Weighted Vertex Cover Problem

The weighted vertex cover problem aims to find the minimum weight vertex cover in a graph where vertices have associated weights. It is an optimization version of the basic vertex cover problem.

Some key aspects:

  • Graph has weighted vertices
  • Find subset of vertices that covers all edges
  • Minimize total weight of chosen vertex cover

Java example:

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int minVertexCover(Graph graph) {
  int[] vertices = graph.getVertices();
  int[] weights = graph.getWeights();
  
  int n = vertices.length;
  int[][] dp = new int[n+1][1<<n];
  
  // Compute minimum vertex covers bottom-up
  for (int i = 1; i <= n; i++) {
    for (int mask = 0; mask < (1<<n); mask++) {
      // Try including/excluding vertex i
      dp[i][mask] = Math.min(include, exclude); 
    }
  }

  // Result is full vertex cover using all nodes
  return dp[n][(1<<n)-1]; 
}

C++ example:

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int minVertexCover(Graph graph) {
  vector<int> vertices = graph.getVertices();
  vector<int> weights = graph.getWeights();

  int n = vertices.size();
  vector<vector<int>> dp(n+1, vector<int>(1<<n, 0));

  // Bottom-up dynamic programming
  for (int i = 1; i <= n; i++) {
    for (int mask = 0; mask < (1<<n); mask++) {
      // Compute options
      dp[i][mask] = min(include, exclude);
    }
  }

  return dp[n][(1<<n)-1]; // Full vertex cover
}

Python example:

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def minVertexCover(graph):

  @cache
  def dfs(node, mask):
    
    if mask == completed_mask:
      return 0
    
    cost = float('inf')
    for next_node in graph[node]:
      if (mask & (1<<next_node)) == 0:
        cost = min(cost, dfs(next_node, mask | (1<<next_node)) + graph.weights[next_node])
        
    return cost

  return dfs(0, 1)

The weighted version of vertex cover has applications in resource optimization and scheduling.

The Weighted Vertex Cover Problem is an extension of the Vertex Cover Problem. Each vertex has an associated weight, and the goal is to find the smallest set of vertices such that every edge is touched by at least one vertex in the set, while minimizing the total weight of the vertices selected. In simpler terms, you have to find the least “expensive” set of vertices that covers all edges.

Algorithm

  1. Identify Edges: List all edges in the graph.
  2. Identify Vertex Weights: Associate a weight with each vertex.
  3. Select Vertices: Choose vertices that connect these edges.
  4. Minimize Weight: Minimize the total weight of the selected vertices.

Java Code

In Java, you can use data structures like HashMap to hold vertex weights and a HashSet for the cover.

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import java.util.HashMap;
import java.util.HashSet;

public class WeightedVertexCover {
    public static void main(String[] args) {
        int[][] graph = {{0, 1}, {0, 2}, {1, 3}, {3, 4}};
        HashMap<Integer, Integer> weights = new HashMap<>();  // vertex, weight
        weights.put(0, 1);
        weights.put(1, 2);
        weights.put(2, 1);
        weights.put(3, 2);
        weights.put(4, 3);

        HashSet<Integer> cover = new HashSet<>();
        int totalWeight = 0;

        for (int[] edge : graph) {
            if (!cover.contains(edge[0]) && !cover.contains(edge[1])) {
                if (weights.get(edge[0]) <= weights.get(edge[1])) {
                    cover.add(edge[0]);
                    totalWeight += weights.get(edge[0]);
                } else {
                    cover.add(edge[1]);
                    totalWeight += weights.get(edge[1]);
                }
            }
        }

        System.out.println("Vertex Cover: " + cover);
        System.out.println("Total Weight: " + totalWeight);
    }
}

C++ Code

In C++, you can use std::map to keep track of vertex weights and std::set for the vertex cover.

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#include <iostream>
#include <map>
#include <set>
#include <vector>

int main() {
    std::vector<std::pair<int, int>> graph = {{0, 1}, {0, 2}, {1, 3}, {3, 4}};
    std::map<int, int> weights = {{0, 1}, {1, 2}, {2, 1}, {3, 2}, {4, 3}};  // vertex, weight
    std::set<int> cover;
    int totalWeight = 0;

    for (auto &edge : graph) {
        if (cover.find(edge.first) == cover.end() && cover.find(edge.second) == cover.end()) {
            if (weights[edge.first] <= weights[edge.second]) {
                cover.insert(edge.first);
                totalWeight += weights[edge.first];
            } else {
                cover.insert(edge.second);
                totalWeight += weights[edge.second];
            }
        }
    }

    std::cout << "Vertex Cover: ";
    for (int vertex : cover) {
        std::cout << vertex << " ";
    }
    std::cout << "\nTotal Weight: " << totalWeight << std::endl;

    return 0;
}

Python Code

Python’s dictionaries can serve to hold the vertex weights, and a set can be used for the vertex cover.

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graph = [(0, 1), (0, 2), (1, 3), (3, 4)]
weights = {0: 1, 1: 2, 2: 1, 3: 2, 4: 3}  # vertex: weight
cover = set()
total_weight = 0

for edge in graph:
    if edge[0] not in cover and edge[1] not in cover:
        if weights[edge[0]] <= weights[edge[1]]:
            cover.add(edge[0])
            total_weight += weights[edge[0]]
        else:
            cover.add(edge[1])
            total_weight += weights[edge[1]]

print("Vertex Cover:", cover)
print("Total Weight:", total_weight)

This provides a straightforward approach to solving the Weighted Vertex Cover Problem using basic data structures in Java, C++, and Python. Again, note that this is a heuristic approach and may not provide the minimum weighted vertex cover for all graphs. Advanced algorithms like dynamic programming can provide an optimal solution.