Reduction from set cover problem to vertex cover problem

Question

Although the reduction from vertex cover problem to set cover problem is quite simple, I did not find anywhere the reduction in the opposite direction. From the similarity in the type of problems, I guess this reduction should be simple too. However, despite trying for some time, I could not develop this. So, any ideas how this reduction can be done?

Answer 1

Edit: Wherever it says "vertex cover", read "dominating set".

Suppose that the sets are $S_1,\ldots,S_n$ and the elements are $x_1,\ldots,x_m$. For each set $S_i$ there will correspond a vertex $S_i$. For each element $x_j$ there will correspond $n$ vertices $x_j^{(1)},\ldots,x_j^{(n)}$. There is also an additional vertex $t$. Finally, assume wlog that the optimal solution for set cover is not $S_1,\ldots,S_n$ (this can be checked in polynomial time).

The instance of vertex cover is as follows. The vertex $t$ is connected to all of $S_1,\ldots,S_n$. Whenever $x_j \in S_i$, the vertex $S_i$ is connected to all of $x_j^{(1)},\ldots,x_j^{(n)}$. If there's a set cover of size $M$, then there is a vertex cover of size $M+1$ (we need to take the extra vertex $t$ so that all set vertices are covered). I believe that the converse should hold as well.

Try to work it out yourself, and if it doesn't work out, try to fix it and let us all know how.

Answer 2

If you make a set of subsets out of the edges of a graph and solve the set cover problem, youll have solved the vertex cover problem. This is true for any graph. If you take a set of subsets,

1. You may not be able to make it into graph.
2. even if you did the vertex cover might not be the set cover.