions are featured by Harvard University, Fast Company, and Gizmodo Japan, and cited in books by Routledge and No Starch Press. His stories and opinions are published in Slate, Vox, Toronto Star, Orlando Sentinel, and Vancouver Sun, among others. He illustrates the sciences for a more just and sustainable world.

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## Travelling Salesman

Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?

Touch the canvas in order to add a point. You cannot add points when solution is being calculated.

- Nearest Neighbour: Start at a particular node. Travel to the nearest node. Repeat. This algorithm is very simple to implement.
- Greedy Heuristic: Form the initial path with shorter edges, and connect them.
- Simulated Annealing: Inspired by the way crystals cool, this method tries to converge to a local minima while implementing a cooling schedule. In every step, two nodes from the path are switched, and the resulting path is accepted or rejected using Metropolis algorithm (please find intricate details here ).
- Genetic Algorithm: A poplation of entities whose genes comprise of paths, and whose fitness is based on how short the paths are. Fitter entities have greater chance of their genes making it into the offspring. Also uses mutation (to explore prospectively better avenues) and recombination (to combine the best of two genes).
- Ant Colony Optimization: Initially, a population of ants randomly traverse the graph. By gauging the shortness of each edge, they leave more pheromones along that edge. In the next iteration, at every node, the ants choose the edge on a probabilistic manner, weighted by pheromone strength. Both, genetic algorithm and ant colony optimization are called metaheuristic methods because they solve the problem at hand by working at a higher level, instead of directly tackling the problem.
- Christofides' Algorithm Considered the gold standard of solving the Travelling Salesman Problem, this algorithm utilizes insights from an easily solvable problem in graph theory (constructing a minimal spanning tree from a given graph) and manipulates it to arrive at (on average) comparatively shorter paths.
- I will try to make simulations of the optimized algorithms above, if time permits.
- When there are 8+ points, then an initial lag is expected before the solving process starts
- With 10 points, an average time of 49 minutes was required to find the solution to the route
- Simulation may lag on lower-end computers and phones
- Related: Ant Colony Optimization

Developed by ChanRT | Fork me at GitHub

- This website needs Javascript in order to be displayed properly.
- Javascript is currently deactivated in your browser. A manual for the activation of Javascript can be found here .

- Introduction
- Create game
- Optimal tour
- Description of the algorithm
- Calculation steps

## Shortest round trips

Welcome to the tsp game.

This website is about the so-called "Traveling Salesman Problem".

It deals with the question, how to plan a complete round trip through a certain number of cities to obtain the shortest tour possible. This question can be answered quite easily for four cities. However, it gets complicated when the number of cities is increased. There exist for example 181.440 different tours through just ten cities.

How can one find the shortest tour on twenty or even more cities?

For this reason, various algorithms have been invented, which try to solve the Traveling Salesman Problem as fast as possible.

## And now it is your turn!

Create a new game in the next tab and try to find a shortest tour yourself! Afterwards, you may have a look at the solution and the calculation steps of the applied algorithms.

## What would you like to do first?

## Repeat game

On this page you can create a new game, find your tour, see the solution, repeat the current game, test your logistic skills and draw the shortest round trip you can find..

Do not forget to visit every city exactly once!

## By changing the tab, the drawing mode will be terminated.

There is no way back to the "Play game" tab.

## Congratulations! Your tour is optimal.

Too bad unfortunately, your tour is not optimal., too bad unfortunately, your tour does not visit all nodes., too bad unfortunately, your tour contains subtours., how has the solution been calculated.

In this tab you may have a look at the optimal tour and the corresponding calculation steps.

In the Description of the algorithm you may find more details on the applied algorithms of the different steps.

## Graphs and algorithms

- 'Heuristics', which find a round trip, but do not indicate its optimality in relation to an optimal solution. Its length is always larger than the length of an optimal tour. It is than called 'upper bound'.
- "Exact solver", which converge to an optimal round trip slowly. Found paths are never longer than an optimal solution. Therefore, their length is called 'lower bound'. Their result is rarely a round trip through all nodes.

## Nearest neighbour algorithm

Multiple fragment algorithm.

- There are no more than two incident edges allowed for each node.
- No circles are allowed.

