高级算法 (Fall 2023)/Problem Set 1

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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]
• [max directed-cut]

Problem 2 (Fingerprinting)

• [Polynomial Identity Testing]
• [Test isomorphism of rooted tree]
• [2D pattern matching]

Problem 3 (Hashing)

• [Bloom filter]
• [Count Distinct Element]

Problem 4 (Concentration of measure)

• [[math]\displaystyle{ k }[/math]-th moment bound]
• [the median trick]
• [cut size in random graph]
• [code rate of boolean code]
• [balls into bins with the "power of two choices"]

Problem 5 (Dimension reduction)

• [inner product]
• [linear separability]
• [sparse vector]

Problem 1 (Lovász Local Lemma)

• [colorable hypergrap]
• [directed cycle]
• [algorithmic LLL]