Approximating Dense Cases of Covering Problems

Abstract:

We study dense cases of several covering problems. An instance of
the set cover problem with $m$ sets is dense if there is $epsilon>0$
such that any element belongs to at least $epsilon m$ sets. We show
that the dense set cover problem can be approximated with the
performance ratio $clog n$ for any $c>0$ and it is unlikely to be
NP-hard. We construct a polynomial-time approximation scheme for the
dense Steiner tree problem in $n$-vertex graphs, i.e. for the case
when each terminal is adjacent to at least $epsilon n$ vertices. We
also study the vertex cover problem in $epsilon$-dense graphs. Though
this problem is shown to be still MAX-SNP-hard as in general graphs,
we find a better approximation algorithm with the performance ratio
$2over{1+epsilon}$. The {em superdense} cases of all these problems
are shown to be solvable in polynomial time.