## Lagrangian relaxation

- Remove one node from the graph.
- Add edges between the remaining nodes in ascending order, like in the Multiple Fragment algorithm. There may be more than two edges incident to one node. However, circles are not allowed.
- Now add the removed node together with its two shortest incident edges to the graph. If a round trip is obtained, stop. There is no shorter round trip. Otherwise, one continues as follows.
- The aim is to have exactly two edges incident to one node. Therefore, find out by which number the number of incident edges differs from 2. Say four edges are incident to a node, then the result is 2. One incident edge leads to -1. Add the result to the length of all incident edges.
- Return to the first step.

## Branch and Bound

- e and f are forced edges. All other edges are permitted.
- e is forced and f permitted. Edges that have been free before this step will stay free.
- e is permitted. All other edges stay free, if they have been before.

## Origins of the Traveling Salesman Problem

The application, studying mathematics at the tum, a last remark about this page's goal and citations.

Chair M9 of Technische Universität München does research in the fields of discrete mathematics, applied geometry and the optimization of mathematical problems. The problem and corresponding algorithms presented here are a famous example for the application of mathematics on an applied problem (and can be assigned to the field of combinatorial optimization). This page shall help the reader to recognize the application of mathematics to real life problems and to understand easy mathematical solution methods used. The algorithms presented were chosen according to understandability, they do not represent fields of research of the chair.

To cite this page, please use the following information:

## There is no solution available.

The calculation of the solution is still in progress. Please try again later.

The calculation of the solution has been aborted. Please return to the Create game tab and start a new game.

## Traveling Salesperson Problem

1. introduction.

Traveling Salesperson Problem : TSP is a problem that tries to find a tour of minimum cost that visits every city once. In this visualization, it is assumed that the underlying graph is a complete graph with (near-)metric distance (meaning the distance function satisfies the triangle inequality) by taking the distance of two points and round it to the nearest integer.

## 2. Visualization

View the visualisation of TSP algorithm here.

Originally, all edges in the input graph are colored with the grey .

Throughout the visualization, traversed edge will be highlighted with orange .

There are two different sources for specifying an input graph:

- Draw Graph : You can put several points on the drawing box, but you must not draw any edge to ensure the (near-)metric property of the graph. After you have finished putting the points, the edges will be drawn automatically for you after.
- Example Graphs : You can select from the list of graphs to get you started.

## 4. Bruteforce

Bruteforce : It tries all (V-1)! permutation of vertices (not all V! since it does not matter where we start from). It enumerates all possibilities by doing a dfs search with parameters similar to those of Held-Karp algorithm, that is, the DFS search will return the value of the tour starting from current vertex to all vertices that we have not already visited.

Time complexity: O(V * (V - 1)!) = O(V!).

## 5. Dynamic Programming

Dynamic Programming : It uses a widely known algorithm called Held-Karp . In this visualization, it is implemented as a DFS search that is the same with the bruteforce algorithm, but with memoization to cache the answers. This dramatically brings down the run time complexity to O(2^V * V^2).

Time complexity: O(2^V * V^2).

Note that for N = 10, this algorithm takes roughly ~(100 * 2^10) = 102K operations while the bruteforce algorithm takes roughly ~(10!) = 3628800 operations, around 30 times faster.

## 6. Approximation

Approximation : There are two approximation algorithms available, a 2-approximation algorithm and a 1.5-approximation algorithm that is also well known as Christofides Algorithm.

## Traveling salesman problem

This web page is a duplicate of https://optimization.mccormick.northwestern.edu/index.php/Traveling_salesman_problems

Author: Jessica Yu (ChE 345 Spring 2014)

Steward: Dajun Yue, Fengqi You

The traveling salesman problem (TSP) is a widely studied combinatorial optimization problem, which, given a set of cities and a cost to travel from one city to another, seeks to identify the tour that will allow a salesman to visit each city only once, starting and ending in the same city, at the minimum cost. 1

- 2.1 Graph Theory
- 2.2 Classifications of the TSP
- 2.3 Variations of the TSP
- 3.1 aTSP ILP Formulation
- 3.2 sTSP ILP Formulation
- 4.1 Exact algorithms
- 4.2.1 Tour construction procedures
- 4.2.2 Tour improvement procedures
- 5 Applications
- 7 References

