On Weighted vs Unweighted Versions of Combinatorial Optimization Problems

We investigate the approximability properties of several weighted problems, by comparing them with the respective unweighted problems. For an appropriate (and very general) definition 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 − o(1). Then we turn our attention to specific problems, and we show that the unweighted versions of Min Vertex Cover, Min Sat, Max Cut, Max DiCut, Max 2Sat, and 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 DiCut, 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.