# What is TABU Search?

Tabu search (TS) is an iterative neighborhood search algorithm, where the neighborhood changes dynamically. Tabu search enhances local search by avoiding points in the search space which are already visited. By avoiding already visited points, loops in search space are avoided and local optima can be escaped.

The main feature of Tabu Search is the use of explicit memory, with two goals:

• To prevent the search from revisiting previously visited solutions.
• To explore the unvisited areas of the solution space.

## How to Optimize an Algorithm Using Tabu Search?

Optimize a solution using the Tabu Search algorithm to find the best solution that minimizes the fitness value of the objective function for a given initial solution, within a specified number of iterations and using a tabu list of limited size.

The code uses Tabu Search to iteratively investigate solutions by assessing their fitness with an objective function and producing nearby solutions with a neighborhood function. Utilizing a tabu list, it attempts to discover the optimal solution with the lowest fitness within a predetermined amount of iterations, avoiding returning to previously explored solutions.

## C++

 `#include ` `#include ` `#include // Added to include accumulate` `#include `   `// Define the objective function` `int` `objective_function(``const` `std::vector<``int``>& solution)` `{ ``// Added const qualifier` `    ``// TODO: Implement your objective function here` `    ``// The objective function should evaluate` `    ``// the quality of a given solution and` `    ``// return a numerical value representing` `    ``// the solution's fitness` `    ``// Example: return std::accumulate(solution.begin(),` `    ``// solution.end(), 0);` `    ``return` `std::accumulate(solution.begin(), solution.end(),` `                           ``0);` `}`   `// Define the neighborhood function` `std::vector >` `get_neighbors(``const` `std::vector<``int``>& solution)` `{ ``// Added const qualifier` `    ``std::vector > neighbors;` `    ``for` `(``size_t` `i = 0; i < solution.size(); i++) {` `        ``for` `(``size_t` `j = i + 1; j < solution.size(); j++) {` `            ``std::vector<``int``> neighbor = solution;` `            ``std::swap(neighbor[i], neighbor[j]);` `            ``neighbors.push_back(neighbor);` `        ``}` `    ``}` `    ``return` `neighbors;` `}`   `// Define the Tabu Search algorithm` `std::vector<``int``>` `tabu_search(``const` `std::vector<``int``>& initial_solution,` `            ``int` `max_iterations, ``int` `tabu_list_size)` `{ ``// Added const qualifier` `    ``std::vector<``int``> best_solution = initial_solution;` `    ``std::vector<``int``> current_solution = initial_solution;` `    ``std::vector > tabu_list;`   `    ``for` `(``int` `iter = 0; iter < max_iterations; iter++) {` `        ``std::vector > neighbors` `            ``= get_neighbors(current_solution);` `        ``std::vector<``int``> best_neighbor;` `        ``int` `best_neighbor_fitness` `            ``= std::numeric_limits<``int``>::max();`   `        ``for` `(``const` `std::vector<``int``>& neighbor : neighbors) {` `            ``if` `(std::find(tabu_list.begin(),` `                          ``tabu_list.end(), neighbor)` `                ``== tabu_list.end()) {` `                ``int` `neighbor_fitness` `                    ``= objective_function(neighbor);` `                ``if` `(neighbor_fitness` `                    ``< best_neighbor_fitness) {` `                    ``best_neighbor = neighbor;` `                    ``best_neighbor_fitness` `                        ``= neighbor_fitness;` `                ``}` `            ``}` `        ``}`   `        ``if` `(best_neighbor.empty()) {` `            ``// No non-tabu neighbors found,` `            ``// terminate the search` `            ``break``;` `        ``}`   `        ``current_solution = best_neighbor;` `        ``tabu_list.push_back(best_neighbor);` `        ``if` `(tabu_list.size() > tabu_list_size) {` `            ``// Remove the oldest entry from the` `            ``// tabu list if it exceeds the size` `            ``tabu_list.erase(tabu_list.begin());` `        ``}`   `        ``if` `(objective_function(best_neighbor)` `            ``< objective_function(best_solution)) {` `            ``// Update the best solution if the` `            ``// current neighbor is better` `            ``best_solution = best_neighbor;` `        ``}` `    ``}`   `    ``return` `best_solution;` `}`   `int` `main()` `{` `    ``// Example usage` `    ``// Provide an initial solution` `    ``std::vector<``int``> initial_solution = { 1, 2, 3, 4, 5 };` `    ``int` `max_iterations = 100;` `    ``int` `tabu_list_size = 10;`   `    ``std::vector<``int``> best_solution = tabu_search(` `        ``initial_solution, max_iterations, tabu_list_size);` `    ``std::cout << ``"Best solution:"``;` `    ``for` `(``int` `val : best_solution) {` `        ``std::cout << ``" "` `<< val;` `    ``}` `    ``std::cout << std::endl;` `    ``std::cout << ``"Best solution fitness: "` `              ``<< objective_function(best_solution)` `              ``<< std::endl;`   `    ``return` `0;` `}`