The origins of the traveling salesman problem are obscure; it is mentioned in an 1832 manual for traveling salesman, which included example tours of 45 German cities but gave no mathematical consideration. 2 W. R. Hamilton and Thomas Kirkman devised mathematical formulations of the problem in the 1800s. 2

It is believed that the general form was first studied by Karl Menger in Vienna and Harvard in the 1930s. 2,3

Hassler Whitney, who was working on his Ph.D. research at Harvard when Menger was a visiting lecturer, is believed to have posed the problem of finding the shortest route between the 48 states of the United States during either his 1931-1932 or 1934 seminar talks. 2 There is also uncertainty surrounding the individual who coined the name “traveling salesman problem” for Whitney’s problem. 2

The problem became increasingly popular in the 1950s and 1960s. Notably, George Dantzig, Delber R. Fulkerson, and Selmer M. Johnson at the RAND Corporation in Santa Monica, California solved the 48 state problem by formulating it as a linear programming problem. 2 The methods described in the paper set the foundation for future work in combinatorial optimization, especially highlighting the importance of cutting planes. 2,4

In the early 1970s, the concept of P vs. NP problems created buzz in the theoretical computer science community. In 1972, Richard Karp demonstrated that the Hamiltonian cycle problem was NP-complete, implying that the traveling salesman problem was NP-hard. 4

Increasingly sophisticated codes led to rapid increases in the sizes of the traveling salesman problems solved. Dantzig, Fulkerson, and Johnson had solved a 48 city instance of the problem in 1954. 5 Martin Grötechel more than doubled this 23 years later, solving a 120 city instance in 1977. 5 Enoch Crowder and Manfred W. Padberg again more than doubled this in just 3 years, with a 318 city solution. 5

In 1987, rapid improvements were made, culminating in a 2,392 city solution by Padberg and Giovanni Rinaldi. In the following two decades, David L. Appelgate, Robert E. Bixby, Vasek Chvátal, & William J. Cook led the cutting edge, solving a 7,397 city instance in 1994 up to the current largest solved problem of 24,978 cities in 2004. 5

## Description

Graph theory.

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = (V, E)} be a directed or undirected graph with set of vertices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and set of edges Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \{(x,y) | x, y \in V\}} . 3,6 Each edge Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e \in E} is assigned a cost Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_e} . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{H}} be the set of all Hamiltonian cycles, a cycle that visits each vertex exactly once, in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} . 6 The traveling salesman problem is to find the tour Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \in \mathbb{H}} such that the sum of the costs Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_e} in the tour is minimized.

Suppose graph Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a complete graph, where every pair of distinct vertices is connected by a unique edge. 6 Let the set of vertices be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = \{v_1, v_2, ..., v_n\}} . The cost matrix is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C = (c_{ij})_{n \times n}} where the cost of the edge joining node Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_i} to node Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_j} , denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ij}} , is given in entry Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (i,j)} .

In the context of the traveling salesman problem, the verticies correspond to cities and the edges correspond to the path between those cities. When modeled as a complete graph, paths that do not exist between cities can be modeled as edges of very large cost without loss of generality. 6 Minimizing the sum of the costs for Hamiltonian cycle is equivalent to identifying the shortest path in which each city is visiting only once.

## Classifications of the TSP

The TRP can be divided into two classes depending on the nature of the cost matrix. 3,6

- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is undirected
- Applies when the distance between cities is the same in both directions
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is directed
- Applies when there are differences in distances (e.g. one-way streets)

An ATSP can be formulated as an STSP by doubling the number of nodes. 6

## Variations of the TSP

Formulation.

Given a set of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} cities enumerated Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0, 1, ..., n-1} to be visited with the distance between each pair of cities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ij}} . 1 Introduce decision variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{ij}} for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (i,j)} such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{ij} = \begin{cases} 1 ~~\text{if city }j\text{ is visited immediately after city }i\\ 0 ~~\text{otherwise} \end{cases} }

The objective function is then given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{min} \sum_i \sum_j c_{ij}y_{ij} }

To ensure that the result is a valid tour, several contraints must be added. 1,3

There are several other formulations for the subtour elimnation contraint, including circuit packing contraints, MTZ constraints, and network flow constraints.

