2 research outputs found
Analysis of d-Hop Dominating Set Problem for Directed Graph with Indegree Bounded by One
Efficient communication between nodes in ad-hoc networks can be established
through repeated cluster formations with designated \textit{cluster-heads}. In
this context minimum d-hop dominating set problem was introduced for cluster
formation in ad-hoc networks and is proved to be NP-complete. Hence, an exact
solution to this problem for certain subclass of graphs (representing an ad-hoc
network) can be beneficial. In this short paper we perform computational
complexity analysis of minimum d-hop dominating set problem for directed graphs
with in-degree bounded by . The optimum solution of the problem can be found
polynomially by exploiting certain properties of the graph under consideration.
For a digraph an solution is provided
to the problem
Statistical Mechanics of the Directed 2-distance Minimal Dominating Set problem
The directed L-distance minimal dominating set (MDS) problem has wide
practical applications in the fields of computer science and communication
networks. Here, we study this problem from the perspective of purely
theoretical interest. We only give results for an Erds
Rnyi (ER) random graph and regular random graph, but this work can
be extended to any type of networks. We develop spin glass theory to study the
directed 2-distance MDS problem. First, we find that the belief propagation
algorithm does not converge when the inverse temperature exceeds a threshold on
either an ER random network or regular random network. Second, the entropy
density of replica symmetric theory has a transition point at a finite inverse
temperature on a regular random graph when the node degree exceeds 4 and on an
ER random graph when the node degree exceeds 6.6; there is no entropy
transition point (or ) in other circumstances. Third, the results
of the replica symmetry (RS) theory are in perfect agreement with those of
belief propagation (BP) algorithm while the results of the belief propagation
decimation (BPD) algorithm are better than those of the greedy heuristic
algorithm