Abstract
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 Hfiastad 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.
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Mishra, S., Sikdar, K. (2000). On the Hardness of Approximating Some NP-Optimization Problems Related to Minimum Linear Ordering Problem. In: van Leeuwen, J., Watanabe, O., Hagiya, M., Mosses, P.D., Ito, T. (eds) Theoretical Computer Science: Exploring New Frontiers of Theoretical Informatics. TCS 2000. Lecture Notes in Computer Science, vol 1872. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44929-9_15
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