TO WEIGHT OR NOT TO WEIGHT: WHERE IS THE QUESTION?

We investigate the approximability properties of several weighted problems, by comparing them with the respective unweighted problems. For an appropriate (and very general) deenition of niceness, we show that if a nice weighted problem is hard to approximate within r, then its polynomially bounded weighted version is hard to approximate within r. Then we turn our attention to specific problems, and we show that the un-Max Exact kSat are exactly as hard to approximate as their weighted versions. We note in passing that Min Vertex Cover is exactly as hard to approximate as Min Sat. In order to prove the reductions for Max 2Sat, Max Cut, Max Directed Cut, and Max E3Sat we introduce the new notion of "mixing" set and we give an explicit construction of such sets. These reductions give new non-approximability results for these problems.