Question

Use the Greedy Heuristic solve the following instance of Min Set Cover Problem. Your solution must contain every step of applying the Heuristic. And answer if the solution you get is the optimal solution.No formal proof is needed.

S={q1,q2,q3,q4,q5,q6}
C={c1={q1,q3,q5},c2={q2,q6},c3={q1,q6},c4={q2,q4,q5},c5={q1,q3,q6},c6={q4,q5,q6}}

Answer 1

The Greedy Heuristic is applied to the Min Set Cover Problem by sequentially selecting sets that cover the most uncovered elements, resulting in the selection of sets c6, c1, and c4 for this instance. The heuristic does not guarantee optimality but provides a feasible solution.

Step-by-step explanation:

The Greedy Heuristic for the Min Set Cover Problem involves selecting the set that covers the most uncovered elements at each step. For the given instance S={q1,q2,q3,q4,q5,q6} and C={c1={q1,q3,q5}, c2={q2,q6}, c3={q1,q6}, c4={q2,q4,q5}, c5={q1,q3,q6}, c6={q4,q5,q6}}, the steps would proceed as follows:

1. Choose set c6 because it covers the most number of elements, which are q4, q5, q6.
2. From the remaining uncovered elements (q1, q2, q3), c1 covers the most (q1, q3, q5) but since q5 is already covered by c6 we look at other sets. c4 covers q2, and c1 covers q1 and q3. Both cover two new elements, but we'll prioritize the one with the larger overall set, thus choose c1.
3. Now, the only uncovered element is q2, which can be covered by selecting c2 or c4. Since c4 is already in our selection, we choose c4.

The sets chosen by the Greedy Heuristic are c6, c1, and c4. This solution covers all elements in S and consists of three sets.