CSCE-658 Randomized Algorithms

1. G0 = G; h = 0; 2. while Gh has more than 2 vertices do randomly pick an edge eh in Gh; Gh+1 = Gh/eh; h = h+ 1; 3. return all edges in Gn−2. As we have shown in section 1, the probability Pr[Eh | h−1 i=1 Ei] that, under the condition that the first (h − 1)-st iterations of the algorithm do not pick an edge in the min-cut C of G, the probability that the h-th iteration does not pick an edge in C is at least 1−2/nh−1 = (n−h−1)/(n−h+1), where nh−1 = n − h + 1 is the number of vertices in the graph Gh−1. When nh−1 is larger, this probability is good. However, when nh−1 gets smaller (for example, when h = n − 2, n(n−2)−1 = nn−3 = 3), the probability 1 − 2/nh−1 becomes bad: for instance, for h = n − 2, 1 − 2/nh−1 = 1/3. As a result, we were only able to show that the probability that the algorithm Contraction returns a min-cut of the input graph is not smaller than 2/(n(n − 1) ≈ 2/n. Thus, we need to repeat Ω(n) times of the algorithm Contraction, in order to have a good probability that the algorithm returns a correct solution, which has a significant impact on the complexity of the algorithm. In order to improve the efficiency of the algorithm for Min-Cut, we can consider how to increase the success probability of the algorithm Contraction. Since the probability (n− h− 1)/(n− h+ 1) gets small when h gets large, if we stop the iteration in the algorithm Contraction earlier when the number nh−1 = n− h+ 1 of vertices in the graph Gh−1 is sufficiently larger, we should end up with a good probability of success. However, then how do we deal with the resulting graph, which still has a significant number of vertices? Karger and Stein’s idea is to work on the small graph Gh−1 multiple times to increase the success probability. Consider the following algorithm: