Bounds and algorithms for limited packings in graphs

Abstract

We consider (closed neighbourhood) packings and their generalization in graphs called limited packings. A vertex set X in a graph G is a k-limited packing if for any vertex $v\in V(G)$, $\left|N[v] \cap X\right| \le k$, where $N[v]$ is the closed neighbourhood of $v$. The k-limited packing number $L_k(G)$ is the largest size of a k-limited packing in G. Limited packing problems can be considered as secure facility location problems in networks. We develop probabilistic and greedy approaches to limited packings in graphs, providing lower bounds for the k-limited packing number, and randomized and greedy algorithms to find k-limited packings satisfying the bounds. Some upper bounds for $L_k(G)$ are given as well. The problem of finding a maximum size k-limited packing is known to be NP-complete even in split or bipartite graphs.