# Traveling Salesman Problem (TSP) Implementation

• Difficulty Level : Medium
• Last Updated : 22 Jul, 2022

Travelling Salesman Problem (TSP) : Given a set of cities and distances between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns to the starting point.
Note the difference between Hamiltonian Cycle and TSP. The Hamiltonian cycle problem is to find if there exists a tour that visits every city exactly once. Here we know that Hamiltonian Tour exists (because the graph is complete) and in fact, many such tours exist, the problem is to find a minimum weight Hamiltonian Cycle.
For example, consider the graph shown in the figure on the right side. A TSP tour in the graph is 1-2-4-3-1. The cost of the tour is 10+25+30+15 which is 80.
The problem is a famous NP-hard problem. There is no polynomial-time known solution for this problem. Examples:

```Output of Given Graph:
minimum weight Hamiltonian Cycle :
10 + 25 + 30 + 15 := 80```

In this post, the implementation of a simple solution is discussed.

1. Consider city 1 as the starting and ending point. Since the route is cyclic, we can consider any point as a starting point.
2. Generate all (n-1)! permutations of cities.
3. Calculate the cost of every permutation and keep track of the minimum cost permutation.
4. Return the permutation with minimum cost.

Below is the implementation of the above idea

## C++

 `// CPP program to implement traveling salesman``// problem using naive approach.``#include ``using` `namespace` `std;``#define V 4` `// implementation of traveling Salesman Problem``int` `travllingSalesmanProblem(``int` `graph[][V], ``int` `s)``{``    ``// store all vertex apart from source vertex``    ``vector<``int``> vertex;``    ``for` `(``int` `i = 0; i < V; i++)``        ``if` `(i != s)``            ``vertex.push_back(i);` `    ``// store minimum weight Hamiltonian Cycle.``    ``int` `min_path = INT_MAX;``    ``do` `{` `        ``// store current Path weight(cost)``        ``int` `current_pathweight = 0;` `        ``// compute current path weight``        ``int` `k = s;``        ``for` `(``int` `i = 0; i < vertex.size(); i++) {``            ``current_pathweight += graph[k][vertex[i]];``            ``k = vertex[i];``        ``}``        ``current_pathweight += graph[k][s];` `        ``// update minimum``        ``min_path = min(min_path, current_pathweight);` `    ``} ``while` `(``        ``next_permutation(vertex.begin(), vertex.end()));` `    ``return` `min_path;``}` `// Driver Code``int` `main()``{``    ``// matrix representation of graph``    ``int` `graph[][V] = { { 0, 10, 15, 20 },``                       ``{ 10, 0, 35, 25 },``                       ``{ 15, 35, 0, 30 },``                       ``{ 20, 25, 30, 0 } };``    ``int` `s = 0;``    ``cout << travllingSalesmanProblem(graph, s) << endl;``    ``return` `0;``}`

## Java

 `// Java program to implement``// traveling salesman problem``// using naive approach.``import` `java.util.*;``class` `GFG{``    ` `static` `int` `V = ``4``;` `// implementation of traveling``// Salesman Problem``static` `int` `travllingSalesmanProblem(``int` `graph[][],``                                    ``int` `s)``{``  ``// store all vertex apart``  ``// from source vertex``  ``ArrayList vertex =``            ``new` `ArrayList();``  ` `  ``for` `(``int` `i = ``0``; i < V; i++)``    ``if` `(i != s)``      ``vertex.add(i);` `  ``// store minimum weight``  ``// Hamiltonian Cycle.``  ``int` `min_path = Integer.MAX_VALUE;``  ``do``  ``{``    ``// store current Path weight(cost)``    ``int` `current_pathweight = ``0``;` `    ``// compute current path weight``    ``int` `k = s;``    ` `    ``for` `(``int` `i = ``0``;``             ``i < vertex.size(); i++)``    ``{``      ``current_pathweight +=``              ``graph[k][vertex.get(i)];``      ``k = vertex.get(i);``    ``}``    ``current_pathweight += graph[k][s];` `    ``// update minimum``    ``min_path = Math.min(min_path,``                        ``current_pathweight);` `  ``} ``while` `(findNextPermutation(vertex));` `  ``return` `min_path;``}` `// Function to swap the data``// present in the left and right