Algorithms, Vol. 13, Pages 31: Constrained Connectivity in Bounded X-Width Multi-Interface Networks (Algorithms)

As technology advances and the spreading of wireless devices grows, the establishment of interconnection networks is becoming crucial. Main activities that involve most of the people concern retrieving and sharing information from everywhere. In heterogeneous networks, devices can communicate by means of multiple interfaces. The choice of the most suitable interfaces to activate (switch-on) at each device results in the establishment of different connections. A connection is established when at its endpoints the devices activate at least one common interface. Each interface is assumed to consume a specific percentage of energy for its activation. This is referred to as the cost of an interface. Due to energy consumption issues, and the fact that most of the devices are battery powered, special effort must be devoted to suitable solutions that prolong the network lifetime. In this paper, we consider the so-called p-Coverage problem where each device can activate at most p of its available interfaces in order to establish all the desired connections of a given network of devices. As the problem has been shown to be NP -hard even for p = 2 and unitary costs of the interfaces, algorithmic design activities have focused in particular topologies where the problem is optimally solvable. Following this trend, we first show that the problem is polynomially solvable for graphs (modeling the underlying network) of bounded treewidth by means of the Courcelle’s theorem. Then, we provide two optimal polynomial time algorithms to solve the problem in two subclasses of graphs with bounded treewidth that are graphs of bounded pathwidth and graphs of bounded carvingwidth. The two solutions are obtained by means of dynamic programming techniques.