## Java

 `import` `java.util.ArrayList;` `import` `java.util.List;`   `public` `class` `TabuSearch {` `    ``// Define the objective function` `    ``public` `static` `int` `objectiveFunction(List solution) {` `        ``// TODO: Implement your objective function here` `        ``// The objective function should evaluate` `        ``// the quality of a given solution and` `        ``// return a numerical value representing` `        ``// the solution's fitness` `        ``// Example: return solution.stream().mapToInt(Integer::intValue).sum();` `        ``return` `solution.stream().mapToInt(Integer::intValue).sum();` `    ``}`   `    ``// Define the neighborhood function` `    ``public` `static` `List> getNeighbors(List solution) {` `        ``List> neighbors = ``new` `ArrayList<>();` `        ``for` `(``int` `i = ``0``; i < solution.size(); i++) {` `            ``for` `(``int` `j = i + ``1``; j < solution.size(); j++) {` `                ``List neighbor = ``new` `ArrayList<>(solution);` `                ``// Swap elements at indices i and j` `                ``int` `temp = neighbor.get(i);` `                ``neighbor.set(i, neighbor.get(j));` `                ``neighbor.set(j, temp);` `                ``neighbors.add(neighbor);` `            ``}` `        ``}` `        ``return` `neighbors;` `    ``}`   `    ``// Define the Tabu Search algorithm` `    ``public` `static` `List tabuSearch(List initialSolution, ` `                                           ``int` `maxIterations, ``int` `tabuListSize) {` `        ``List bestSolution = ``new` `ArrayList<>(initialSolution);` `        ``List currentSolution = ``new` `ArrayList<>(initialSolution);` `        ``List> tabuList = ``new` `ArrayList<>();`   `        ``for` `(``int` `iter = ``0``; iter < maxIterations; iter++) {` `            ``List> neighbors = getNeighbors(currentSolution);` `            ``List bestNeighbor = ``new` `ArrayList<>();` `            ``int` `bestNeighborFitness = Integer.MAX_VALUE;`   `            ``for` `(List neighbor : neighbors) {` `                ``if` `(!tabuList.contains(neighbor)) {` `                    ``int` `neighborFitness = objectiveFunction(neighbor);` `                    ``if` `(neighborFitness < bestNeighborFitness) {` `                        ``bestNeighbor = ``new` `ArrayList<>(neighbor);` `                        ``bestNeighborFitness = neighborFitness;` `                    ``}` `                ``}` `            ``}`   `            ``if` `(bestNeighbor.isEmpty()) {` `                ``// No non-tabu neighbors found, terminate the search` `                ``break``;` `            ``}`   `            ``currentSolution = ``new` `ArrayList<>(bestNeighbor);` `            ``tabuList.add(bestNeighbor);`   `            ``if` `(tabuList.size() > tabuListSize) {` `                ``// Remove the oldest entry from the tabu list if it exceeds the size` `                ``tabuList.remove(``0``);` `            ``}`   `            ``if` `(objectiveFunction(bestNeighbor) < objectiveFunction(bestSolution)) {` `                ``// Update the best solution if the current neighbor is better` `                ``bestSolution = ``new` `ArrayList<>(bestNeighbor);` `            ``}` `        ``}`   `        ``return` `bestSolution;` `    ``}`   `    ``public` `static` `void` `main(String[] args) {` `        ``// Example usage` `        ``// Provide an initial solution` `        ``List initialSolution = List.of(``1``, ``2``, ``3``, ``4``, ``5``);` `        ``int` `maxIterations = ``100``;` `        ``int` `tabuListSize = ``10``;`   `        ``List bestSolution = ` `          ``tabuSearch(initialSolution, maxIterations, tabuListSize);` `        ``System.out.print(``"Best solution:"``);` `        ``for` `(``int` `val : bestSolution) {` `            ``System.out.print(``" "` `+ val);` `        ``}` `        ``System.out.println();` `        ``System.out.println(``"Best solution fitness: "` `+ objectiveFunction(bestSolution));` `    ``}` `}`

