We study hardness of approximating several minimaximal and maximinimal NP-optimizationproblems related to the minimum linear ordering problem (MINLOP). MINLOP is to find a minimum weight acyclic tournament in a given arc-weighted completedigraph. MINLOP is APX-hard but its unweighted version is polynomial timesolvable. We prove that MIN-MAX-SUBDAG problem, which is a generalization ofMINLOP and requires to find a minimum cardinality maximal acyclic subdigraphof a given digraph, is, however, APX-hard. Using results of Håstad concerning hardness of approximating independence number of a graph we then prove similar results concerning MAX-MIN-VC (respectively, MAX-MIN-FVS) which requires to find a maximum cardinality minimal vertex cover in a given graph (respectively,a maximum cardinality minimal feedback vertex set in a given digraph). We alsoprove APX-hardness of these and several related problems on various degree bounded graphs and digraphs.
