Dealing with residual energy when transmitting data in energy-constrained capacitated networks

This paper addresses several problems relating to the energy available after the transmission of a given amount of data in a capacitated network. The arcs have an associated parameter representing the energy consumed during the transmission along the arc and the nodes have limited power to transmit data. In the first part of the paper, we consider the problem of designing a path which maximizes the minimum of the residual energy remaining at the nodes. After formulating the problem and proving the main theoretical results, a polynomial time algorithm is proposed based on computing maxmin paths in a sequence of non-capacitated networks. In the second part of the paper, the problem of obtaining a quickest path in this context is analyzed. First, the bi-objective variant of this problem is considered in which we aim to minimize the transmission time and to maximize the minimum residual energy. An exact polynomial time algorithm is proposed to find a minimal complete set of efficient solutions which amounts to solving shortest path problems. Second, the problem of computing an energy-constrained quickest path which guarantees at least a given residual energy at the nodes is reformulated as a variant of the energy-constrained quickest path problem. The algorithms are tested on a set of benchmark problems providing the optimal solution or the Pareto front within reasonable computing times.