高级算法 (Fall 2019)/Problem Set 4

• 作业电子版于2020/1/14 23:59 之前提交到邮箱 njuadvalg@163.com
• 每道题目的解答都要有完整的解题过程。中英文不限。

Problem 1

Modify the Karger's Contraction algorithm so that it works for the weighted min-cut problem. Prove that the modified algorithm returns a weighted minimum cut with probability at least [math]\displaystyle{ \frac{2}{n(n-1)} }[/math]. The weighted min-cut problem is defined as follows.

• Input: an undirected weighted graph [math]\displaystyle{ G(V, E) }[/math], where every edge [math]\displaystyle{ e \in E }[/math] is associated with a positive real weight [math]\displaystyle{ w_e }[/math];
• Output: a cut [math]\displaystyle{ C }[/math] in [math]\displaystyle{ G }[/math] such that [math]\displaystyle{ \sum_{e \in C} w_e }[/math] is minimized.