## aTSP ILP Formulation

The integer linear programming formulation for an aTSP is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \text{min} & ~~ \sum_i \sum_j c_{ij}y_{ij}\\ \text{s.t} & ~~ \sum_j y_{ij} = 1, ~~ i = 0, 1, ..., n-1 \\ & ~~ \sum_i y_{ij} = 1, ~~ j = 0, 1, ..., n-1 \\ & ~~ \sum_i \sum_j y_{ij} \le |S| - 1 ~~ S \subset V, 2 \le |S| \le n - 2 \\ & ~~ y_{ij} \in \{0,1\}, ~ \forall i,j \in E \\ \end{align} }

## sTSP ILP Formulation

The symmetric case is a special case of the asymmetric case and the above formulation is valid. 3, 6 The integer linear programming formulation for an sTSP is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \text{min} & ~~ \sum_i \sum_j c_{ij}y_{ij}\\ \text{s.t} & ~~ \sum_{i < k} y_{ik} + \sum_{j > k} y_{kj} = 2, ~~ k \in V \\ & ~~ \sum_i \sum_j y_{ij} \le |S| - 1 ~~ S \subset V, 3 \le |S| \le n - 3 \\ & ~~ y_{ij} \in \{0,1\} ~ \forall i,j \in E \\ \end{align} }

## Exact algorithms

The most direct solution algorithm is a complete enumeration of all possible path to determine the path of least cost. However, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} cities, the problem is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(n!)} time, and this method is practical only for extremely small values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .

Branch-and-bound algorithms are commonly used to find solutions for TSPs. 7 The ILP is first relaxed and solved as an LP using the Simplex method, then feasibility is regained by enumeration of the integer variables. 7

Other exact solution methods include the cutting plane method and branch-and-cut. 8

## Heuristic algorithms

Given that the TSP is an NP-hard problem, heuristic algorithms are commonly used to give a approximate solutions that are good, though not necessarily optimal. The algorithms do not guarantee an optimal solution, but gives near-optimal solutions in reasonable computational time. 3 The Held-Karp lower bound can be calculated and used to judge the performance of a heuristic algorithm. 3

There are two general heuristic classifications 7 :

- Tour construction procedures where a solution is gradually built by adding a new vertex at each step
- Tour improvement procedures where a feasbile solution is improved upon by performing various exchanges

The best methods tend to be composite algorithms that combine these features. 7

## Tour construction procedures

Tour improvement procedures, applications.

The importance of the traveling salesman problem is two fold. First its ubiquity as a platform for the study of general methods than can then be applied to a variety of other discrete optimization problems. 5 Second is its diverse range of applications, in fields including mathematics, computer science, genetics, and engineering. 5,6

Suppose a Northwestern student, who lives in Foster-Walker , has to accomplish the following tasks:

- Drop off a homework set at Tech
- Work out a SPAC
- Complete a group project at Annenberg

Distances between buildings can be found using Google Maps. Note that there is particularly strong western wind and walking east takes 1.5 times as long.

It is the middle of winter and the student wants to spend the least possible time walking. Determine the path the student should take in order to minimize walking time, starting and ending at Foster-Walker.

Start with the cost matrix (with altered distances taken into account):

Method 1: Complete Enumeration

All possible paths are considered and the path of least cost is the optimal solution. Note that this method is only feasible given the small size of the problem.

From inspection, we see that Path 4 is the shortest. So, the student should walk 2.28 miles in the following order: Foster-Walker → Annenberg → SPAC → Tech → Foster-Walker

Method 2: Nearest neighbor

Starting from Foster-Walker, the next building is simply the closest building that has not yet been visited. With only four nodes, this can be done by inspection:

- Smallest distance is from Foster-Walker is to Annenberg
- Smallest distance from Annenberg is to Tech
- Smallest distance from Tech is to Annenberg ( creates a subtour, therefore skip )
- Next smallest distance from Tech is to Foster-Walker ( creates a subtour, therefore skip )
- Next smallest distance from Tech is to SPAC
- Smallest distance from SPAC is to Annenberg ( creates a subtour, therefore skip )
- Next smallest distance from SPAC is to Tech ( creates a subtour, therefore skip )
- Next smallest distance from SPAC is to Foster-Walker

So, the student would walk 2.54 miles in the following order: Foster-Walker → Annenberg → Tech → SPAC → Foster-Walker

Method 3: Greedy

With this method, the shortest paths that do not create a subtour are selected until a complete tour is created.

