高级算法 (Fall 2023)/Problem Set 2

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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{ {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.

• [Max directed-cut] In the maximum directed cut (MAX-DICUT) problem, we are given as input a directed graph [math]\displaystyle{ G(V,E) }[/math]. Each directed arc [math]\displaystyle{ (i, j) \in E }[/math] has nonnegative weight [math]\displaystyle{ w_{ij} \ge 0 }[/math]. The goal is to partition [math]\displaystyle{ V }[/math] into disjoint sets [math]\displaystyle{ U }[/math] and [math]\displaystyle{ W=V\setminus U }[/math] so as to maximize the total weight of the arcs going from [math]\displaystyle{ U }[/math] to [math]\displaystyle{ W }[/math]. Give a randomized [math]\displaystyle{ 1/4 }[/math]-approximation algorithm for this problem.