A primal-dual approximation algorithm for the Minimum Cost Stashing problem in wireless sensor networks

We study the problem of computing an energy-efficient data delivery scheme in wireless sensor networks that leverages the knowledge of a set of trajectories of mobile sinks in the network to route data from the sensors to the mobile sinks. Sensors collect data from the environment and instead of directly sending them to the mobile sinks (henceforth simply “sinks”), they route data to a number of selected nodes (we call them relay or stashing nodes) in the network. These stashing nodes lie on the trajectories of the sinks, and relay the received data directly to the sinks on behalf of the sensors. Assuming a set of p different applications being executed on each sensor node, we consider the following problem: Given a set of p trajectories T₁, T₂, ..., T_P corresponding to p sinks, where sink Ti is dedicated to collect i-th application data, node u selects at least k stashing nodes from T_i such that it can forward at least k copies of its i-application data to them. The goal of u is to minimize the total routing cost to send all its p application data to the corresponding stashing nodes of p trajectories. We use the expected number of transmissions on a link as the routing cost of the link and the routing cost for a path is the sum of all the link costs of that path. We wish to minimize the sum of the total routing costs of all the nodes. We call this the Minimum Cost Stashing problem and formulate this as a primal-dual problem. We present a 1/2(2f -k + 1)-approximation algorithm using the primal-dual method for approximation algorithms, where f is the maximum size of a trajectory.