We study
hardness of approximating several minimaximal and maximinimal NP-optimization
problems related to the minimum linear ordering problem (MINLOP). MINLOP is
to find a minimum weight acyclic tournament in a given arc-weighted complete
digraph. MINLOP is APX-hard but its unweighted version is polynomial time
solvable. We prove that MIN-MAX-SUBDAG problem, which is a generalization of
MINLOP and requires to find a minimum cardinality maximal acyclic subdigraph
of 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 also
prove APX-hardness of these and several related problems on various degree
bounded graphs and digraphs.