Max-min Fair Rate Allocation and Routing in Energy Harvesting Networks: Algorithmic Analysis

This paper considers max-min fair rate allocation and routing in energy harvesting networks where fairness is required among both the nodes and the time slots . Unlike most previous work on fairness, we focus on multihop topologies and consider different routing methods . We assume a predictable energy profile and focus on the design of efficient and optimal algorithms that can serve as benchmarks for distributed and approximate algorithms. We first develop an algorithm that obtains a max-min fair rate assignment for any routing that is specified at the input. We then turn to the problem of determining a “good” routing. For time-invariable unsplittable routing , we develop an algorithm that finds routes that maximize the minimum rate assigned to any node in any slot. For fractional routing , we derive a combinatorial algorithm for the time-invariable case with constant rates. We show that the time-variable case is at least as hard as the 2-commodity feasible flow problem and design an FPTAS to combat the high running time. Finally, we show that finding an unsplittable routing or a routing tree that provides lexicographically maximum rate assignment (i.e., the best in the max-min fairness terms) is NP-hard, even for a time horizon of a single slot. Our analysis provides insights into the problem structure and can be applied to other related fairness problems.