The Travelling Salesman Problem (TSP) is one of the most famous optimization problems in computer science and operations research. Our solution provides advanced algorithms to find the shortest possible route that visits each location exactly once and returns to the starting point.

How to Use the Tool

1. Define Cities: Start by adding your desired locations (cities) to the input list. You can manually enter X and Y coordinates, which represent the city's position in a 2D plane. The solver calculates distances between cities based on these coordinates.
2. Select Algorithm: Choose one of the available algorithms based on your needs. For smaller problems (up to 10 cities), exact algorithms like Brute Force can be used. For larger problems, heuristic and metaheuristic algorithms provide near-optimal solutions efficiently.
3. Solve TSP: Click the "Solve TSP" button to execute the chosen algorithm. The tool will calculate the optimal or near-optimal route.
4. View Results: The output section at the bottom will display the total distance, execution time, the optimized route sequence, and a detailed breakdown of distances between stops.

Algorithms Supported

Our solver offers a comprehensive range of algorithms, from exact methods for smaller instances to advanced heuristics and metaheuristics for larger, more complex problems. Each algorithm has its own strengths and is suited for different scenarios.

Exact Algorithms

Brute Force Algorithm

The brute force approach systematically examines every possible permutation of cities to find the absolute shortest route. It works by generating all possible tours, calculating the total distance for each tour, and selecting the one with the minimum distance. This method guarantees finding the globally optimal solution since it explores the entire solution space exhaustively.

• Pros: Guarantees the optimal solution, simple to understand and implement.
• Cons: Computationally prohibitive for more than 10-12 cities due to factorial time complexity, becomes impractical very quickly as problem size increases.
• Complexity: O(n!) time complexity, where n is the number of cities.

Branch and Bound Algorithm

Branch and bound is an optimization technique that systematically explores the solution space while intelligently pruning branches that cannot lead to better solutions. It maintains a bound (best solution found so far) and uses lower bound estimates to eliminate partial solutions that cannot improve upon the current best. This approach significantly reduces the search space compared to brute force.

• Pros: More efficient than brute force, still guarantees optimal solution, can handle slightly larger problems.
• Cons: Still exponential in worst case, implementation complexity is higher, performance depends heavily on the quality of bounds.
• Complexity: O(n²2ⁿ) in worst case, but often performs much better in practice.

Heuristic & Metaheuristic Algorithms

Nearest Neighbor Algorithm

The nearest neighbor algorithm is a greedy heuristic that constructs a tour by starting at an arbitrary city and repeatedly visiting the closest unvisited city until all cities have been visited, then returning to the starting city. This approach makes locally optimal choices at each step, hoping to find a globally good solution.

• Pros: Very fast execution, simple to implement, provides reasonable solutions for many instances, good starting point for other algorithms.
• Cons: Does not guarantee optimal solution, can produce poor results for certain city configurations, greedy nature can lead to suboptimal choices.
• Complexity: O(n²) time complexity.

2-opt Heuristic Algorithm

The 2-opt heuristic is a local search improvement algorithm that starts with an initial tour (often from nearest neighbor) and iteratively improves it by removing two edges and reconnecting the path in a different way. It examines all possible pairs of edges and makes swaps that reduce the total tour length, continuing until no further improvements can be made.

• Pros: Significantly improves initial solutions, relatively fast, easy to implement, often produces high-quality results.
• Cons: Can get stuck in local optima, quality depends on initial solution, no guarantee of global optimum.
• Complexity: O(n²) per iteration, typically converges quickly.

Genetic Algorithm

Genetic algorithms are inspired by natural selection and evolution. They maintain a population of candidate solutions (tours) and evolve them over generations using operations like selection, crossover, and mutation. The algorithm selects better solutions for reproduction, combines them to create offspring, and introduces random changes to maintain diversity and explore new areas of the solution space.

• Pros: Can escape local optima, handles large problem sizes well, parallelizable, robust across different problem instances.
• Cons: No guarantee of optimal solution, requires parameter tuning, can be computationally expensive, convergence can be slow.
• Complexity: O(g × p × n²) where g is generations, p is population size, and n is number of cities.

Simulated Annealing Algorithm

Simulated annealing is inspired by the metallurgical process of annealing, where materials are heated and then slowly cooled to reduce defects. The algorithm starts with a random solution and iteratively makes small changes, accepting improvements and occasionally accepting worse solutions with a probability that decreases over time (as the "temperature" cools). This allows the algorithm to escape local optima early in the search.

• Pros: Can escape local optima, simple to implement, works well for many optimization problems, requires minimal memory.
• Cons: Requires careful parameter tuning (cooling schedule), can be slow to converge, no guarantee of optimal solution, sensitive to initial temperature and cooling rate.
• Complexity: O(k × n) where k is the number of iterations and n is the number of cities.

Ant Colony Optimization Algorithm

Ant Colony Optimization (ACO) mimics the foraging behavior of ants, who find optimal paths between their colony and food sources using pheromone trails. In the TSP context, artificial ants construct tours probabilistically, with the probability of choosing an edge influenced by both the distance (shorter is better) and the pheromone concentration (more pheromone indicates a good path). Successful tours deposit more pheromone, while pheromone evaporates over time.

• Pros: Excellent for finding high-quality solutions, naturally parallelizable, adapts well to dynamic problems, good balance between exploration and exploitation.
• Cons: Many parameters to tune, can be computationally expensive, theoretical convergence guarantees are limited, requires careful balance of pheromone parameters.
• Complexity: O(i × m × n²) where i is iterations, m is number of ants, and n is number of cities.