Completeness in approximation classes beyond APX

We present a reduction that allows us to establish completeness results for several approximation classes mainly beyond APX. Using it, we extend one of the basic results of S. Khanna, R. Motwani, M. Sudan, and U. Vazirani (On syntactic versus computational views of approximability, SIAM J. Comput., 28:164–191, 1998) by proving the existence of complete problems for the whole Log-APX, the class of problems approximable within ratios that are logarithms of the size of the instance. We also introduce a new approximability class, called Poly-APX(∆), dealing with graph-problems approximable with ratios functions of the maximum degree ∆ of the input graph. For this class also, using the reduction propose, we establish complete problems.