indices``public` `static` `ArrayList swap(``              ``ArrayList data,``              ``int` `left, ``int` `right)``{``  ``// Swap the data``  ``int` `temp = data.get(left);``  ``data.set(left, data.get(right));``  ``data.set(right, temp);` `  ``// Return the updated array``  ``return` `data;``}``  ` `// Function to reverse the sub-array``// starting from left to the right``// both inclusive``public` `static` `ArrayList reverse(``              ``ArrayList data,``              ``int` `left, ``int` `right)``{``  ``// Reverse the sub-array``  ``while` `(left < right)``  ``{``    ``int` `temp = data.get(left);``    ``data.set(left++,``             ``data.get(right));``    ``data.set(right--, temp);``  ``}` `  ``// Return the updated array``  ``return` `data;``}``  ` `// Function to find the next permutation``// of the given integer array``public` `static` `boolean` `findNextPermutation(``                      ``ArrayList data)``{ ``  ``// If the given dataset is empty``  ``// or contains only one element``  ``// next_permutation is not possible``  ``if` `(data.size() <= ``1``)``    ``return` `false``;` `  ``int` `last = data.size() - ``2``;` `  ``// find the longest non-increasing``  ``// suffix and find the pivot``  ``while` `(last >= ``0``)``  ``{``    ``if` `(data.get(last) <``        ``data.get(last + ``1``))``    ``{``      ``break``;``    ``}``    ``last--;``  ``}` `  ``// If there is no increasing pair``  ``// there is no higher order permutation``  ``if` `(last < ``0``)``    ``return` `false``;` `  ``int` `nextGreater = data.size() - ``1``;` `  ``// Find the rightmost successor``  ``// to the pivot``  ``for` `(``int` `i = data.size() - ``1``;``           ``i > last; i--) {``    ``if` `(data.get(i) >``        ``data.get(last))``    ``{``      ``nextGreater = i;``      ``break``;``    ``}``  ``}` `  ``// Swap the successor and``  ``// the pivot``  ``data = swap(data,``              ``nextGreater, last);` `  ``// Reverse the suffix``  ``data = reverse(data, last + ``1``,``                 ``data.size() - ``1``);` `  ``// Return true as the``  ``// next_permutation is done``  ``return` `true``;``}` `// Driver Code``public` `static` `void` `main(String args[])``{``  ``// matrix representation of graph``  ``int` `graph[][] = {{``0``, ``10``, ``15``, ``20``},``                   ``{``10``, ``0``, ``35``, ``25``},``                   ``{``15``, ``35``, ``0``, ``30``},``                   ``{``20``, ``25``, ``30``, ``0``}};``  ``int` `s = ``0``;``  ``System.out.println(``  ``travllingSalesmanProblem(graph, s));``}``}` `// This code is contributed by adityapande88`

## Python3

 `# Python3 program to implement traveling salesman``# problem using naive approach.``from` `sys ``import` `maxsize``from` `itertools ``import` `permutations``V ``=` `4` `# implementation of traveling Salesman Problem``def` `travellingSalesmanProblem(graph, s):` `    ``# store all vertex apart from source vertex``    ``vertex ``=` `[]``    ``for` `i ``in` `range``(V):``        ``if` `i !``=` `s:``            ``vertex.append(i)` `    ``# store minimum weight Hamiltonian Cycle``    ``min_path ``=` `maxsize``    ``next_permutation``=``permutations(vertex)``    ``for` `i ``in` `next_permutation:` `        ``# store current Path weight(cost)``        ``current_pathweight ``=` `0` `        ``# compute current path weight``        ``k ``=` `s``        ``for` `j ``in` `i:``            ``current_pathweight ``+``=` `graph[k][j]``            ``k ``=` `j``        ``current_pathweight ``+``=` `graph[k][s]` `        ``# update minimum``        ``min_path ``=` `min``(min_path, current_pathweight)``        ` `    ``return` `min_path`  `# Driver Code``if` `__name__ ``=``=` `"__main__"``:` `    ``# matrix representation of graph``    ``graph ``=` `[[``0``, ``10``, ``15``, ``20``], [``10``, ``0``, ``35``, ``25``],``            ``[``15``, ``35``, ``0``, ``30``], [``20``, ``25``, ``30``, ``0``]]``    ``s ``=` `0``    ``print``(travellingSalesmanProblem(graph, s))`