## Python

 `import` `random`   `# Define the objective function`     `def` `objective_function(solution):` `    ``# TODO: Implement your objective function here` `    ``# The objective function should evaluate` `    ``# the quality of a given solution and` `    ``# return a numerical value representing` `    ``# the solution's fitness` `    ``# Example: return sum(solution)` `    ``return` `sum``(solution)`   `# Define the neighborhood function`     `def` `get_neighbors(solution):`   `    ``# TODO: Implement your neighborhood function here` `    ``# The neighborhood function should generate` `    ``# a set of neighboring solutions based on a given solution` `    ``# Example: Generate neighboring solutions by` `    ``# swapping two elements in the solution`   `    ``neighbors ``=` `[]` `    ``for` `i ``in` `range``(``len``(solution)):` `        ``for` `j ``in` `range``(i ``+` `1``, ``len``(solution)):` `            ``neighbor ``=` `solution[:]` `            ``neighbor[i], neighbor[j] ``=` `neighbor[j], neighbor[i]` `            ``neighbors.append(neighbor)` `    ``return` `neighbors`   `# Define the Tabu Search algorithm`     `def` `tabu_search(initial_solution, max_iterations, tabu_list_size):` `    ``best_solution ``=` `initial_solution` `    ``current_solution ``=` `initial_solution` `    ``tabu_list ``=` `[]`   `    ``for` `_ ``in` `range``(max_iterations):` `        ``neighbors ``=` `get_neighbors(current_solution)` `        ``best_neighbor ``=` `None` `        ``best_neighbor_fitness ``=` `float``(``'inf'``)`   `        ``for` `neighbor ``in` `neighbors:` `            ``if` `neighbor ``not` `in` `tabu_list:` `                ``neighbor_fitness ``=` `objective_function(neighbor)` `                ``if` `neighbor_fitness < best_neighbor_fitness:` `                    ``best_neighbor ``=` `neighbor` `                    ``best_neighbor_fitness ``=` `neighbor_fitness`   `        ``if` `best_neighbor ``is` `None``:`   `            ``# No non-tabu neighbors found,` `            ``# terminate the search` `            ``break`   `        ``current_solution ``=` `best_neighbor` `        ``tabu_list.append(best_neighbor)` `        ``if` `len``(tabu_list) > tabu_list_size:`   `            ``# Remove the oldest entry from the` `            ``# tabu list if it exceeds the size` `            ``tabu_list.pop(``0``)`   `        ``if` `objective_function(best_neighbor) < objective_function(best_solution):` `            ``# Update the best solution if the` `            ``# current neighbor is better` `            ``best_solution ``=` `best_neighbor`   `    ``return` `best_solution`     `# Example usage` `# Provide an initial solution` `initial_solution ``=` `[``1``, ``2``, ``3``, ``4``, ``5``]` `max_iterations ``=` `100` `tabu_list_size ``=` `10`   `best_solution ``=` `tabu_search(initial_solution, max_iterations, tabu_list_size)` `print``("Best solution: {}".``format``(best_solution))` `print``("Best solution fitness: {}".``format``(objective_function(best_solution)))`

