Approximating k-Connected m-Dominating Sets

A subset $S$ of nodes in a graph $G$ is a $k$-connected $m$-dominating set ($(k,m)$-cds) if the subgraph $G[S]$ induced by $S$ is $k$-connected and every $v \in V \setminus S$ has at least $m$ neighbors in $S$. In the $k$-Connected $m$-Dominating Set ($(k,m)$-CDS) problem the goal is to find a minimum weight $(k,m)$-cds in a node-weighted graph. For $m \geq k$ we obtain the following approximation ratios. For general graphs our ratio $O(k \ln n)$ improves the previous best ratio $O(k^2 \ln n)$ and matches the best known ratio for unit weights. For unit disc graphs we improve the ratio $O(k \ln k)$ to $\min\left\{\frac{m}{m-k},k^{2/3}\right\} \cdot O(\ln^2 k)$ -- this is the first sublinear ratio for the problem, and the first polylogarithmic ratio $O(\ln^2 k)/\epsilon$ when $m \geq (1+\epsilon)k$; furthermore, we obtain ratio $\min\left\{\frac{m}{m-k},\sqrt{k}\right\} \cdot O(\ln^2 k)$ for uniform weights. These results are obtained by showing the same ratios for the Subset $k$-Connectivity problem when the set $T$ of terminals is an $m$-dominating set with $m \geq k$

Latency-bounded target set selection in signed networks

It is well-documented that social networks play a considerable role in information spreading. The dynamic processes governing the diffusion of information have been studied in many fields, including epidemiology, sociology, economics, and computer science. A widely studied problem in the area of viral marketing is the target set selection: in order to market a new product, hoping it will be adopted by a large fraction of individuals in the network, which set of individuals should we “target” (for instance, by offering them free samples of the product)? In this paper, we introduce a diffusion model in which some of the neighbors of a node have a negative influence on that node, namely, they induce the node to reject the feature that is supposed to be spread. We study the target set selection problem within this model, first proving a strong inapproximability result holding also when the diffusion process is required to reach all the nodes in a couple of rounds. Then, we consider a set of restrictions under which the problem is approximable to some extent

Bounds on the 2-domination number

In a graph G, a set D⊆V(G) is called 2-dominating set if each vertex not in D has at least two neighbors in D. The 2-domination number γ2(G) is the minimum cardinality of such a set D. We give a method for the construction of 2-dominating sets, which also yields upper bounds on the 2-domination number in terms of the number of vertices, if the minimum degree δ(G) is fixed. These improve the best earlier bounds for any 6≤δ(G)≤21. In particular, we prove that γ2(G) is strictly smaller than n/2, if δ(G)≥6. Our proof technique uses a weight-assignment to the vertices where the weights are changed during the procedure. © 2017 Elsevier B.V

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 times larger α 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( ∆ + k) + 1 < ln(∆) + 1.7, with ∆ 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.