Approximating Fault-Tolerant Domination in General Graphs

In this paper we study the NP-complete problem of finding small k-dominating sets in general graphs, which allow k---1 nodes to fail and still dominate the graph. The classic minimum dominating set problem is a special case with k = 1. We show that the approach of having at least k dominating nodes in the neighborhood of every node is not optimal. For each α > 1 it can give solutions k/α times larger than a minimum k-dominating set. We also study lower bounds on possible approximation ratios. We show that it is NP-hard to approximate the minimum k-dominating set problem with a factor better than (0.2267/k) ln(n/k). Furthermore, a result for special finite sums allows us to use a greedy approach for k-domination with an approximation ratio of ln(Δ + k) + 1 < ln(Δ) + 1.7, with Δ being the maximum node-degree. We also achieve an approximation ratio of ln(n)+1.7 for h-step k-domination, where nodes do not need to be direct neighbors of dominating nodes, but can be h steps away.