  Given a graph $G = (V, E)$ and an integer $k$, the Minimum Membership
Dominating Set problem asks to compute a set $S \subseteq V$ such that for each
$v \in V$, $1 \leq |N[v] \cap S| \leq k$. The problem is known to be
NP-complete even on split graphs and planar bipartite graphs. In this paper, we
approach the problem from the algorithmic standpoint and obtain several
interesting results. We give an $\mathcal{O}^*(1.747^n)$ time algorithm for the
problem on split graphs. Following a reduction from a special case of 1-in-3
SAT problem, we show that there is no sub-exponential time algorithm running in
time $\mathcal{O}^*(2^{o(n)})$ for bipartite graphs, for any $k \geq 2$. We
also prove that the problem is NP-complete when $\Delta = k+2$, for any $k\geq
5$, even for bipartite graphs. We investigate the parameterized complexity of
the problem for the parameter twin cover and the combined parameter distance to
cluster, membership($k$) and prove that the problem is fixed-parameter
tractable. Using a dynamic programming based approach, we obtain a linear-time
algorithm for trees.
