We have introduced and discussed below Karger’s algorithm in set 1.

1) Initialize contracted graph CG as copy of original graph 2) While there are more than 2 vertices. a) Pick a random edge (u, v) in the contracted graph. b) Merge (or contract) u and v into a single vertex (update the contracted graph). c) Remove self-loops 3) Return cut represented by two vertices.

As discussed in the previous post, Karger’s algorithm doesn’t always find min cut. In this post, the probability of finding min-cut is discussed.

* Probability that the cut produced by Karger’s Algorithm is Min-Cut is greater than or equal to 1/(n^{2})*

**Proof:**

Let there be a unique Min-Cut of given graph and let there be C edges in the Min-Cut and the edges be {e_{1}, e_{2}, e_{3}, .. e_{c}}. The Karger’s algorithm would produce this Min-Cut if and only if none of the edges in set {e_{1}, e_{2}, e_{3}, .. e_{c}} is removed in iterations in the main while loop of above algorithm.

c is number of edges in min-cut m is total number of edges n is total number of vertices S_{1}= Event that one of the edges in {e_{1}, e_{2}, e_{3}, .. e_{c}} is chosen in 1^{st}iteration. S_{2}= Event that one of the edges in {e_{1}, e_{2}, e_{3}, .. e_{c}} is chosen in 2^{nd}iteration. S_{3}= Event that one of the edges in {e_{1}, e_{2}, e_{3}, .. e_{c}} is chosen in 3^{rd}iteration. .................. .................. The cut produced by Karger's algorithm would be a min-cut if none of the above events happen. So the required probability is P[S_{1}^{'}∩ S_{2}^{'}∩ S_{3}^{'}∩ ............]

**Probability that a min-cut edge is chosen in first iteration:**

Let us calculate P[S_{1}^{'}] P[S_{1}] = c/m P[S_{1}^{'}] = (1 - c/m)Above value is in terms of m (or edges), let us convert it in terms of n (or vertices) using below 2 facts.. 1) Since size of min-cut is c, degree of all vertices must be greater than or equal to c. 2) As per Handshaking Lemma, sum of degrees of all vertices = 2m From above two facts, we can conclude below. n*c <= 2m m >= nc/2 P[S_{1}] <= c / (cn/2) <= 2/n P[S_{1}] <= c / (cn/2) <= 2/n P[S_{1}^{'}] >= (1-2/n) ------------(1)

**Probability that a min-cut edge is chosen in second iteration:**

P[S_{1}^{'}∩ S_{2}^{'}] = P[S_{2}^{'}| S_{1}^{'}] * P[S_{1}^{'}] In the above expression, we know value of P[S_{1}^{'}] >= (1-2/n) P[S_{2}^{'}| S_{1}^{'}] is conditional probability that is, a min cut is not chosen in second iteration given that it is not chosen in first iteration Since there are total (n-1) edges left now and number of cut edges is still c, we can replace n by n-1 in inequality (1). So we get. P[S_{2}^{'}| S_{1}^{'}] >= (1 - 2/(n-1)) P[S_{1}^{'}∩ S_{2}^{'}] >= (1-2/n) x (1-2/(n-1))

**Probability that a min-cut edge is chosen in all iterations:**

P[S_{1}^{'}∩ S_{2}^{'}∩ S_{3}^{'}∩.......... ∩ S_{n-2}^{'}] >= [1 - 2/n] * [1 - 2/(n-1)] * [1 - 2/(n-2)] * [1 - 2/(n-3)] *... ... * [1 - 2/(n - (n-4)] * [1 - 2/(n - (n-3)] >= [(n-2)/n] * [(n-3)/(n-1)] * [(n-4)/(n-2)] * .... 2/4 * 2/3 >= 2/(n * (n-1)) >= 1/n^{2}

**How to increase probability of success?**

The above probability of success of basic algorithm is very less. For example, for a graph with 10 nodes, the probability of finding the min-cut is greater than or equal to 1/100. The probability can be increased by repeated runs of basic algorithm and return minimum of all cuts found.

**Applications:**

**1)** In war situation, a party would be interested in finding minimum number of links that break communication network of enemy.

**2) **The min-cut problem can be used to study reliability of a network (smallest number of edges that can fail).

**3) **Study of network optimization (find a maximum flow).

**4)** Clustering problems (edges like associations rules) Matching problems (an NC algorithm for min-cut in directed graphs would result in an NC algorithm for maximum matching in bipartite graphs)

**5)** Matching problems (an NC algorithm for min-cut in directed graphs would result in an NC algorithm for maximum matching in bipartite graphs)

**Sources: **

https://www.youtube.com/watch?v=-UuivvyHPas

http://disi.unal.edu.co/~gjhernandezp/psc/lectures/02/MinCut.pdf

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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