  In this article, we study the $d$-distance $m$-tuple ($\ell, r$)-domination
problem. Given a simple undirected graph $G=(V, E)$, and positive integers $d,
m, \ell$ and $r$, a subset $V' \subseteq V$ is said to be a $d$-distance
$m$-tuple ($\ell, r$)-dominating set if it satisfies the following conditions:
(i) each vertex $v \in V$ is $d$-distance dominated by at least $m$ vertices in
$V'$, and (ii) each $r$ size subset $U$ of $V$ is $d$-distance dominated by at
least $\ell$ vertices in $V'$. Here, a vertex $v$ is $d$-distance dominated by
another vertex $u$ means the shortest path distance between $u$ and $v$ is at
most $d$ in $G$. A set $U$ is $d$-distance dominated by a set of $\ell$
vertices means size of the union of the $d$-distance neighborhood of all
vertices of $U$ in $V'$ is at least $\ell$. The objective of the $d$-distance
$m$-tuple ($\ell, r$)-domination problem is to find a minimum size subset $V'
\subseteq V$ satisfying the above two conditions.
  We prove that the problem of deciding whether a graph $G$ has (i) a
1-distance $m$-tuple ($\ell, r$)-dominating set for each fixed value of $m,
\ell$, and $r$, and (ii) a $d$-distance $m$-tuple ($\ell, 2$)-dominating set
for each fixed value of $d (> 1), m$, and $\ell$ of cardinality at most $k$
(here $k$ is a positive integer) are NP-complete. We also prove that for any
$\varepsilon>0$, the 1-distance $m$-tuple $(\ell, r)$-domination problem and
the $d$-distance $m$-tuple $(\ell,2)$-domination problem cannot be approximated
within a factor of $(\frac{1}{2}- \varepsilon)\ln |V|$ and $(\frac{1}{4}-
\varepsilon)\ln |V|$, respectively, unless $P = NP$.
