Towards a Polynomial Equivalence Between $\{k\}$-Packing Functions and k-Limited Packings in Graphs

Abstract

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. This notion was recently introduced as a variation of the k-limited packing problem (kLP) introduced in 2010, where the function was supposed to assign a value in $\{0,1\}$. For all the graph classes explored up to now, $\{k\}$PF and kLP have the same computational complexity. It is an open problem to determine a graph class where one of them is NP-complete and the other, polynomially solvable. In this work, we first prove that $\{k\}$PF is NP-complete for bipartite graphs, as kLP is known to be. We also obtain new graph classes where the complexity of these problems would coincide.