## C#

 `using` `System;` `using` `System.Collections.Generic;` `using` `System.Linq;`   `class` `Program` `{` `      ``// Define the objective function` `    ``static` `int` `ObjectiveFunction(List<``int``> solution)` `    ``{` `     ``// Added const qualifier` `    ``// TODO: Implement your objective function here` `    ``// The objective function should evaluate` `    ``// the quality of a given solution and` `    ``// return a numerical value representing` `    ``// the solution's fitness` `    ``// Example: return std::accumulate(solution.begin(),` `    ``// solution.end(), 0);` `        ``return` `solution.Sum();` `    ``}`   `      ``// Define the neighborhood function` `    ``static` `List> GetNeighbors(List<``int``> solution)` `    ``{` `          ``// Added const qualifier` `        ``List> neighbors = ``new` `List>();` `        ``for` `(``int` `i = 0; i < solution.Count; i++)` `        ``{` `            ``for` `(``int` `j = i + 1; j < solution.Count; j++)` `            ``{` `                ``List<``int``> neighbor = ``new` `List<``int``>(solution);` `                ``neighbor[i] = solution[j];` `                ``neighbor[j] = solution[i];` `                ``neighbors.Add(neighbor);` `            ``}` `        ``}` `        ``return` `neighbors;` `    ``}`   `      ``// Define the Tabu Search algorithm` `    ``static` `List<``int``> TabuSearch(List<``int``> initial_solution, ` `                                ``int` `max_iterations, ` `                                ``int` `tabu_list_size)` `            ``{` `      `  `    ``// Added const qualifier` `    ``List<``int``> best_solution = initial_solution;` `    ``List<``int``> current_solution = initial_solution;` `    ``List> tabu_list = ``new` `List>();` `    ``for` `(``int` `iter = 0; iter < max_iterations; iter++)` `    ``{` `        ``List> neighbors = GetNeighbors(current_solution);` `        ``List<``int``> best_neighbor = ``new` `List<``int``>();` `        ``int` `best_neighbor_fitness = ``int``.MaxValue;` `        ``foreach` `(List<``int``> neighbor ``in` `neighbors)` `        ``{` `            ``if` `(!tabu_list.Contains(neighbor))` `            ``{` `                ``int` `neighbor_fitness = ObjectiveFunction(neighbor);` `                ``if` `(neighbor_fitness < best_neighbor_fitness)` `                ``{` `                    ``best_neighbor = neighbor;` `                    ``best_neighbor_fitness = neighbor_fitness;` `                ``}` `            ``}` `        ``}` `        ``if` `(best_neighbor.Count == 0)` `        ``{` `              ``// No non-tabu neighbors found,` `            ``// terminate the search` `            ``break``;` `        ``}` `        ``current_solution = best_neighbor;` `        ``tabu_list.Add(best_neighbor);` `        ``if` `(tabu_list.Count > tabu_list_size)` `        ``{` `              ``// Remove the oldest entry from the` `            ``// tabu list if it exceeds the size` `            ``tabu_list.RemoveAt(0);` `        ``}` `        ``if` `(ObjectiveFunction(best_neighbor) < ObjectiveFunction(best_solution))` `        ``{` `              ``// Update the best solution if the` `            ``// current neighbor is better` `            ``best_solution = best_neighbor;` `        ``}` `    ``}` `    ``return` `best_solution;` `}` `      ``static` `void` `Main()` `    ``{` `          ``// Example usage` `        ``// Provide an initial solution` `        ``List<``int``> initial_solution = ``new` `List<``int``> { 1, 2, 3, 4, 5 };` `        ``int` `max_iterations = 100;` `        ``int` `tabu_list_size = 10;` `        ``List<``int``> best_solution = TabuSearch(` `            ``initial_solution, max_iterations, tabu_list_size);` `        ``Console.Write(``"Best solution:"``);` `        ``foreach` `(``int` `val ``in` `best_solution)` `        ``{` `            ``Console.Write(``" "` `+ val);` `        ``}` `        ``Console.WriteLine();` `        ``Console.WriteLine(``"Best solution fitness: "` `+` `            ``ObjectiveFunction(best_solution));` `    ``}`   `}`

