Title:

Approximating nodeweighted Steiner subgraphs for multicast communication in wireless networks

We are motivated by the problem of computing multicast routing structures in wireless adhoc networks modelled by special classes of graphs including unit disk graphs, quasiunit disk graphs and (λ + 1)clawfree graphs. Multicast communication can be established by a tree known as Steiner tree. Wireless adhoc networks must operate using limited resources, therefore, the suitability of nodes for inclusion in a Steiner tree can vary widely between different nodes. We model this by assuming that each node of the network is assigned a weight that represents the cost of including it in the Steiner tree. Our goal is to compute a Steiner tree with minimum total node weight. However, in scenarios where nodes and links are not reliable, a tree has the drawback that it can be disconnected by the failure of even a single link or node in the network. Therefore, we also consider various faulttolerant routing structures called 2edgeconnected Steiner subgraphs, kedgeconnected Steiner subgraphs, 2vertexconnected Steiner subgraphs, and 2edgeconnected group Steiner subgraphs. The problems we consider are NPhard, so we are interested in algorithms that compute provably good approximate solutions in polynomial time. We present a generalization of Steiner subgraph problems referred to as the nodeweighted δSteiner subgraph problem, where δ represents connectivity requirements. We present an algorithm with approximation ratio 0.5dρ for the nodeweighted δSteiner subgraph problem, where d is the bounded maximum degree of the solution subgraph, and ρ is the approximation ratio of the edgeweighted version of the δSteiner subgraph problem. We then shown how to construct solution subgraphs of bounded maximum degree d in several graph classes for our problem variants. As a result, we obtain algorithms for the problems we consider, on graph classes that admit subgraphs of small degree, whose approximation ratios are better than the best known ratios for the same problems on general graphs.
