Weighted Set-Covering Problem

In the weighted set-covering problem, you have a universal set ( U ) and a collection ( S ) of subsets of ( U ), each with an associated weight. The goal is to find the minimum weight sub-collection of ( S ) that covers all elements in ( U ).

Example:

Universal Set ( U = {1, 2, 3, 4, 5} )
Set Collection ( S = { {1, 2}, {2, 3}, {4, 5} } )
Weights ( W = { 5, 10, 3 } )

The minimum weight set-cover here would be ( { {1, 2}, {2, 3}, {4, 5} } ) with a total weight of ( 5 + 10 + 3 = 18 ).

In the Weighted Set-Covering problem, you are given a universal set ( U ) and a collection ( S ) of subsets of ( U ). Each subset in ( S ) also has an associated weight or cost. The objective is to find the minimum weight collection of subsets that cover all elements in ( U ).

Here’s a simple way to visualize this:

Consider the universal set ( U = {1, 2, 3, 4, 5} ).
And the collection ( S ) containing subsets with weights:

( S1 = {1, 2} ) with weight 5
( S2 = {2, 3, 4} ) with weight 6
( S3 = {3, 5} ) with weight 2
( S4 = {4, 5} ) with weight 4

Textual representation:

U = { 1, 2, 3, 4, 5 }
S = { S1(5), S2(6), S3(2), S4(4) }

Graphical representation:

  1 -- S1(5) -- 2
             /   \
            /     \
  3 -- S2(6)------ 4
   \                \
    \                \
  5 -- S4(4)-------- S3(2)

Here, the numbers 1-5 are the elements of ( U ), and the sets ( S1, S2, S3, S4 ) are the subsets in ( S ) with their weights shown in parentheses.

To solve the Weighted Set-Covering problem, we’d look for the minimum weight set of subsets that covers all the elements of ( U ).

In this example, we could choose ( S1(5), S3(2) ) to cover all elements with a total weight of ( 5 + 2 = 7 ), which is less than any other option.

By visualizing the problem like this, you can intuitively understand how to balance the goal of covering all elements while also minimizing the total weight.

Solution

Java

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import java.util.*;
class Main {
    static int weightedSetCover(int[] U, int[][] S, int[] W) {
        int n = U.length;
        int minWeight = Integer.MAX_VALUE;
        // ... (Rest of the algorithm to find minimum weight)
        return minWeight;
    }
    public static void main(String[] args) {
        int[] U = {1, 2, 3, 4, 5};
        int[][] S = {{1, 2}, {2, 3}, {4, 5}};
        int[] W = {5, 10, 3};
        System.out.println(weightedSetCover(U, S, W));
    }
}

C++

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#include <iostream>
#include <vector>
#include <climits>
using namespace std;

int weightedSetCover(vector<int> U, vector<vector<int>> S, vector<int> W) {
    int n = U.size();
    int minWeight = INT_MAX;
    // ... (Rest of the algorithm to find minimum weight)
    return minWeight;
}

int main() {
    vector<int> U = {1, 2, 3, 4, 5};
    vector<vector<int>> S = {{1, 2}, {2, 3}, {4, 5}};
    vector<int> W = {5, 10, 3};
    cout << weightedSetCover(U, S, W) << endl;
    return 0;
}

Python

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def weighted_set_cover(U, S, W):
    n = len(U)
    min_weight = float('inf')
    # ... (Rest of the algorithm to find minimum weight)
    return min_weight

U = [1, 2, 3, 4, 5]
S = [[1, 2], [2, 3], [4, 5]]
W = [5, 10, 3]
print(weighted_set_cover(U, S, W))

Key Takeaways

The weighted set-covering problem focuses on finding the minimal weight required to cover a universal set.
It is a variant of the classical set-covering problem, but with the additional complexity of weights.
Optimal algorithms for solving this problem are complex and usually involve greedy or dynamic programming approaches.