85 research outputs found
Optimal partisan districting on planar geographies
We show that optimal partisan districting in the plane with geographical constraints is an NP-complete problem
Gerrymandering and Compactness: Implementation Flexibility and Abuse
The shape of an electoral district may suggest whether it was drawn with
political motivations, or gerrymandered. For this reason, quantifying the shape
of districts, in particular their compactness, is a key task in politics and
civil rights. A growing body of literature suggests and analyzes compactness
measures mathematically, but little consideration has been given to how these
scores should be calculated in practice. Here, we consider the effects of a
number of decisions that must be made in interpreting and implementing a set of
popular compactness scores. We show that the choices made in quantifying
compactness may themselves become political tools, with seemingly innocuous
decisions leading to disparate scores. We show that when the full range of
implementation flexibility is used, it can be abused to make clearly
gerrymandered districts appear quantitatively reasonable. This complicates
using compactness as a legislative or judicial standard to counteract unfair
redistricting practices. This paper accompanies the release of packages in C++,
Python, and R which correctly, efficiently, and reproducibly calculate a
variety of compactness scores.Comment: 10 pages, 17 figures, 1 tabl
Strategic districting for the mitigation of educational segregation : a pilot model for school district optimization in Helsinki
Helsingin kaupunkirakenne on eriytynyt viimeisten vuosikymmenien aikana merkittävästi sosiaalisilla mittareilla tarkasteltuna. Kehitys on heijastunut kouluihin oppilaspohjien ja oppimistuloksien erojen kasvuna, minkä lisäksi Helsingissä on löydetty viitteitä myös itsenäisistä kouluvaikutuksista. Koulujen eriytymiskehityksen pelätään mainevaikutuksen kautta kiihdyttävän alueellista segregaatiota ja siten oppilaspohjien eriytymistä entisestään. Oppilaspohjien eroihin on kuitenkin mahdollista vaikuttaa määrittämällä oppilasalueet uudelleen tavalla, joka minimoi oppilasalueiden välisiä sosiaalisia eroja mahdollisimman tehokkaasti. Tätä varten tarvitaan uudenlaisia, koneoppimiseen perustuvia optimointityökaluja.
Tämän opinnäytetyön päätavoitteena on tutkia mahdollisuutta optimoida Helsingin oppilasalueita väestöpohjiltaan sisäisesti heterogeenisemmiksi ja keskenään homogeenisemmiksi. Tavoitetta varten olen kehittänyt työssäni automatisoidun optimointimallin, joka minimoi sosiaalisten muuttujien varianssia oppilasalueiden välillä oppilasalueiden muotoa varioimalla. Mallin pilottisovelluksessa optimoin Helsingin oppilaaksiottoalueita tasaisemmiksi käyttäen optimoitavana muuttujana vieraskielisen väestön osuutta. Olemassa olevaa kouluverkostoa eli koulujen sijaintia, oppilasalueiden maantieteellistä yhtenäisyyttä, enimmäisoppilasmääriä koulukohtaisella marginaalilla sekä koulukohtaista koulumatkan enimmäispituutta on käytetty mallissa alueiden muodostamista rajoittavina tekijöinä.
Tutkimukseni keskeinen löydös on, että oppilasaluerajoja siirtelemällä oppilasalueiden sosiaalisen pohjan eroihin voidaan vaikuttaa Helsingissä merkittävästi. Malli vaatii kuitenkin vielä perusteellista jatkokehittämistä soveltuakseen aluejakojen käytännön suunnitteluun, ja tässä vaiheessa sen merkittävimmät kehityskohteet liittyvät optimoitujen alueiden muotoon, mallin laskennalliseen vaativuuteen ja koulumatkojen turvallisuutta mittaavan optimointiparametrin puuttumiseen.The social urban structure of Helsinki has experienced a significant rise in spatial differences during the last two decades. This development has reflected on schools as rising differences between schools’ student compositions and learning outcomes. Additionally, signs of independent school effects have been observed in several studies. The differentiation of student compositions is feared to exacerbate residential segregation and differentiate schools’ operating environments further. It is possible, however, to intervene this development by drawing the school attendance districts such that the social differences between schools’ student compositions are effectively minimized. For this purpose, new machine learning based optimization tools are needed.
The main objective of this master’s thesis study is to examine the possibility to optimize Helsinki’s school districts toward more internally heterogeneous and externally homogeneous social compositions. For this purpose, I have developed an optimization model that minimizes the variance of social variables between school districts by iteratively redrawing the districts’ borders. In a pilot application of the model I optimize the school districts of Helsinki by using the share of population with immigrant background as the optimization variable, while existing school infrastructure (the school locations and student capacities), spatial contiguity of the districts, and school-specific maximum travel distances are used as constraints restricting the shapes that the districts can take.
