Problem 3.
a) Define the notion of polynomial time factor c-approximation.
b) Design a polynomial time factor 3-approximaiton for minimum weighted vertex cover
problem assuming the weight of each vertex is either 1 or 2. Give and prove the
computational time and approximation ratio. You may use the method from slides, which
is based on matching.
c) Can you design a polynomial time factor 2-approximation algorithm for minimum
weighted vertex cover problem such that each vertex has a positive weight? Give and
prove the computational time and approximation ratio.
The minimum weighted vertex cover problem: given a graph G that each vertex i has a
positive weight wi, find a subset S of vertices in G such that the sum of weights in the S
is the least, and each edge of G is adjacent to at least one vertex in S.
For example, the weight of vertex 1 is 10, and the weights of vertices 2, 3, and 4 are
equal to 1 in the graph below. Both {1} and {2,3,4} are vertex cover solutions. The
minimum vertex cover is {2, 3, 4} instead of {1}. The sum of weights of vertices in {2, 3,
4} is 1+1+1=3, which is less than the weight 10 for vertex 1.
w2=1
2
w1=10
1
3
w3=1
4
w4=1
