Design and Analysis - Vertex Cover Problem

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Have you ever wondered about the placement of traffic cameras? That how they are efficiently placed without wasting too much budget from the government? The answer to that comes in the form of vertex-cover algorithm. The positions of the cameras are chosen in such a way that one camera covers as many roads as possible, i.e., we choose junctions and make sure the camera covers as much area as possible.

A vertex-cover of an undirected graph G = (V,E) is the subset of vertices of the graph such that, for all the edges (u,v) in the graph,u and v ∈ V. The junction is treated as the node of a graph and the roads as the edges. The algorithm finds the minimum set of junctions that cover maximum number of roads.

It is a minimization problem since we find the minimum size of the vertex cover – the size of the vertex cover is the number of vertices in it. The optimization problem is an NP-Complete problem and hence, cannot be solved in polynomial time; but what can be found in polynomial time is the near optimal solution.

Vertex Cover Algorithm

The vertex cover approximation algorithm takes an undirected graph as an input and is executed to obtain a set of vertices that is definitely twice as the size of optimal vertex cover.

The vertex cover is a 2-approximation algorithm.

Algorithm

Step 1 − Select any random edge from the input graph and mark all the edges that are incident on the vertices corresponding to the selected edge.

Step 2 − Add the vertices of the arbitrary edge to an output set.

Step 3 − Repeat Step 1 on the remaining unmarked edges of the graph and add the vertices chosen to the output until there’s no edge left unmarked.

Step 4 − The final output set achieved would be the near-optimal vertex cover.

Pseudocode

APPROX-VERTEX_COVER (G: Graph)
c ← { }
E’ ← E[G]
while E’ is not empty do
   Let (u, v) be an arbitrary edge of E’
   c ← c U {u, v}
   Remove from E’ every edge incident on either u or v
return c

Example

The set of edges of the given graph is −

{(1,6),(1,2),(1,4),(2,3),(2,4),(6,7),(4,7),(7,8),(3,5),(8,5)}

Now, we start by selecting an arbitrary edge (1,6). We eliminate all the edges, which are either incident to vertex 1 or 6 and we add edge (1,6) to cover.

In the next step, we have chosen another edge (2,3) at random.

Now we select another edge (4,7).

We select another edge (8,5).

Hence, the vertex cover of this graph is {1,6,2,3,4,7,5,8}.

Analysis

It is easy to see that the running time of this algorithm is O(V + E), using adjacency list to represent E'.