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Chapter 14: Problem 55

Describe the Brute Force Method of solving traveling salesperson problems.

Short Answer

Expert verified
The Brute force method of solving the Traveling Salesman Problem (TSP) consists of generating all (n-1)!/2 permutations of the cities, calculating the distance for each permutation and choosing the shortest. It guarantees a correct solution but is computationally expensive with a time complexity of \(O(n!)\), becoming impractical for larger number of cities.

Step by step solution

01

Understanding the Brute force method

The brute force method, also known as exhaustive search, is a problem-solving strategy where you enumerate all possible candidates for the solution and check which one satisfies the problem's conditions. It does not require any special problem-solving skills or mathematical knowledge, beyond being able to generate the set of candidates and test the 'works?' predicate.
02

Applying Brute Force to TSP

In the case of the Traveling Salesman Problem (TSP), the brute force approach will generate all (n-1)!/2 permutations of the cities (excluding the first city and halving because direction doesn't matter), calculate the distance of each permutation, and then return the list of cities that gives the shortest trip.
03

Efficiency of Brute Force

While the brute force method guarantees finding the shortest possible route, it is seldom used for TSP because it is computationally expensive. Since it generates all the permutations, the number of possibilities to check grows rapidly with the number of cities. The algorithm has a time complexity of \(O(n!)\), where n is the number of cities. This makes it feasible only for small problems (around 10 to 20 cities). For larger numbers, it becomes impractical as the time to solve the TSP increases exponentially.

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