On Mathematical Optimization for the Visualization of Frequencies and Adjacencies as Rectangular Maps

Emilio Carrizosa, Vanesa Guerrero, Dolores Romero Morales

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

In this paper we address the problem of visualizing a frequency distribution and an adjacency relation attached to a set of individuals. We represent this information using a rectangular map, i.e., a subdivision of a rectangle into rectangular portions so that each portion is associated with one individual, their areas reflect the frequencies, and the adjacencies between portions represent the adjacencies between the individuals. Due to the impossibility of satisfying both area and adjacency requirements, our aim is to fit as well as possible the areas, while representing as many adjacent individuals as adjacent rectangular portions as possible and adding as few false adjacencies, i.e., adjacencies between rectangular portions corresponding to non-adjacent individuals, as possible. We formulate this visualization problem as a Mixed Integer Linear Programming (MILP) model. We propose a matheuristic that has this MILP model at its core. Our experimental results demonstrate that our matheuristic provides rectangular maps with a good fit in both the frequency distribution and the adjacency relation.
In this paper we address the problem of visualizing a frequency distribution and an adjacency relation attached to a set of individuals. We represent this information using a rectangular map, i.e., a subdivision of a rectangle into rectangular portions so that each portion is associated with one individual, their areas reflect the frequencies, and the adjacencies between portions represent the adjacencies between the individuals. Due to the impossibility of satisfying both area and adjacency requirements, our aim is to fit as well as possible the areas, while representing as many adjacent individuals as adjacent rectangular portions as possible and adding as few false adjacencies, i.e., adjacencies between rectangular portions corresponding to non-adjacent individuals, as possible. We formulate this visualization problem as a Mixed Integer Linear Programming (MILP) model. We propose a matheuristic that has this MILP model at its core. Our experimental results demonstrate that our matheuristic provides rectangular maps with a good fit in both the frequency distribution and the adjacency relation.
LanguageEnglish
JournalEuropean Journal of Operational Research
Volume265
Issue number1
Pages290-302
ISSN0377-2217
DOIs
StatePublished - 2018

Bibliographical note

Published online: 13. July 2017

Keywords

  • Mixed integer linear programming
  • Visualization
  • Multidimensional scaling
  • Rectangular maps
  • Frequencies and adjacencies

Cite this

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title = "On Mathematical Optimization for the Visualization of Frequencies and Adjacencies as Rectangular Maps",
abstract = "In this paper we address the problem of visualizing a frequency distribution and an adjacency relation attached to a set of individuals. We represent this information using a rectangular map, i.e., a subdivision of a rectangle into rectangular portions so that each portion is associated with one individual, their areas reflect the frequencies, and the adjacencies between portions represent the adjacencies between the individuals. Due to the impossibility of satisfying both area and adjacency requirements, our aim is to fit as well as possible the areas, while representing as many adjacent individuals as adjacent rectangular portions as possible and adding as few false adjacencies, i.e., adjacencies between rectangular portions corresponding to non-adjacent individuals, as possible. We formulate this visualization problem as a Mixed Integer Linear Programming (MILP) model. We propose a matheuristic that has this MILP model at its core. Our experimental results demonstrate that our matheuristic provides rectangular maps with a good fit in both the frequency distribution and the adjacency relation.",
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On Mathematical Optimization for the Visualization of Frequencies and Adjacencies as Rectangular Maps. / Carrizosa, Emilio; Guerrero, Vanesa; Morales, Dolores Romero.

In: European Journal of Operational Research, Vol. 265, No. 1, 2018, p. 290-302.

Research output: Contribution to journalJournal articleResearchpeer-review

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N2 - In this paper we address the problem of visualizing a frequency distribution and an adjacency relation attached to a set of individuals. We represent this information using a rectangular map, i.e., a subdivision of a rectangle into rectangular portions so that each portion is associated with one individual, their areas reflect the frequencies, and the adjacencies between portions represent the adjacencies between the individuals. Due to the impossibility of satisfying both area and adjacency requirements, our aim is to fit as well as possible the areas, while representing as many adjacent individuals as adjacent rectangular portions as possible and adding as few false adjacencies, i.e., adjacencies between rectangular portions corresponding to non-adjacent individuals, as possible. We formulate this visualization problem as a Mixed Integer Linear Programming (MILP) model. We propose a matheuristic that has this MILP model at its core. Our experimental results demonstrate that our matheuristic provides rectangular maps with a good fit in both the frequency distribution and the adjacency relation.

AB - In this paper we address the problem of visualizing a frequency distribution and an adjacency relation attached to a set of individuals. We represent this information using a rectangular map, i.e., a subdivision of a rectangle into rectangular portions so that each portion is associated with one individual, their areas reflect the frequencies, and the adjacencies between portions represent the adjacencies between the individuals. Due to the impossibility of satisfying both area and adjacency requirements, our aim is to fit as well as possible the areas, while representing as many adjacent individuals as adjacent rectangular portions as possible and adding as few false adjacencies, i.e., adjacencies between rectangular portions corresponding to non-adjacent individuals, as possible. We formulate this visualization problem as a Mixed Integer Linear Programming (MILP) model. We propose a matheuristic that has this MILP model at its core. Our experimental results demonstrate that our matheuristic provides rectangular maps with a good fit in both the frequency distribution and the adjacency relation.

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