Uniform and balanced sampling of connected planar graph partitions is hard unless RP=NP, the flip walk mixes exponentially slowly on explicit triangulation families, and tractable cases include series-parallel and bounded-treewidth graphs.
Quantifying Gerrymandering in North Carolina
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abstract
Using an ensemble of redistricting plans, we evaluate whether a given political districting faithfully represents the geo-political landscape. Redistricting plans are sampled by a Monte Carlo algorithm from a probability distribution that adheres to realistic and non-partisan criteria. Using the sampled redistricting plans and historical voting data, we produce an ensemble of elections that reveal geo-political structure within the state. We showcase our methods on the two most recent districtings of NC for the U.S. House of Representatives, as well as a plan drawn by a bipartisan redistricting panel. We find the two state enacted plans are highly atypical outliers whereas the bipartisan plan accurately represents the ensemble both in partisan outcome and in the fine scale structure of district-level results.
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Complexity and Geometry of Sampling Connected Graph Partitions
Uniform and balanced sampling of connected planar graph partitions is hard unless RP=NP, the flip walk mixes exponentially slowly on explicit triangulation families, and tractable cases include series-parallel and bounded-treewidth graphs.