On the hardness of approximating some NP-optimization problems related to minimum linear ordering problem

Mishra, Sounaka; Sikdar, Kripasindhu

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) no. 3, pp. 287-309.

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.

Classification : 68Q17, 68R01, 68W25
Mots clés : NP-optimization problems, minimaximal and maximinimal NP-optimization problems, approximation algorithms, hardness of approximation, APX-hardness, AP-reduction, L-reduction, S-reduction

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