## C#

 `// C# program to implement``// traveling salesman problem``// using naive approach.``using` `System;``using` `System.Collections.Generic;` `class` `GFG {` `    ``static` `int` `V = 4;` `    ``// implementation of traveling Salesman Problem``    ``static` `int` `travllingSalesmanProblem(``int``[, ] graph,``                                        ``int` `s)``    ``{``        ``List<``int``> vertex = ``new` `List<``int``>();` `        ``for` `(``int` `i = 0; i < V; i++)``            ``if` `(i != s)``                ``vertex.Add(i);` `        ``// store minimum weight``        ``// Hamiltonian Cycle.``        ``int` `min_path = Int32.MaxValue;` `        ``do` `{``            ``// store current Path weight(cost)``            ``int` `current_pathweight = 0;``            ``int` `k = s;` `            ``// compute current path weight``            ``for` `(``int` `i = 0; i < vertex.Count; i++) {``                ``current_pathweight += graph[k, vertex[i]];``                ``k = vertex[i];``            ``}` `            ``current_pathweight += graph[k, s];` `            ``// update minimum``            ``min_path``                ``= Math.Min(min_path, current_pathweight);` `        ``} ``while` `(findNextPermutation(vertex));` `        ``return` `min_path;``    ``}` `    ``// Function to swap the data resent in the left and``    ``// right indices``    ``public` `static` `List<``int``> swap(List<``int``> data, ``int` `left,``                                 ``int` `right)``    ``{``        ``// Swap the data``        ``int` `temp = data[left];``        ``data[left] = data[right];``        ``data[right] = temp;` `        ``// Return the updated array``        ``return` `data;``    ``}` `    ``// Function to reverse the sub-array starting from left``    ``// to the right both inclusive``    ``public` `static` `List<``int``> reverse(List<``int``> data,``                                    ``int` `left, ``int` `right)``    ``{``        ``// Reverse the sub-array``        ``while` `(left < right) {``            ``int` `temp = data[left];``            ``data[left++] = data[right];``            ``data[right--] = temp;``        ``}` `        ``// Return the updated array``        ``return` `data;``    ``}` `    ``// Function to find the next permutation of the given``    ``// integer array``    ``public` `static` `bool` `findNextPermutation(List<``int``> data)``    ``{``        ``// If the given dataset is empty``        ``// or contains only one element``        ``// next_permutation is not possible``        ``if` `(data.Count <= 1)``            ``return` `false``;``        ``int` `last = data.Count - 2;` `        ``// find the longest non-increasing``        ``// suffix and find the pivot``        ``while` `(last >= 0) {``            ``if` `(data[last] <  data[last + 1])``                ``break``;``            ``last--;``        ``}` `        ``// If there is no increasing pair``        ``// there is no higher order permutation``        ``if` `(last < 0)``            ``return` `false``;``        ``int` `nextGreater = data.Count - 1;` `        ``// Find the rightmost successor``        ``// to the pivot``        ``for` `(``int` `i = data.Count - 1; i > last; i--) {``            ``if` `(data[i] >  data[last]) {``                ``nextGreater = i;``                ``break``;``            ``}``        ``}` `        ``// Swap the successor and``        ``// the pivot``        ``data = swap(data, nextGreater, last);` `        ``// Reverse the suffix``        ``data = reverse(data, last + 1, data.Count - 1);` `        ``// Return true as the``        ``// next_permutation is done``        ``return` `true``;``    ``}` `    ``// Driver Code``    ``public` `static` `void` `Main(``string``[] args)``    ``{``        ``//   matrix representation of graph``        ``int``[, ] graph``            ``= ``new` `int``[4, 4] { { 0, 10, 15, 20 },``                              ``{ 10, 0, 35, 25 },``                              ``{ 15, 35, 0, 30 },``                              ``{ 20, 25, 30, 45 } };``        ``int` `s = 0;``        ``Console.WriteLine(``            ``travllingSalesmanProblem(graph, s));``    ``}``}` `// This code is contributed by Tapesh(tapeshdua420)`

Output

`80`

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