Capacitated Dominating Set on Planar Graphs
  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. 
Amariah Becker
cs.DS