Approximation Algorithm for the Minimum Hub Cover Set Problem

A subset
${\mathcal{ S}}\subseteq V$
of vertices of an undirected graph
$G=(V,E)$
is a hub cover when for each edge
$(u,v) \in E$
, at least one of its endpoints belongs to
${\mathcal{ S}}$
, or there exists a vertex
$r \in {\mathcal{ S}}$
that is a neighbor of both
$u$
and
$v$
. The problem of computing a minimum hub cover set in arbitrary graphs is NP-hard. This problem has applications for indexing large databases. This paper proposes
$\Psi $
-MHC, the first approximation algorithm for the minimum hub cover set in arbitrary graphs to the best of our knowledge. The approximation ratio of this algorithm is
$\ln \mu $
, where
$\mu $
is upper bounded by
$\min \left\{{\frac {1}{2}(\Delta +1)^{2}, \vert E\vert }\right\}$
and
$\Delta $
is the degree of
$G$
. The execution time of
$\Psi $
-MHC is
$O((\Delta + 1) \vert E \vert + \vert {\mathcal{ S}} \vert \vert V \vert)$
. Experimental results show that
$\Psi $
-MHC far outperforms the theoretical approximation ratio for the input graph instances.