Fairly Wired: Towards Leximin-Optimal Division of Electricity

In many parts of the world - particularly in developing countries - the demand for electricity exceeds the available supply. In such cases, it is impossible to provide electricity to all households simultaneously. This raises a fundamental question: how should electricity be allocated fairly? In this paper, we explore this question through the lens of egalitarianism - a principle that emphasizes equality by prioritizing the welfare of the worst-off households. One natural rule that aligns with this principle is to maximize the egalitarian welfare - the smallest utility across all households. We show that computing such an allocation is NP-hard, even under strong simplifying assumptions. Leximin is a stronger fairness notion that generalizes the egalitarian welfare: it also requires to maximize the smallest utility, but then, subject to that, the second-smallest, then the third, and so on. The hardness results extends directly to leximin as well. Despite this, we present a Fully Polynomial-Time Approximation Scheme (FPTAS) for leximin in the special case where the network connectivity graph is a tree. This means that we can efficiently approximate leximin - and, in particular, the egalitarian welfare - to any desired level of accuracy.