How Do You Solve a Traveling Salesman Problem?

Different Approaches to Look Out For

Given a set of cities & the distance between every pair of cities as an adjacency matrix, find shortest route that visits each city once & returns to the starting point.

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Problem Statement

1. Choose any city as the starting and ending point.   2. Generate all possible permutations of cities that are (n-1)!.  3. Calculate the cost of each permutation.

Approach 1: Simple Approach

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Time complexity: O(N!), Where N is the number of cities.  Space complexity: O(1).

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In this, subsets of required cities, distances between them & starting city are taken as inputs. Using recursive calls, the cost function for each subset is calculated.

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Approach 2: Dynamic Programming

Time Complexity: O(N^2*2^N)

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1. Create a list to hold indices of cities & an array for results. 2. Traverse adjacency matrix & update cost if a cheaper path is found.  3. Generate minimum path cycle & return minimum cost.

Approach 3: Greedy Approach

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Time complexity: O(N^2*logN), Where N is the number of cities.  Space complexity: O(N).

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