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Traveling Salesman Problem (TSP) Implementation
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.
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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.
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Traveling Salesman Problem (TSP) Implementation
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.
<img alt="">
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.
Check all answers on Geeksforgeeks
2/5

Traveling Salesman Problem using Genetic Algorithm
// 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<Integer> vertex = new ArrayList<Integer>(); 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<Integer> swap( ArrayList<Integer> 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;...
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Traveling Salesman Problem (TSP) in Python
The Traveling Salesman Problem (TSP) is a classic algorithmic problem in the fields of computer science and operations research. It involves finding the shortest possible route that visits a set of cities and returns to the origin city. The challenge of TSP lies in its NP-hard nature, meaning that as the number of cities increases, the problem becomes exponentially more complex to solve optimally.
What is Traveling Salesman Problem (TSP)?In TSP, a “salesman” starts at a home city, visits a list of given cities exactly once, and returns to the home city, minimizing the total travel distance or cost. This problem can be applied to various practical scenarios, including logistics, planning, and manufacturing.
Approaches to Solve Traveling Salesman Problem (TSP):There are several approaches to solving TSP, ranging from exact algorithms that guarantee an optimal solution to heuristic and metaheuristic methods that provide approximate solutions. Some common methods include:
- Brute Force: Checking all possible permutations of cities to find the shortest route. This method guarantees an optimal solution but is computationally infeasible for large datasets due to its factorial time complexity.
- Dynamic Programming: The Held-Karp algorithm uses dynamic programming to reduce the time complexity to 𝑂(𝑛2⋅2𝑛)O(n2⋅2n), which is still exponential but more efficient than brute force for moderately sized problems.
- Greedy Algorithms: These provide quick but often suboptimal solutions by making locally optimal choices at each step.
- Genetic Algorithms: These use principles of natural selection and genetics to find approximate solutions.
- Simulated Annealing: This probabilistic technique approximates the global optimum of a given function by iterative improvement.
Implementing of Traveling Salesman Problem (TSP) in PythonLet’s implement a simple solution using dynamic programming (Held-Karp algorithm) in Python. This method involves breaking the problem into smaller subproblems and solving each subproblem only once, storing the results to avoid...
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