Optimal redistricting under geographical constraints: Why "pack and crack" does not work

We show that optimal partisan redistricting with geographical constraints is a computationally intractable (NP-complete) problem. In particular, even when voter's preferences are deterministic, a solution is generally not obtained by concentrating opponent's supporters in \unwinnable" districts ("packing") and spreading one's own supporters evenly among the other districts in order to produce many slight marginal wins ("cracking").

A computational approach to unbiased districting

In the context of discrete districting problems with geographical constraints, we demonstrate that determining an (ex post) unbiased districting, which requires that the number of representatives of a party should be proportional to its share of votes, turns out to be a computationally intractable (NP-complete) problem. This raises doubts as to whether an independent jury will be able to come up with a “fair” redistricting plan in case of a large population, that is, there is no guarantee for finding an unbiased districting (even if such exists). We also show that, in the absence of geographical constraints, an unbiased districting can be implemented by a simple alternating-move game among the two parties.