Approximation Algorithms 1.1 Minimum Vertex Cover

1 Approximation Algorithms Any known algorithm that finds the solution to an NP-hard optimization problem has exponential running time. However, sometimes polynomial time algorithms exist which find a " good " solution instead of an optimum solution. Given a minimization problem and an approximation algorithm, we can evaluate the algorithm as follows. First, we find a lower bound on the optimum solution. Then we compare the algorithm's performance against the lower bound. For a maximization problem, we would find an upper bound and compare the solutions found by our approximation algorithm with that. Remember a vertex cover is a set of vertices that touch all the edges in the graph. The Minimum Vertex Cover Problem is to find the least-cardinality vertex cover. A lower bound on the minimum vertex cover is given by a maximal matching. Since no two edges in a matching share the same vertex, there must be at least one vertex in the vertex cover for each edge in the matching. Also, notice that the set of all matched vertices in a maximum matching is a vertex cover. This follows as any edge whose end-vertices are both unmatched may be added to the matching, contradicting the maximality of the matching. Clearly this algorithm contains twice as many vertices as our lower bound, which is the number of edges in a maximal matching. So the algorithm is within twice optimal. Two issues are of interest here: how good is our lower bound with respect to the optimal solution, and how good is our final solution with respect to the optimal solution. First we show that the lower bound can be a factor 2 away from optimal. Consider the complete graph with n edges. The maximal matching has n 2 edges, so our lower bound is n 2 .