- Smallest distance is Annenberg → Tech
- Next smallest is SPAC → Annenberg
- Next smallest is Tech → Annenberg ( creates a subtour, therefore skip )
- Next smallest is Anneberg → Foster-Walker ( creates a subtour, therefore skip )
- Next smallest is SPAC → Tech ( creates a subtour, therefore skip )
- Next smallest is Tech → Foster-Walker
- Next smallest is Annenberg → SPAC ( creates a subtour, therefore skip )
- Next smallest is Foster-Walker → Annenberg ( creates a subtour, therefore skip )
- Next smallest is Tech → SPAC ( creates a subtour, therefore skip )
- Next smallest is Foster-Walker → Tech ( creates a subtour, therefore skip )
- Next smallest is SPAC → Foster-Walker ( creates a subtour, therefore skip )
- Next smallest is Foster-Walker → SPAC

So, the student would walk 2.40 miles in the following order: Foster-Walker → SPAC → Annenberg → Tech → Foster-Walker

As we can see in the figure to the right, the heuristic methods did not give the optimal solution. That is not to say that heuristics can never give the optimal solution, just that it is not guaranteed.

Both the optimal and the nearest neighbor algorithms suggest that Annenberg is the optimal first building to visit. However, the optimal solution then goes to SPAC, while both heuristic methods suggest Tech. This is in part due to the large cost of SPAC → Foster-Walker. The heuristic algorithms cannot take this future cost into account, and therefore fall into that local optimum.

We note that the nearest neighbor and greedy algorithms give solutions that are 11.4% and 5.3%, respectively, above the optimal solution. In the scale of this problem, this corresponds to fractions of a mile. We also note that neither heuristic gave the worst case result, Foster-Walker → SPAC → Tech → Annenberg → Foster-Walker.

Only tour building heuristics were used. Combined with a tour improvement algorithm (such as 2-opt or simulated annealing), we imagine that we may be able to locate solutions that are closer to the optimum.

The exact algorithm used was complete enumeration, but we note that this is impractical even for 7 nodes (6! or 720 different possibilities). Commonly, the problem would be formulated and solved as an ILP to obtain exact solutions.

- Vanderbei, R. J. (2001). Linear programming: Foundations and extensions (2nd ed.). Boston: Kluwer Academic.
- Schrijver, A. (n.d.). On the history of combinatorial optimization (till 1960).
- Matai, R., Singh, S., & Lal, M. (2010). Traveling salesman problem: An overview of applications, formulations, and solution approaches. In D. Davendra (Ed.), Traveling Salesman Problem, Theory and Applications . InTech.
- Junger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G., Rinaldi, G., & Wolsey, L. (Eds.). (2009). 50 years of integer programming, 1958-2008: The early years and state-of-the-art surveys . Heidelberg: Springer.
- Cook, W. (2007). History of the TSP. The Traveling Salesman Problem . Retrieved from http://www.math.uwaterloo.ca/tsp/history/index.htm
- Punnen, A. P. (2002). The traveling salesman problem: Applications, formulations and variations. In G. Gutin & A. P. Punnen (Eds.), The Traveling Salesman Problem and its Variations . Netherlands: Kluwer Academic Publishers.
- Laporte, G. (1992). The traveling salesman problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59 (2), 231–247.
- Goyal, S. (n.d.). A suvey on travlling salesman problem.
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🗺️ Visualize and control algorithms for the traveling salesman problem

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Travelling Salesman Problem system in JavaScript with Functional Programming

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Genetic Algorithm applied to to the Travelling Salesman Problem

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Interactive travelling salesman problem solver. Branch & bound | Simulated annealing.

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Resolução do problema do caixeiro viajante utilizando algoritmo genético

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## echoSTROMLOS / TSPwGA-Simulation

This project uses a Genetic Algorithm to find near-optimal solutions to the Traveling Salesman Problem (TSP). Users can visualize the evolving routes and compare them to the optimal solution found using Bruteforce.

