A widespread belief about county splits in political districting plans is wrong

Consider the task of dividing a state into k contiguous political districts whose populations must not differ by more than one person, following current practice for congressional districting in the USA. A widely held belief among districting experts is that this task requires at least k-1 county splits. This statement has appeared in expert testimony, special master reports, and Supreme Court oral arguments. In this paper, we seek to dispel this belief. To illustrate, we find plans for several states that use zero county splits, i.e., all counties are kept whole, despite satisfying contiguity and 1-person deviation. This is not a rare phenomenon; Montana admits 30,223 such plans. In practice, mapmakers may need to satisfy additional criteria, like compactness, minority representation, and partisan fairness, which may lead them to believe k-1 splits to be minimum. Again, this need not be true. To illustrate, we conduct short case studies for North Carolina (for partisan fairness) and Alabama (for minority representation). Contrary to expert testimony and Supreme Court oral arguments from Allen v. Milligan (2023), we find that fewer than k-1 county splits suffices, even when subjected to these additional criteria. This demonstrates our narrow point that k-1 county splits should not be assumed minimum and also suggests that districting criteria do not conflict as much as people sometimes believe. The optimization methods proposed in this paper are flexible and can assist mapmakers in satisfying them.

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