The core finding of this study is that in Helsinki, the social compositions of school districts can be significantly evened out by redrawing the school district borders. However, for the model to be suitable for district planning in practice it needs further development. At this stage, the main limitations of the model are related to the shapes of the optimized districts, the model’s time complexity and the lack of a constraint or optimization parameter that accounts for the safety of children’s school trips
Bringing Spatial Interaction Measures into Multi-Criteria Assessment of Redistricting Plans Using Interactive Web Mapping
Redistricting is the process by which electoral district boundaries are
drawn, and a common normative assumption in this process is that districts
should be drawn so as to capture coherent communities of interest (COIs). While
states rely on various proxies for community illustration, such as compactness
metrics and municipal split counts, to guide redistricting, recent legal
challenges and scholarly works have shown the failings of such proxy measures
and the difficulty of balancing multiple criteria in district plan creation. To
address these issues, we propose the use of spatial interaction communities to
directly quantify the degree to which districts capture the underlying COIs.
Using large-scale human mobility flow data, we condense spatial interaction
community capture for a set of districts into a single number, the interaction
ratio (IR), which can be used for redistricting plan evaluation. To compare the
IR to traditional redistricting criteria (compactness and fairness), and to
explore the range of IR values found in valid districting plans, we employ a
Markov chain-based regionalization algorithm (ReCom) to produce ensembles of
valid plans, and calculate the degree to which they capture spatial interaction
communities. Furthermore, we propose two methods for biasing the ReCom
algorithm towards different IR values. We perform a multi-criteria assessment
of the space of valid maps, and present the results in an interactive web map.
The experiments on Wisconsin congressional districting plans demonstrate the
effectiveness of our methods for biasing sampling towards higher or lower IR
values. Furthermore, the analysis of the districts produced with these methods
suggests that districts with higher IR and compactness values tend to produce
district plans that are more proportional with regards to seats allocated to
each of the two major parties.Comment: 12 figure
Spanning tree methods for sampling graph partitions
In the last decade, computational approaches to graph partitioning have made
a major impact in the analysis of political redistricting, including in U.S.
courts of law. Mathematically, a districting plan can be viewed as a balanced
partition of a graph into connected subsets. Examining a large sample of valid
alternative districting plans can help us recognize gerrymandering against an
appropriate neutral baseline. One algorithm that is widely used to produce
random samples of districting plans is a Markov chain called recombination (or
ReCom), which repeatedly fuses adjacent districts, forms a spanning tree of
their union, and splits that spanning tree with a balanced cut to form new
districts. One drawback is that this chain's stationary distribution has no
known closed form when there are three or more districts. In this paper, we
modify ReCom slightly to give it a property called reversibility, resulting in
a new Markov chain, RevReCom. This new chain converges to the simple, natural
distribution that ReCom was originally designed to approximate: a plan's
stationary probability is proportional to the product of the number of spanning
trees of each district. This spanning tree score is a measure of district
"compactness" (or shape) that is also aligned with notions of community
structure from network science. After deriving the steady state formally, we
present diagnostic evidence that the convergence is efficient enough for the
method to be practically useful, giving high-quality samples for full-sized
problems within several hours. In addition to the primary application of
benchmarking of redistricting plans (i.e., describing a normal range for
statistics), this chain can also be used to validate other methods that target
the spanning tree distribution
Districting Problems - New Geometrically Motivated Approaches
This thesis focuses on districting problems were the basic areas are represented by points or lines. In the context of points, it presents approaches that utilize the problem\u27s underlying geometrical information. For lines it introduces an algorithm combining features of geometric approaches, tabu search, and adaptive randomized neighborhood search that includes the routing distances explicitly. Moreover, this thesis summarizes, compares and enhances existing compactness measures
Supervised regionalization methods, a survey.
This paper reviews almost four decades of contributions on the subject of supervised regionalization methods. These methods aggregate a set of areas into a predefined number of spatially contiguous regions while optimizing certain aggregation criteria. The authors present a taxonomic scheme that classifies a wide range of regionalization methods into eight groups, based on the strategy applied for satisfying the spatial contiguity constraint. The paper concludes by providing a qualitative comparison of these groups in terms of a set of certain characteristics, and by suggesting future lines of research for extending and improving these methods.regionalization, constrained clustering, analytical regions.
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