高级算法 (Fall 2021)/Problem Set 1

• 每道题目的解答都要有完整的解题过程。中英文不限。

Problem 1

Recall that in class we show by the probabilistic method how to deduce a [math]\displaystyle{ \frac{n(n-1)}{2} }[/math] upper bound on the number of distinct min-cuts in any multigraph [math]\displaystyle{ G }[/math] with [math]\displaystyle{ n }[/math] vertices from the [math]\displaystyle{ \frac{2}{n(n-1)} }[/math] lower bound for success probability of Karger's min-cut algorithm.

Also recall that the [math]\displaystyle{ FastCut }[/math] algorithm taught in class guarantees to return a min-cut with probability at least [math]\displaystyle{ \Omega(1/\log n) }[/math]. Does this imply a much tighter [math]\displaystyle{ O(\log n) }[/math] upper bound on the number of distinct min-cuts in any multigraph [math]\displaystyle{ G }[/math] with [math]\displaystyle{ n }[/math] vertices? Prove your improved upper bound if your answer is "yes", and give a satisfactory explanation if your answer is "no".