Constrained clustering via diagrams: A unified theory and its application to electoral district design

Brieden, Andreas; Gritzmann, Peter; Klemm, Fabian

doi:10.1016/j.ejor.2017.04.018

Cited by 22 publications

(17 citation statements)

References 51 publications

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“…The concept of GBPDs can be viewed as structure-driven weight balanced clusterings; see [171,172]. For j ∈ [l] let s j denote the center of the jth grain, and let κ − j , κ + j be lower and upper bounds for its volume, respectively.…”

Section: Tomographic Grain Mappingmentioning

confidence: 99%

“…We remark that the clustering approach described above was previously applied (in an 'isotropic' fashion) in the context of farmland consolidation [11,170]. For an application in designing electoral districts where municipalities of a state have to be grouped into districts of nearly equal population while obeying certain politically motivated requirements, see [172].…”

Section: Tomographic Grain Mappingmentioning

confidence: 99%

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13. On the reconstruction of static and dynamic discrete structures

Alpers¹,

Gritzmann²

2019

The Radon Transform

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We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the 'classical' task of reconstructing finite point sets in R d ). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems.

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Section: Tomographic Grain Mappingmentioning

confidence: 99%

Section: Tomographic Grain Mappingmentioning

confidence: 99%

13. On the reconstruction of static and dynamic discrete structures

Alpers¹,

Gritzmann²

2019

The Radon Transform

Self Cite

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“…Instead of spending a lot of computing time exhausting the possible division results and then discarding most of the unqualified, it is better to do cluster planning in advance to save system resources. Constrained clustering is proved as an efficient computation to pursue various conflicting demands [17]. Thus, preliminary clustering could be used to reduce the complexity of subsequent partitioning.…”

Section: Integrated Computation and Geo-information Applicationsmentioning

confidence: 99%

“…Voronoi diagrams were applied to electoral districting for the completeness and symmetry of districts [14,15]. Cluster analysis was also applied in the design of a constituency to solve the problem of seats and electoral redistricting [16,17]. However, there is no specific protocol for implementing redistricting with the current constituencies using such political principles and computing theories.…”

Section: Introductionmentioning

confidence: 99%

Using Intelligent Clustering to Implement Geometric Computation for Electoral Districting

Hung

Chen

2019

IJGI

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Traditional electoral districting is mostly carried out by artificial division. It is not only time-consuming and labor-intensive, but it is also difficult to maintain the principles of fairness and consistency. Due to specific political interests, objectivity is usually distorted and controversial in a proxy-election. In order to reflect the spirit of democracy, this study uses computing technologies to automatically divide the constituency and use the concepts of “intelligent clustering” and “extreme arrangement” to conquer many shortcomings of traditional artificial division. In addition, various informational technologies are integrated to obtain the most feasible solutions within the maximum capabilities of the computing system, yet without sacrificing the global representation of the solutions. We take Changhua County, Taiwan as an example of complete electoral districting, and find better results relative to the official version, which obtained a smaller difference in the population of each constituency, more complete and symmetrical constituencies, and fewer regional controversies. Our results demonstrate that multidimensional algorithms using a geographic information system could solve many problems of block districting to make decisions based on different needs.

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“…In recent years, the MSSC has attracted a lot of attention in the area of data science and operations research (Gambella et al, 2021;Bengio et al, 2021), and it has been extended to integrate background knowledge on objects and/or on clusters through user constraints (Basu et al, 2008;Brieden et al, 2017;Gançarski et al, 2020). Incorporating such constraints into the clustering process transforms the MSSC from an unsupervised learning problem to a semisupervised one.…”

Section: Introductionmentioning

confidence: 99%

An Exact Algorithm for Semi-supervised Minimum Sum-of-Squares Clustering

Piccialli,

Russo,

Sudoso

2021

Preprint

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The minimum sum-of-squares clustering (MSSC), or k-means type clustering, is traditionally considered an unsupervised learning task. In recent years, the use of background knowledge to improve the cluster quality and promote interpretability of the clustering process has become a hot research topic at the intersection of mathematical optimization and machine learning research.The problem of taking advantage of background information in data clustering is called semisupervised or constrained clustering. In this paper, we present a new branch-and-bound algorithm for semi-supervised MSSC, where background knowledge is incorporated as pairwise must-link and cannot-link constraints. For the lower bound procedure, we solve the semidefinite programming relaxation of the MSSC discrete optimization model, and we use a cutting-plane procedure for strengthening the bound. For the upper bound, instead, by using integer programming tools, we propose an adaptation of the k-means algorithm to the constrained case. For the first time, the proposed global optimization algorithm efficiently manages to solve real-world instances up to 800 data points with different combinations of must-link and cannot-link constraints and with a generic number of features. This problem size is about four times larger than the one of the instances solved by state-of-the-art exact algorithms.

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Constrained clustering via diagrams: A unified theory and its application to electoral district design

Cited by 22 publications

References 51 publications

13. On the reconstruction of static and dynamic discrete structures

13. On the reconstruction of static and dynamic discrete structures

Using Intelligent Clustering to Implement Geometric Computation for Electoral Districting

An Exact Algorithm for Semi-supervised Minimum Sum-of-Squares Clustering

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