NP-completeness of the {k}-packing function problem in graphs

Given a positive integer k, the {k}-packing function problem ({k}PF) is to find in a given graph G, a function f of maximum weight that assigns a non-negative integer to the vertices of G in such a way that the sum of f(v) over each closed neighborhood is at most k. In this work we prove that {k}PF is NP-complete for general graphs. We also expand the set of instances where it is known that {k}PF is linear time solvable, by proving that it is so in dually chordal graphs.