Travelling Salesman Problem | Part 2

Solution Approach(Dynamic programming)

A dynamic programming paradigm is used when we have problems that can be decomposed into similar sub-problems. In this paradigm, the results of the subproblems are stored to reuse later. These algorithms are primarily used for optimization.

As we saw earlier, TSP is an optimization problem. To apply the DP technique, we need to find out if TSP can be divided into similar subproblems or not. 

Let's take the above example. The graph and its cost adjacency matrix is given below:

We can observe that the adjacency cost matrix is symmetric. By symmetric, we mean the distance between cities 2 to 3 is the same as cities 3 to 2.
Let’s take a tour T from city 1 to any of the cities 2, 3, 4 as T (1,{2,3,4}). There are three possible ways to complete this tour. 

Case 1: If he visits from City 1 to 2, this further depends on a subproblem, i.e. visiting from City 2 to City 3 or 4.

Case 2: If he visits from City 1 to 3, this further depends on a subproblem, i.e. visiting from City 3 to City 2 or 4.

Case 3: If he visits from City 1 to 4, this further depends on a subproblem, i.e. visiting from City 4 to City 2 or 3.

Here, we can observe that TSP can be divided into subproblems. This means we can use dynamic programming techniques to solve this problem.

Now, let’s see the dynamic programming approach in detail of the tour mentioned above:

Let the equation 1 be: 

The above problem depends on some new subproblems. Let’s solve them first.

Now putting the values of these subproblems in equation 1

Here, the tour having the path 1 -> 2 -> 4 -> 3 -> 1 and 1 -> 3 -> 4 -> 2 -> 1 are same and having the minimum cost as 80.

Implementation

Here, we are using the DP array to store the results of the subproblems. We are using bit masking techniques to keep track of the cities that are visited.

#include<bits/stdc++.h>
using namespace std;

#define MAX 999999
//Number of Cities
int n=4;
//Adjacency Matrix of the graph
int Adj_matrix[10][10] = {
        {0,10,15,20},
        {10,0,35,25},
        {15,35,0,30},
        {20,25,30,0}
};
//Bit Mask to keep track of visited Cities
int Visited = (1<<n) - 1;
//DP Array to store the results of subproblem
int dp[16][4];

int  TSP_DP(int mask,int pos){
//If the city is already visited
    if(mask==Visited){
		return Adj_matrix[pos][0];
	}
//Lookup Case
	if(dp[mask][pos]!=-1){
	  return dp[mask][pos];
	}
//Now, from the current node, we will try to go to every 
      //other node and take the min ans
	int ans = MAX;
	//Visit all the unvisited cities and take the best route
	for(int city=0;city<n;city++){
		if((mask&(1<<city))==0){
			int newAns = Adj_matrix[pos][city] + TSP_DP( mask|(1<<city), city);
			ans = min(ans, newAns);
			}
	}
	return dp[mask][pos] = ans;
} 

int main(){

    //Initialisation of DP Array

    for(int i=0;i<(1<<n);i++)
    {
        for(int j=0;j<n;j++)
        {
            dp[i][j] = -1;
        }
    }
    //Starting city: 1, BitMask: 0
	cout<<"Minimum Cost is: "<<TSP_DP(1,0);
	return 0;
}

You can also try this code with Online C++ Compiler

Run Code

Output

Minimum Cost is: 80

Time and Space Complexity

• The time complexity is O (n²2ⁿ), where n is the number of cities. 

Reason: While solving recursive equations, we will get a total (n-1) 2^(n-2)  sub-problems, which is O (n2ⁿ). Each sub-problem will take O (n) time to find a path to the remaining (n-1) cities.

Therefore total time complexity is O (n2ⁿ) * O (n) = O (n²2ⁿ)

• Space Complexity is O (n2ⁿ).

The implementation of the travelling salesman problem using the brute force approach is explained in Part-1. So, go check it out!

Check out Longest Common Substring

Frequently Asked Questions

What is TSP?

TSP is the travelling salesman problem consists of a salesperson and his travel to various cities. The salesperson must travel to each of the cities, beginning and ending in the same. 

What type of problem is TSP?

TSP is an optimization problem. In this, we have to optimize the route of travel.

How is TSP related to the minimum spanning tree?

TSP and MST are two algorithmic problems that are closely connected. In particular, an open-loop TSP solution is a spanning tree, although it is not always the shortest spanning tree.

Conclusion

In this article, we ran you through the travelling salesman problem. We saw a dynamic programming approach to solve the problem. 

There are various algorithmic paradigms such as Dynamic Programming and Backtracking to solve this problem. We saw earlier that TSP is somewhat related to the minimum spanning tree. You can visit these two outstanding articles on Prims and Kruskal's algorithm.

Check out this problem - Minimum Coin Change Problem 

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