Question

A salesperson wants to visit 25 cities while minimizing the total number of miles she must drive. Because she has studied computer science, she decides to design an algorithm to determine the optimal order in which to visit the cities to (1) keep her driving distance to a minimum, and (2) visit each city exactly once. The algorithm that she has devised is the following: The computer first lists all possible ways to visit the 25 cities and then, for each one, determines the total mileage associated with that particular ordering. (Assume that the computer has access to data that gives the distances between all cities.) After determining the total mileage for each possible trip, the computer searches for the ordering with the minimum mileage and prints out the list of cities on that optimal route, that is, the order in which the salesperson should visit her destinations. If a computer could analyze \(10,000,000\) separate paths per second, how long would it take to determine the optimal route for visiting these 25 cities? On the basis of your answer, do you think this is a feasible algorithm? If it is not, can you think of a way to obtain a reasonable solution to this problem?

Short answer

Brute-force is infeasible (\(25!\) is too large). Use approximation algorithms for practical solutions.

Step-by-step solution

Define the problem using permutations

The problem requires finding the optimal route to visit 25 cities with the minimum driving distance. This can be stated as finding the shortest path that visits each city exactly once, known as the Traveling Salesperson Problem (TSP). The possible routes for 25 cities are given by the permutations of the cities, which is calculated as \(25!\) (read as 25 factorial).

Calculate the number of permutations

Calculate \(25!\), which is the total number of permutations or different paths the salesperson can take. The factorial of 25 is: \[ 25! = 25 \times 24 \times 23 \times \ldots \times 2 \times 1 \] Computing this gives approximately \(15,511,210,043,330,985,984,000,000\) possible paths.

Compute the processing time

Determine how long it would take the computer to evaluate each path assuming it processes each path in one second per every 10,000,000 paths:\[ \frac{15,511,210,043,330,985,984,000,000}{10,000,000} \] Calculating this division gives approximately \(1,551,121,004,333,099\) seconds. Convert this into years to understand the feasibility:\[ \text{For seconds to years, divide by } 60 \times 60 \times 24 \times 365 \approx 31,557,600 \] Resulting in roughly \(49,156,049\) years.

Evaluate feasibility and alternative solutions

Given the massive processing time, this brute-force algorithm is infeasible. A better approach is to apply heuristic or approximation algorithms, such as Nearest Neighbor, Genetic Algorithms, or other optimization techniques like Simulated Annealing, which provide near-optimal solutions in significantly less time.

Key concepts

Algorithm Design

Algorithm design involves creating a step-by-step method to solve problems effectively. The Traveling Salesperson Problem (TSP) is a classic example requiring such design, where the goal is to find the shortest possible route that visits a set of cities and returns to the origin city. In TSP for 25 cities, the salesperson needs a well-planned algorithm to minimize travel distance and ensure each city is visited once.
A naive method is to list all permutations of the cities and evaluate each for the shortest route. However, this results in an immense number of possibilities – specifically, 25 factorial ( 25! ), which is over 15 septillion possible routes.
Since listing all permutations is computationally taxing, algorithm design for TSP often involves making strategic decisions to reduce the problem's complexity. This includes leveraging mathematical insights or computational shortcuts that help manage extensive possibilities more efficiently.

Optimization Techniques

Optimization techniques are crucial in solving real-world problems like the TSP in a practical manner. Instead of evaluating every possible route, which is inefficient, optimization methods aim to find the best solution efficiently.
Some common optimization techniques used in TSP include:

• Branch and Bound: A method that can eliminate large portions of possibilities without a full enumeration.
• Dynamic Programming: Breaks down problems into simpler subproblems, storing results to avoid repeated calculations.
• Linear Programming: A mathematical approach to determine the best outcome, often used in TSP for smaller instances.

These techniques work by narrowing down the field of possible solutions, making it feasible to find an optimal or near-optimal solution without excessive computations.

Heuristic Algorithms

Heuristic algorithms present a practical solution when exact algorithms are too slow due to computational complexity. They provide a way to find good enough answers quickly for large-scale problems like TSP.
Some popular heuristic methods for solving the TSP are:

• Nearest Neighbor: Always visit the nearest unvisited city, which is simple but might not always give the best solution.
• Genetic Algorithms: Mimic natural selection, where paths are "bred" to evolve into better solutions over successive generations.
• Simulated Annealing: Uses a probabilistic technique to escape local minima by occasionally allowing less optimal solutions temporarily.

They are designed to provide reasonable paths in much less time compared to brute-force methods, making them highly valued for their efficiency and practicality in real-world applications.

Computational Complexity

Computational complexity measures how the time or space requirements of an algorithm grow with input size. For TSP, understanding its complexity helps to grasp why some methods are favored over others.
The TSP is classified as NP-hard, meaning no known polynomial-time solution exists. As the number of cities increases, the number of potential solutions grows factorially, leading to exponential time requirements with traditional methods.
Given that computing all possible routes for 25 cities would take billions of years with current technology, studying computational complexity directs researchers and practitioners to develop more feasible algorithms, balancing between accuracy and efficiency.