Completeness in standard and differential approximation classes: Poly-(D)APX- and (D)PTAS-completeness

Several problems are known to be APX-, DAPX-, PTAS-, or Poly-APX-PB-complete under suitably defined approximation-preserving reductions. But, to our knowledge, no natural problem is known to be PTAS-complete and no problem at all is known to be Poly-APX-complete. On the other hand, DPTAS- and Poly-DAPX-completeness have not been studied until now. We first prove in this paper the existence of natural Poly-APX- and Poly-DAPX-complete problems under the well known PTAS-reduction and under the DPTAS-reduction (defined in “G. Ausiello, C. Bazgan, M. Demange, and V. Th. Paschos, Completeness in differential approximation classes, MFCS’03”), respectively. Next, we deal with PTAS- and DPTAS-completeness. We introduce approximation preserving reductions, called FT and DFT, respectively, and prove that, under these new reductions, natural problems are PTAS-complete, or DPTAS-complete. Then, we deal with the existence of intermediate problems under our reductions and we partially answer this question showing that the existence of NPO-intermediate problems under Turing-reductions is a sufficient condition for the existence of intermediate problems under both FT- and DFT-reductions. Finally, we show that MIN COLORING is DAPX-complete under DPTAS-reductions. This is the first DAPX-complete problem that is not simultaneously APX-complete.