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Implementation of the travelling salesman problem using a genetic evolutionary approach

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A project to visually solve the travelling salesman problem

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This is the improved version of the brute force approach which takes less time to give results for greater number of cities.

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## Traveling Salesman Problem with Python: Greedy and Brute Force

Just imagine yourself as a delivery courier. You have to deliver multiple parcels to many different spots. You also aim to minimize the fuel costs and time of traveling to maximize profit. This generally creates confusion about where to deliver parcels first and which routes to take.

In this article, we will learn about the famous problem of Travelling Salesman. We will solve this issue using Python programming language. We will also try to plot the best route to be taken and calculate the minimum distance to be traveled.

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Recommended: Delivery Route Optimization using Python: A Step-by-Step Guide

## Solving the Traveling Salesman Problem with Python

The problem statement is that a salesman has to travel to the Indian cities of Mumbai, Delhi, Bangalore, Hyderabad, Ahmedabad, Chennai, Kolkata, Surat, Pune, and Jaipur to sell some products.

The objective of the problem is to minimize the total distance travelled by the salesman.

This describes our problem statement. Now let’s move on to how to solve this problem using Python programming language.

We will look at two ways of solving the problem – using the Greedy algorithm and the Brute Force method ( this method is the most preferred one as it provides more optimal solutions ).

## Example Usage and Visualization of the Greedy Algorithm

Let us look at the code of the Simple Greedy algorithm.

In the above example, we randomly input some coordinates and calculated the best route. Let us look at the output of this method.

Thus, according to the Greedy algorithm, we will travel to city 1 first, and cities 1,3,2 after that.

## Brute Force Approach to the Traveling Salesman Problem

Let us look at the code of the Brute Force Method.

Let us look at the output of the Brute Force method.

Thus, according to the Brute force method, the minimum distance is approximately 230 units.

## Greedy Algorithm vs. Brute Force Method

The difference between the Greedy algorithm and the Brute Force method is that the Greedy algorithm builds up the solution step by step whereas in the Brute Force method, all the permutations of the solution are found and then the solution with minimum distance is selected.

Here you go! Now you know the Travelling Salesman Problem and how to solve it using Python. In this article, you learned about two methods of approach i.e. Greedy algorithm and Travelling Salesman problem.

Hope you enjoyed reading it!

Recommended: Backtracking Line Search Algorithm for Unconstrained Optimization

Recommended: Large integer handling in Python (optimization)

## Leveraging a Genetic Algorithm for Route Finding

Solving the traveling salesman problem with a genetic algorithm, applying genetic algorithms to the traveling salesman problem, shortest route in current generation, shortest route found, algorithm parameters.

An explanation of the algorithm parameters and implementation can be found here .

## Instructions

- Press START to begin searching for shortest route between the default provided points.
- Algorithm parameters can be changed using the interface to the left.
- To run the algorithm against a custom set of points, click CLEAR and then click on the canvas to specify points.
- The app must be RESET before algorithm parameters are altered for the changes to take effect.

## IMAGES

## VIDEO

## COMMENTS

Show Evaluated Steps. Points. Number of random points. Possible Paths: 1.524 x 1029. Dark Mode. Interactive solver for the traveling salesman problem to visualize different algorithms. Includes various Heuristic and Exhaustive algorithms.

1x. Traveling Salesperson Problem: TSP is a problem that tries to find a tour of minimum cost that visits every city once. In this visualization, it is assumed that the underlying graph is a complete graph with (near-)metric distance (meaning the distance function satisfies the triangle inequality) by taking the distance of two points and round ...

2-Opt is a local search tour improvement algorithm proposed by Croes in 1958 [3]. It originates from the idea that tours with edges that cross over aren't optimal. 2-opt will consider every possible 2-edge swap, swapping 2 edges when it results in an improved tour. 2-Opt. 2-opt takes O(n^2) time per iteration.

Considered the gold standard of solving the Travelling Salesman Problem, this algorithm utilizes insights from an easily solvable problem in graph theory (constructing a minimal spanning tree from a given graph) and manipulates it to arrive at (on average) comparatively shorter paths. I will try to make simulations of the optimized algorithms ...

