  Redistricting is the problem of partitioning a set of geographical units into
a fixed number of districts, subject to a list of often-vague rules and
priorities. In recent years, the use of randomized methods to sample from the
vast space of districting plans has been gaining traction in courts of law for
identifying partisan gerrymanders, and it is now emerging as a possible
analytical tool for legislatures and independent commissions. In this paper, we
set up redistricting as a graph partition problem and introduce a new family of
Markov chains called Recombination (or ReCom) on the space of graph partitions.
The main point of comparison will be the commonly used Flip walk, which
randomly changes the assignment label of a single node at a time. We present
evidence that ReCom mixes efficiently, especially in contrast to the
slow-mixing Flip, and provide experiments that demonstrate its qualitative
behavior. We demonstrate the advantages of ReCom on real-world data and explain
both the challenges of the Markov chain approach and the analytical tools that
it enables. We close with a short case study involving the Virginia House of
Delegates.
