## Tuesday, March 4, 2008

### k-Set-Cover

Jeremiah asked me today about the status of the problem "k-Set-Cover" for small k. This is the Min-Set-Cover problem where all the sets have cardinality at most k.

I wasn't exactly sure on the status, actually. On the homework we showed that the greedy algorithm gives an $H_k$ ($k$th harmonic number) factor approximation. I guessed that this was known to be more or less sharp.

Turns out that is true to some extent. With some augmentations to Feige's result, Trevisan showed that the problem is hard to approximate within a factor of $\ln k - \Omega(\ln \ln k)$.

While this looks pretty sharp, it's actually useless for the relatively interesting case of small constant $k$. For $k = 2$ the problem is AKA "Edge Cover", and it's solvable in P.

For $k \geq 3$ the problem is known to be NP-hard, and furthermore known to be "APX-hard", meaning that it is hard to approximate within a factor of $1 + \epsilon$ for some universal positive $\eps > 0$. (Result due to Papadimitriou and Yannakakis?) Improving this to an explicit decent value of $\epsilon$, depending on $k$, would be a nice problem.

On the positive side, the case of most interest to Jeremiah was $k = 3$. Here the best result is due to Duh and Fürer: a factor 4/3 algorithm. Actually, they get a factor $H_k - 1/2$ algorithm for any $k$.

There have been subsequent marginal improvements. A result slightly better than 19/12 is known for $k = 4$, and the world record is $H_k - .59$ for all $k \geq 6$; that appears in this recent paper of Athanassopoulos, Caragiannis, and Kaklamanis.