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In the process of redistricting, one important metric is the number of competitive districts, that is, districts where both parties have a reasonable chance of winning a majority of votes. Competitive districts are important for achieving proportionality, responsiveness, and other desirable qualities; some states even directly list competitiveness in their legally-codified districting requirements. In this work, we discuss the problem of drawing plans with at least a fixed number of competitive districts. In addition to the standard, ``vote-band'' measure of competitivenesss (i.e., how close was the last election?), we propose a measure that explicitly considers ``swing voters'' - the segment of the population that may choose to vote either way, or not vote at all, in a given election. We present two main, contrasting results. First, from a computational complexity perspective, we show that the task of drawing plans with competitive districts is NP-hard, even on very natural instances where the districting task itself is easy (e.g., small rectangular grids of population-balanced cells). Second, however, we show that a simple hill-climbing procedure can in practice find districtings on real states in which all the districts are competitive. We present the results of the latter on the precinct-level graphs of the U.S. states of North Carolina and Arizona, and discuss trade-offs between competitiveness and other desirable qualities., Comment: To be published in FORC 2024

When auditing a redistricting plan, a persuasive method is to compare the plan with an ensemble of neutrally drawn redistricting plans. Ensembles are generated via algorithms that sample distributions on balanced graph partitions. To audit the partisan difference between the ensemble and a given plan, one must ensure that the non-partisan criteria are matched so that we may conclude that partisan differences come from bias rather than, for example, levels of compactness or differences in community preservation. Certain sampling algorithms allow one to explicitly state the policy-based probability distribution on plans, however, these algorithms have shown poor mixing times for large graphs (i.e. redistricting spaces) for all but a few specialized measures. In this work, we generate a multiscale parallel tempering approach that makes local moves at each scale. The local moves allow us to adopt a wide variety of policy-based measures. We examine our method in the state of Connecticut and succeed at achieving fast mixing on a policy-based distribution that has never before been sampled at this scale. Our algorithm shows promise to expand to a significantly wider class of measures that will (i) allow for more principled and situation-based comparisons and (ii) probe for the typical partisan impact that policy can have on redistricting., Comment: 26 pages with appendix; 11 figures