The Delay Constrained Relay Node Placement Problem (DCRNPP) frequently arises in the Wireless Sensor Network (WSN) design. In WSN, Sensor Nodes are placed across a target geographical region to detect relevant signals. These signals are communicated to a central location, known as the Base Station, for further processing. The DCRNPP aims to place the minimum number of additional Relay Nodes at a subset of Candidate Relay Node locations in such a manner that signals from various Sensor Nodes can be communicated to the Base Station within a pre-specified delay bound. In this paper, we study the structure of the projection polyhedron of the problem and develop valid inequalities in form of the node-cut inequalities. We also derive conditions under which these inequalities are facet defining for the projection polyhedron. We formulate a branch-and-cut algorithm, based upon the projection formulation, to solve DCRNPP optimally. A Lagrangian relaxation based heuristic is used to generate a good initial solution for the problem that is used as an initial incumbent solution in the branch-and-cut approach. Computational results are reported on several randomly generated instances to demonstrate the efficacy of the proposed algorithm. 
Optimal relay node placement in delay constrained wireless sensor network design