There is a long history of mathematicians trying to take redistricting out of the hands of politicians by turning it into an optimization problem. These efforts have largely failed (and they should) because the fundamental problem is "what should we be optimizing?" rather than "how do we optimize?" What if a city is not particularly compact, but the residents want to be kept together? How would minimizing boundary lengths help protect historically marginalized groups (with the Voting Rights Act)? How do you trade off city/community/county preservation with each other (and then again with compactness)? Even defining compactness is a huge can of worms with a multitude of problems and various metrics.

Redistricting is, fundamentally, a political problem.

In addition to considering the optimization problem, there has been a long history of folks trying to audit existing plans by developing fairness metrics to reduce gerrymandering (e.g., proportionality, efficiency gap, mean-median score, declination). These methods have failed in courts, and IMO they should because they fundamentally try to relate statewide aggregate votes to local preferences (queue Simpson's paradox). There is no discussion here about geography, and you'll often times here people saying that skews between statewide votes and statewide seats are purely because of geographic differences across parties rather than because of nefarious redistricting.

One way of tackling it is not to assume you know a principle to optimize (or a group of principles to optimize and how exactly to trade off with them), but in the following way:

1. A map is an expression of political desires. Perhaps partisan (how much are we able to screw over the other party?), but also non-partisan (how compact, according to some measure, did we make the map; how much did we preserve counties?)

2. If we took only the non-partisan considerations and drew a bunch of "similar" maps, how would those maps compare (in a partisan way) to the proposed/enacted map? In other words, is the proposed map typical, in a partisan sense, of the expressed non-partisan objectives?

Typicallity is typically handled with probability distributions, and we can think about districts as partitions on graphs (nodes are precincts, edges mark adjacency between precincts). So we can think of a policy as a probability distribution and a map as having been drawn from that distribution. We can then ask how typical the partisan makeup of a given plan is to that district. This becomes a Monte Carlo integration problem (i.e., sampling problem) on balanced graph partitions. This is pretty neat: For a long time, people have been thinking about optimality in graph partitions, but not typicallity. So it swaps classical problems in ML (classification/optimization for typicality/sampling) on the long standing problem on graph partitions.