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https://github.com/sleekpanther/minimum-weighted-vertex-cover-approximation-algorithm

Approximation Algorithm for the NP-Complete problem of finding a vertex cover of minimum weight in a graph with weighted vertices. Guarantees an answers at most 2 times the optimal minimum weighted vertex cover
https://github.com/sleekpanther/minimum-weighted-vertex-cover-approximation-algorithm

algorithm algorithm-design approximation approximation-algorithms edge-cost graph minimum-weighted-vertex-cover noah noah-patullo noahpatullo np-complete pattullo pattulo patullo patulo pricing-method vertex vertex-cover vertices weighted-vertex-cover

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Approximation Algorithm for the NP-Complete problem of finding a vertex cover of minimum weight in a graph with weighted vertices. Guarantees an answers at most 2 times the optimal minimum weighted vertex cover

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README 

        
# Minimum Weighted Vertex Cover - Pricing Method (Approximation Algorithm)

Approximation Algorithm for the NP-Complete problem of finding a **vertex cover of minimum weight** in a graph with weighted vertices. Guarantees an answers at most **2 times** the optimal minimum weighted vertex cover (2-approximation algorithm, [see references for the proof](#references)).



## Problem Statement

- Given an **undirected graph** with each vertex weighted > 0

- Find a vertex cover **S ⊆ V** (where each edge has at least 1 edge in **S**)

- And the vertex cover has the **minimum total weight** (when adding weights of the selected vertices)

#### Graph1




#### Graph 1 Optimal Minimum Weighted Vertex Cover




Vertex Cover = a, c 


Total Weight = 4+5 = 9




#### Graph 1 Algorithm's Weighted Vertex Cover (sub-optimal)




Vertex Cover = a, b, d 


Total Weight = 3+4+3 = 10




The algorithm **does not find the optimal solution**, but the answer given is **10**, which is less than **twice the optimal value** which would be **2 * 9 = 18**  



## Algorithm Solution

The problem is **NP-Complete** but this algorithm is a polynomial-time 2-approximation algorithm.  

The answer found is at most **2 times** the weight of the **Optimal** Minimum Weighted Vertex Cover.



### Pricing Strategy (Fairness)

- Each edge ***e=(i, j)*** must pay a price Pe > 0 to the vertices **i** and **j**

- **Fairness:** an edge cannot pay more than the remaining weight of either vertex

- **Tight:** a vertex is tight when it has no remaining weight  

![](images/pseudocode.png)  



## Input Graphs

#### Graph 2




#### Graph2 Algorithm's Weighted Vertex Cover (optimal)




Vertex Cover = a, d, b 


Total Weight = 2+2+4 = 8




#### Graph 3




#### Graph 3 Optimal Minimum Weighted Vertex Cover




Vertex Cover = a, c, f 


Total Weight = 2+4+9 = 15




#### Graph 3 Algorithm's Weighted Vertex Cover (sub-optimal)




Vertex Cover = a, b, c, e 


Total Weight = 2+3+4+7 = 16




#### Graph 4




#### Graph 4 Algorithm's Weighted Vertex Cover (optimal)




Vertex Cover = d, c, b 


Total Weight = 1+1+1 = 3




## Usage

- Create a `int[] weights` array  

`weights[0]` is the weight of **vertex `0`**, `weights[1]` is the weight of **vertex `1`**, etc.

- *(Optional)* Create `String` array for vertex names (0=a, 1=b, 2=c etc. in the example but this is arbitrary & it works as long as the underlyng graph created with integers is valid)

- Create the graph by adding undirected edges (**only add each edge once**)  

*e.g.*  

`graph1.add(new Edge(0,1, vertexNames1));;`  

`graph1.add(new Edge(0,3, vertexNames1));`  

*etc.*

- Vertex names can be left out (A names array is created automatically if none is provided using the String representation of the integer). There's an example for `graph4` which doesn't pass a names array

- **The order of edges added and the names of the vertices can result in a different vertex cover**

- Call `MinimumWeightedVertexCover.findMinimumWeightedVertexCoverApprox(graph1, weights1, vertexNames1);`



## References

- [Approximation Algorithms - Kevin Wayne](https://www.cs.princeton.edu/~wayne/kleinberg-tardos/pdf/11ApproximationAlgorithms.pdf#page=25)

- BAD [Vertex Cover Problem - GeeksForGeeks](http://www.geeksforgeeks.org/vertex-cover-problem-set-1-introduction-approximate-algorithm-2/)  

**This was a terrible mistake! This algorithm is very different from the approximation algorithm and had me confused on my initial attempt**