## Javascript

 `function` `objectiveFunction(solution) {` `    ``// TODO: Implement your objective function here` `    ``// The objective function should evaluate` `    ``// the quality of a given solution and` `    ``// return a numerical value representing` `    ``// the solution's fitness` `    ``// Example: return solution.reduce((sum, val) => sum + val, 0);` `    ``return` `solution.reduce((sum, val) => sum + val, 0);` `}`   `function` `getNeighbors(solution) {` `    ``const neighbors = [];` `    ``for` `(let i = 0; i < solution.length; i++) {` `        ``for` `(let j = i + 1; j < solution.length; j++) {` `            ``const neighbor = [...solution];` `            ``// Swap elements at indices i and j` `            ``[neighbor[i], neighbor[j]] = [neighbor[j], neighbor[i]];` `            ``neighbors.push(neighbor);` `        ``}` `    ``}` `    ``return` `neighbors;` `}`   `function` `tabuSearch(initialSolution, maxIterations, tabuListSize) {` `    ``let bestSolution = [...initialSolution];` `    ``let currentSolution = [...initialSolution];` `    ``const tabuList = [];`   `    ``for` `(let iter = 0; iter < maxIterations; iter++) {` `        ``const neighbors = getNeighbors(currentSolution);` `        ``let bestNeighbor = [];` `        ``let bestNeighborFitness = Number.MAX_VALUE;`   `        ``for` `(const neighbor of neighbors) {` `            ``if` `(!tabuList.some(entry => JSON.stringify(entry) === JSON.stringify(neighbor))) {` `                ``const neighborFitness = objectiveFunction(neighbor);` `                ``if` `(neighborFitness < bestNeighborFitness) {` `                    ``bestNeighbor = [...neighbor];` `                    ``bestNeighborFitness = neighborFitness;` `                ``}` `            ``}` `        ``}`   `        ``if` `(bestNeighbor.length === 0) {` `            ``// No non-tabu neighbors found, terminate the search` `            ``break``;` `        ``}`   `        ``currentSolution = [...bestNeighbor];` `        ``tabuList.push([...bestNeighbor]);`   `        ``if` `(tabuList.length > tabuListSize) {` `            ``// Remove the oldest entry from the tabu list if it exceeds the size` `            ``tabuList.shift();` `        ``}`   `        ``if` `(objectiveFunction(bestNeighbor) < objectiveFunction(bestSolution)) {` `            ``// Update the best solution if the current neighbor is better` `            ``bestSolution = [...bestNeighbor];` `        ``}` `    ``}`   `    ``return` `bestSolution;` `}`   `// Example usage` `const initialSolution = [1, 2, 3, 4, 5];` `const maxIterations = 100;` `const tabuListSize = 10;`   `const bestSolution = tabuSearch(initialSolution, maxIterations, tabuListSize);` `console.log(``"Best solution:"``, bestSolution.join(``" "``));` `console.log(``"Best solution fitness:"``, objectiveFunction(bestSolution));`

Output

```Best solution: [1, 2, 3, 4, 5]
Best solution fitness: 15

```

NOTE:  We can replace the objective_function implementation with your specific objective function and provide a suitable initial_solution to ensure the code works as intended.

### Explanation of code:

• Implementing the objective function and the neighborhood function unique to your situation is required by the code.
• A solution’s fitness is quantified numerically by the objective function, which assesses a solution’s quality.
• Using a given solution as a starting point, the neighborhood function creates neighboring solutions.
• The best non-tabu neighbor is considered as the search space is iteratively explored using the Tabu Search algorithm.
• According to the specifications of your problem, you should adjust the starting solution, maximum iterations, and tabu list size.
• Depending on how difficult your problem is, the code provides a basic framework for implementing Tabu Search and might need more customization.
• Don’t forget to replace the TODO placeholder sections with your actual implementation for your particular problem, describing the solution representation and fitness calculation in the objective function.

### Complexity Analysis:

As Tabu Search’s temporal complexity varies depending on the particular problem being solved and the scope of the search area, there is no universal method for calculating it.

• The worst-case time complexity of a Tabu Search problem, which is commonly expressed as O(2^n), where n is the size of the search space, nevertheless, can be exponential in the magnitude of the problem.
• Although it depends on the size of the search space, Tabu Search’s space complexity is often substantially less than its worst-case time complexity. The size of the search space is denoted as O(k*n), where n is the size of the search space and k is the size of the Tabu List or the number of candidate solutions being assessed and stored in memory.

It’s important to keep in mind that the actual time and space complexity of Tabu Search might vary significantly depending on the particular problem being solved, the algorithm parameters chosen (such as the size of the Tabu List, the aspiration criteria, the diversification strategies, etc.), and the effectiveness of the implementation. To discover the greatest performance for a particular task, it is therefore frequently required to experiment with various parameter values and implementation methodologies.

### Examples of Problems to Solve with Tabu Search:

• N-Queens Problem
• Traveling Salesman Problem (TSP)
• Minimum Spanning Tree (MST)
• Assignment Problems
• Vehicle Routing
• DNA Sequencing

• Can be applied to both discrete and continuous solutions.
• The tabu list can be used to resist cycles and revert to old solutions.