高级算法 (Fall 2023)/Problem Set 1

Problem 1 (Min-cut/Max-cut)

[counting [math]\displaystyle{ \alpha }[/math]-approximate min-cut] For any [math]\displaystyle{ \alpha \ge 1 }[/math], a cut is called an [math]\displaystyle{ \alpha }[/math]-approximate min-cut in a multigraph [math]\displaystyle{ G }[/math] if the number of edges in it is at most [math]\displaystyle{ \alpha }[/math] times that of the min-cut. Prove that the number of [math]\displaystyle{ \alpha }[/math]-approximate min-cuts in a multigraph [math]\displaystyle{ G }[/math] is at most [math]\displaystyle{ n^{2\alpha} / 2 }[/math]. Hint: Run Karger's algorithm until it has [math]\displaystyle{ \lceil 2\alpha \rceil }[/math] supernodes. What is the chance that a particular [math]\displaystyle{ \alpha }[/math]-approximate min-cut is still available? How many possible cuts does this collapsed graph have?
[weighted min-cut problem] 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.