We focus our attention on the class RMAX(2) of NP optimization problems. Owing to recent developments in interactive proof techniques, RMAX(2) was shown to be the lowest class of logical classification that contains problems hard to approximate. Namely, the RMAX(2)-complete problem MAX CLIQUE (of finding the size of the largest clique in a graph) is not approximable in polynomial time within any constant factor unless NP=P.
We are interested in problems inside RMAX(2) that are not known to be complete but are still hard to approximate. We point out that one such problem is MAXlog n, n, considered by Berman and Schnitger: given m conjunctions, each of them consisting of log m propositional variables or their negations, find the maximal number of simultaneously satisfiable conjunctions. We also obtain the approximation hardness results for some other problems in RMAX(2). Finally, we discuss the question of whether or not the problems under consideration are RMAX(2)-complete.