  Capacitated Domination generalizes the classic Dominating Set problem by
specifying for each vertex a required demand and an available capacity for
covering demand in its closed neighborhood. The objective is to find a
minimum-sized set of vertices that can cover all of the graph's demand without
exceeding any of the capacities. In this paper we look specifically at
domination with hard-capacities, where the capacity and cost of a vertex can
contribute to the solution at most once. Previous complexity results suggest
that this problem cannot be solved (or even closely approximated) efficiently
in general. In this paper we present a polynomial-time approximation scheme for
Capacitated Domination in unweighted planar graphs when the maximum capacity
and maximum demand are bounded. We also show how this result can be extended to
the closely-related Capacitated Vertex Cover problem.
