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I was reading about the $\mathcal{O}(\frac{1}{\log n})$ approximation algorithm for the Unique Coverage Problem from these notes.

The gist of the algorithm is as follows:

  • Partition the elements into $\log n$ classes according to their degrees, i.e., the number of sets that cover the element. So class $i$ contains all the elements which are covered with at least $2^i$ and at most $2^{i+1}$ sets.
  • Let $i$ be the class of the maximum cardinality.
  • Choose any set with probability $\frac1{2i}$$\frac1{2^i}$

Then in Lemma 3, they go on to show that

Lemma 3: The expected number of the elements uniquely covered from class $i$ is $\frac{1}{e^2}$ fraction of the element in class $i$.

I was able to follow the algorithm till here but I am completely stumped as to how they arrived at the statement immediately following lemma 3

Therefore, the total profit of uniquely covered elements is at least $\frac{1}{e^2 \log n} \times \text{opt}$

How did they arrive to this conclusion? They didn't even show what the upper bound is for this maximization algorithm. What am I overlooking?

It is trivial to prove that the expected number of unique elements covered by the algorithms is $\frac{n}{e^2}$ since it directly follows Lemma 3 but that is about all that I am able to observe :/

I was reading about the $\mathcal{O}(\frac{1}{\log n})$ approximation algorithm for the Unique Coverage Problem from these notes.

The gist of the algorithm is as follows:

  • Partition the elements into $\log n$ classes according to their degrees, i.e., the number of sets that cover the element. So class $i$ contains all the elements which are covered with at least $2^i$ and at most $2^{i+1}$ sets.
  • Let $i$ be the class of the maximum cardinality.
  • Choose any set with probability $\frac1{2i}$

Then in Lemma 3, they go on to show that

Lemma 3: The expected number of the elements uniquely covered from class $i$ is $\frac{1}{e^2}$ fraction of the element in class $i$.

I was able to follow the algorithm till here but I am completely stumped as to how they arrived at the statement immediately following lemma 3

Therefore, the total profit of uniquely covered elements is at least $\frac{1}{e^2 \log n} \times \text{opt}$

How did they arrive to this conclusion? They didn't even show what the upper bound is for this maximization algorithm. What am I overlooking?

It is trivial to prove that the expected number of unique elements covered by the algorithms is $\frac{n}{e^2}$ since it directly follows Lemma 3 but that is about all that I am able to observe :/

I was reading about the $\mathcal{O}(\frac{1}{\log n})$ approximation algorithm for the Unique Coverage Problem from these notes.

The gist of the algorithm is as follows:

  • Partition the elements into $\log n$ classes according to their degrees, i.e., the number of sets that cover the element. So class $i$ contains all the elements which are covered with at least $2^i$ and at most $2^{i+1}$ sets.
  • Let $i$ be the class of the maximum cardinality.
  • Choose any set with probability $\frac1{2^i}$

Then in Lemma 3, they go on to show that

Lemma 3: The expected number of the elements uniquely covered from class $i$ is $\frac{1}{e^2}$ fraction of the element in class $i$.

I was able to follow the algorithm till here but I am completely stumped as to how they arrived at the statement immediately following lemma 3

Therefore, the total profit of uniquely covered elements is at least $\frac{1}{e^2 \log n} \times \text{opt}$

How did they arrive to this conclusion? They didn't even show what the upper bound is for this maximization algorithm. What am I overlooking?

It is trivial to prove that the expected number of unique elements covered by the algorithms is $\frac{n}{e^2}$ since it directly follows Lemma 3 but that is about all that I am able to observe :/

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Approximation Algorithm for the Unique Coverage Problem

I was reading about the $\mathcal{O}(\frac{1}{\log n})$ approximation algorithm for the Unique Coverage Problem from these notes.

The gist of the algorithm is as follows:

  • Partition the elements into $\log n$ classes according to their degrees, i.e., the number of sets that cover the element. So class $i$ contains all the elements which are covered with at least $2^i$ and at most $2^{i+1}$ sets.
  • Let $i$ be the class of the maximum cardinality.
  • Choose any set with probability $\frac1{2i}$

Then in Lemma 3, they go on to show that

Lemma 3: The expected number of the elements uniquely covered from class $i$ is $\frac{1}{e^2}$ fraction of the element in class $i$.

I was able to follow the algorithm till here but I am completely stumped as to how they arrived at the statement immediately following lemma 3

Therefore, the total profit of uniquely covered elements is at least $\frac{1}{e^2 \log n} \times \text{opt}$

How did they arrive to this conclusion? They didn't even show what the upper bound is for this maximization algorithm. What am I overlooking?

It is trivial to prove that the expected number of unique elements covered by the algorithms is $\frac{n}{e^2}$ since it directly follows Lemma 3 but that is about all that I am able to observe :/