{"id":27012,"date":"2021-09-03T06:00:00","date_gmt":"2021-09-03T13:00:00","guid":{"rendered":"https:\/\/insidebigdata.com\/?p=27012"},"modified":"2021-09-04T09:09:41","modified_gmt":"2021-09-04T16:09:41","slug":"redistricting-with-optimization","status":"publish","type":"post","link":"https:\/\/insidebigdata.com\/2021\/09\/03\/redistricting-with-optimization\/","title":{"rendered":"Redistricting with Optimization"},"content":{"rendered":"\n

Introduction<\/strong><\/p>\n\n\n\n

The US Census Bureau conducts a full census of all states in the USA every ten years. When the data is collected, collated, checked and released it must be used to redistrict each sate for the election of representatives to the US Congress, State Senate and State House. This is an expensive and politically fraught process often involving significant litigation. Gerrymandering is commonplace and causes political disquiet in its undermining of the democratic process.<\/p>\n\n\n\n

We propose to use optimization to accomplish these redistricting tasks. For each state these assignment problems can be formulated as a Mixed Integer Programs (MIPs) [1]. The MIP ensures district contiguity, minimizes district diameter and attempts to balance district population size. Furthermore it needs to ensure compliance with the Voting Rights Act (1965 and renewed) in that if a district with a majority of a minority group can be created, it should be.<\/p>\n\n\n\n

The MIP models are necessarily very large and challenging even for a state like Michigan which is by no means the most populous. We use ODH|CPLEX from Optimization Direct Inc. and IBM Corp. as a tool for solving these models. Typically it ensures that district population deviation from the average is less than half a percent and that the districts are appropriately shaped \u2013 without holes, not winding and generally compact.<\/p>\n\n\n\n

The merit of this method is that it is entirely non-political having no Republican or Democrat bias whatsoever. It is both transparent and auditable. It is configurable in that it can reflect state priorities in trading, for example, district diameter off with population balance.<\/p>\n\n\n\n

Methodology<\/strong><\/p>\n\n\n\n

Each redistricting task is an assignment problem in which each of a large number of indivisible voter tracts need to be assigned to one of a much smaller number of districts. Voters in a district elect a representative for the US Congress, State Senate or State House according to the redistricting task. The assignment must yield districts which are approximately equally sized in terms of population, contiguous, without holes and compact.<\/p>\n\n\n\n

There are between 132 and 8057 voter tracts in a state depending on its population size (2010 Census). The number of districts varies from 1 (congressional district in Wyoming) to 400 (for the State House in New Hampshire). Typically there are around 2813 voter tracts, 13 congressional districts, 38 State Senate districts and 110 State House districts (for Michigan, 2010 Census).<\/p>\n\n\n\n

Because the geographical position of the voter tracts needs to be considered, we allow every voter tract to be a potential district center. Thus in the case of Michigan, e.g. there are potentially 7.9 million combinations that need to be represented. Removing implausible assignments reduces this by 60% – 90% depending on the model so, e.g. in the case of Michigan\u2019s congressional districting, only about 2.7 million plausible assignments remain. However, the main contribution to the model\u2019s size is from constraints that need to be imposed on these potential assignments to ensure contiguity and proscribe district holes, such as a donut shaped district with others in its center. Their number depends the state\u2019s physical geography but is typically around 65 for every potential assignment.<\/p>\n\n\n\n

Technically this is a multiway number partitioning<\/em> problem [4] with side constraints. It has received some attention in the technical literature: [5] is fairly comprehensive: it reviews and refines formulations and algorithms for the districting problem but the methods are not readily auditable, since special-purpose algorithms need to be used to handle an otherwise impractical number of constraints. It does not proscribe holes and whilst the majority-minority issue is recognized, it is not addressed in the numerical experiments. The authors state their intent only \u2018to compare the performance of the contiguity formulations on realistic instances\u2014and not to create \u201cgood\u201d plans\u2019.<\/p>\n\n\n\n

We perform the optimization in two stages. In the first we minimize a weighted combination of the maximum population deviation of the districts and the sum of the district diameters and ignore the voter tract ethnic composition. In the second, we account for the ethnic composition of the voter tracts and constrain the district population deviation, minimizing the weighted sum of the district diameters and penalties for minority-minority districts.<\/p>\n\n\n\n

Weights and the population deviation constraint can be chosen to capture the state\u2019s priorities and goals in this task. For example, we would normally use a population deviation constraint of 0.5%, but the population deviation can usually be reduced to 0.05% or less at the expense of less compact districts. Sates can also prioritize not splitting counties across congressional districts.<\/p>\n\n\n\n

The MIP models are very large \u2013 typically around 80M constraints, 1.3M variables (nearly all of which are binary) and 160M matrix elements. Just solving the root LP of the MIP can take 12 or more hours with IBM ILOG CPLEX 20.10 [2] running on a 24 core Intel Xeon server. To get solutions we used Optimization Direct\u2019s ODH|CPLEX 6.05 [3] solver which can find solutions whilst the underlying CPLEX solver tightens the dual bound. The solution to the first model is used as a starting solution for the second.<\/p>\n\n\n\n

An Example: Michigan\u2019s Congressional Districts<\/strong><\/p>\n\n\n\n

We chose Michigan as a test example using the 2010 Census data for the following reasons:<\/p>\n\n\n\n