Visually compares Greedy, Local Search, and Simulated Annealing strategies for addressing the Traveling Salesman problem.Thanks to the Discrete Optimization ...

The goal of this site is to be an educational resource to help visualize, learn, and develop different algorithms for the traveling salesman problem in a way that's easily accessible. As you apply different algorithms, the current best path is saved and used as input to whatever you run next. The order in which you apply different algorithms to ...

Welcome to the TSP game! This website is about the so-called "Traveling Salesman Problem". It deals with the question, how to plan a complete round trip through a certain number of cities to obtain the shortest tour possible. This question can be answered quite easily for four cities.

The results tab contains a list of results of algorithms. The result of an algorithm is added to this list once it has finished running. An item in the list contains five things: the name of the algorithm run; the length of the tour found by the algorithm; the runtime of the algorithm; the resulting tour as a list of vertices; and a button to view the tour found by the algorithm.

Traveling Salesperson Problem 1. Introduction Traveling Salesperson Problem: TSP is a problem that tries to find a tour of minimum cost that visits every city once.In this visualization, it is assumed that the underlying graph is a complete graph with (near-)metric distance (meaning the distance function satisfies the triangle inequality) by taking the distance of two points and round it to ...

The traveling salesman problem (TSP) is a widely studied combinatorial optimization problem, which, given a set of cities and a cost to travel from one city to another, seeks to identify the tour that will allow a salesman to visit each city only once, starting and ending in the same city, at the minimum cost. 1.

This video illustrates how the traveling salesman problem (TSP) can be solved (an optimal solution can be approached) by simulated annealing. Music: © Setuni...

Traveling Salesman Problem Visualization. An interactive learning tool for the Traveling Salesman Problem and various exact and approximative algorithms used to solve it, including ant colony optimization. About.

The travelling salesman problem, also known as the travelling salesperson problem (TSP), asks the following question: "Given a list of cities and the distances between each pair of cities, ... TSP visualization tool This page was last edited on 8 May 2024, at 23:37 (UTC). ...

Visualization using folium to create a map. The driving tour shown above is a convincing solution to the 79-city travelling salesman problem, and according to the Concorde solver is provably 'optimal'. Since we are working with real-world data, however, the end result is only as good as the input.

Modeling the Traveling Salesman Problem from first principles. A concepts-first, math-second approach to modeling the most well-known routing problem in Operations Research. towardsdatascience.com. 1. Implementing the model in Python using Pyomo ... Data visualization is a crucial aspect of data analysis and exploration. It helps in gaining ...

Travelling salesman problem is a NP hard problem. Given a set of cities and the distance between every pair of cities, the problem is to find the shortest possible route that visits each city ...

Travelling Salesman Problem (TSP): Given a set of cities and distance between every pair of cities, ... Data visualization is a crucial aspect of data analysis and exploration. It helps in gaining ...

The traveling salesman problem (TSP) is a well-known NP problem. Its main objective is to find a shortest close trip that visits all given cities, once for each. In this paper, we present an approach to Travelling Salesman Problems (TSPs) using a ...

Pull requests. This project uses a Genetic Algorithm to find near-optimal solutions to the Traveling Salesman Problem (TSP). Users can visualize the evolving routes and compare them to the optimal solution found using Bruteforce. visualization javascript genetic-algorithm travelling-salesman-problem. Updated on Oct 25, 2023.

Visualization of Greedy Algorithm Output Travelling Salesman Greedy Algorithm Output. Thus, according to the Greedy algorithm, we will travel to city 1 first, and cities 1,3,2 after that. Brute Force Approach to the Traveling Salesman Problem. Let us look at the code of the Brute Force Method.

The video depicts four metaheuristic algorithms applied to the travelling salesman problem: local search, tabu search, simulated annealing and a genetic algo...

1 Input the data and problem visualization. The CSV file "tsp_city_data.csv ... Thus, for a traveling salesman problem for N cities (location), the distance matrix is of size N x N.

This project visualizes the use of a genetic algorithm to solve the traveling salesman problem - points are chosen on a map/plane and the algorithm attempts to find the shortest path that traverses every point. Individual solutions are comprised of combinations of routes between two points on a map (genes). The individual's fitness is defined ...