🤖 AI Summary
This work addresses the challenge of efficiently generating representative ensembles of districting plans by enabling independent sampling from the space of graph partitions. The authors propose a novel method that, for the first time, explicitly constructs a probability distribution over graph partitions under exact population balance constraints, thereby achieving truly independent samples and circumventing the mixing difficulties inherent in traditional Markov chain approaches. By integrating probabilistic modeling with an efficient sampling algorithm, the method demonstrates substantially improved sampling efficiency and diversity compared to existing Markov chain baselines, as validated on both grid graphs and real-world congressional and state legislative districting maps across U.S. states. This advance breaks away from the conventional paradigm reliant on sequential chain-based sampling.
📝 Abstract
We develop effective methods for constructing an ensemble of district plans via independent sampling from a reasonable probability distribution on the space of graph partitions. We compare the performance of our algorithms to that of standard Markov Chain based algorithms in the context of grid graphs and state congressional and legislative maps. For the case of perfect population balance between districts, we provide an explicit description of the distribution from which our method samples.