The probabilistic approach to limited packings in graphs

We consider (closed neighbourhood) packings and their generalization in graphs. 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 $L_k(G)$ of a graph 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. In this paper, we develop a new probabilistic approach to limited packings in graphs, resulting in lower bounds for the k-limited packing number and a randomized algorithm to find k-limited packings satisfying the bounds. In particular, we prove that for any graph G of order n with maximum vertex degree $\Delta$, $$L_k(G) \ge {kn \over (k+1)\sqrt[k]{\pmatrix{\Delta \cr k} (\Delta +1)}}.$$ The problem of finding a maximum size k-limited packing is known to be NP-complete even in split or bipartite graphs.