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.