I'll return them on Tuesday. The average was 79%. Some comments:

1a) Several people erroneously claimed a log*|D| approximation here. (In fact, no factor approximation is possible.) The reason the AKC algorithm does not work is as follows: Assume we have correctly guessed R = Opt. At first, we indeed know that there k vertices which R-cover all of D. So the first Set-Cover approximation returns some set of vertices A R-covering D. But now, we do not know that there are k vertices which R-cover A. So we are stuck.

2. This was surprisingly not done so well. Most people seemed to get the key point that ($c_0$, $s_0$) hardness implies hardness along the two line segments (in your diagram) joining this point to (1,1) and to (0,0). But people seemed to have a very hard time deducing the easy region (due to the Vertex-Cover algorithm), which is the triangle with vertices (1,1), (1, 0), and (1-1/k, 0).

3. Most everybody got this one, except for failing to point out in part b the simple but key fact that the reduction preserves Opt = 1.

4. As mentioned earlier, this question was kind of screwed up. In honor of that, everyone received 3 bonus points. Mainly, we just wanted you to revisit the Max-Coverage proof, pay attention to the parameters, and see how the parameters from #3 fit in.

5. Almost no one tried this, although it is not too hard.

6. Most people who tried this more or less got it, although some failed to argue why shifting preserves the weight of the set. For a solution, see the original paper.

## Sunday, March 16, 2008

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