International Journal of Systems Science Vol・ 40, No・ 5, May 2009, 47ト477 ㊨ Taylor & FranCis Ta〆or i FrarTCLS Group A minirmax spaming forest approach to the polit Takeo Yamada* Department of Computer Science, The National Defense Acade (Received 23 September 2007; final version received ll No We fomulate the problem of political districtlng aS amini-max spann search-based heuristics to solve the problem approximately・ Throu performance of the developed algorithms・ We also glVe a Case Study o the Lower House Members of the National Diet・ We observe that 'hype solutions, With the resulting districts all connected and usually balan Keywords: political districtlng; heuristic algorithm; spannlng fores 1. 1mtroduction Gerrymandering (Morris 2006) is a form of districting in which electoral district or constituency boundaries are manlpulated fわr an electoral advantage. To prevent this and to design more fair and reasonable districting, mathematical methods have been explored (Balinski and Young 1982) ・ For example, the split-line algorithm (Smith and Kok 2008) and the Voronoi method (Balinski, Brams, and Pukelsheim 2004) are based on geographical configuration of the region and divided into constituencies with artificial, pleCeWise linear boundaries. Cluster analysis (Romesburg 2004) and statistical physics have been applied to this problem. These methods introduce such measures as compactness quotients (Nguyen and Kreinovich 1999; Bottman, Essig, and Whittle 2007) or HamiltoniaT energy (Chou and Li 2006) to evaluate the approprlaten.esp Of the resulting districts and try to find the distrlCtlng that maximises (or mini.mies) the sum of such measures over all constituencleS. Lush, Gamez, and Kreinovich (2007) observed that naive cluste空g approach can lead to a disproportional representatlOn・ In the mathematical programmlng approach cost is associated with each possible district, and the problem is usually formulated as a sort of the set packing/coverL'ng problem (Lawler 1976) to miTimise the sum or these costs over all possible combination Or districts・ Some heuristic algorithms have been pro- posed to solve this 0-1 linear programmlng problem approximately, using tabu search (Bozkaya, Erkut, and Laporte 2003) or GRASP (Rios-Mercado and Fernandez 2009) methods. Exact also been explored to solve this pr These include techniques such as and branch-and-price (Mehrot Nemhauser 1998), network optim Lamar, and Wallace 1997), and cap tation problem (Hojati 1996). In all these works, the objective or some measure or approprlateneSS encies・ Unfわrtunately, the resultin unbalanced; we may have some ver together with some small ones・AIso, approach, unless district costs are we may obtain disconnected distri optimal solution. The purpose or this article is to and balance or constituencies expli and present a mathematical met problem to some polnt Of satisfac in Section 2 We fわrmulate the pro mini-max spanning forest prob explain the relation or this fわ MMSFPs studied in earlier Takahashi, and Kataoka 1996, 1 presents two kind or heuristic algo problem approximately・ After a num a smalトsized example in Section 4 a summary or numerical experim artificial instances・ Finally, Sectio study of Kanagawa Prefecture, Japa or the Lower House Members or t Through these we observe that *Email: [email protected]lSSN 0020 7721 prLnt/ISSN 1464-5319 onllne ◎ 2009 Taylor 良 Francls DOI 10.1080/00207720802645246 http //www lnformaworld com
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A minirmax spaming forest approach to the …a summary or numerical experiments fわr larger artificial instances・ Finally, Section 6 describes a case study of Kanagawa Prefecture,
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International Journal of Systems Science
Vol・ 40, No・ 5, May 2009, 47ト477 ㊨ Taylor & FranCisTa〆or i FrarTCLS Group
A minirmax spaming forest approach to the politiCaldistricting problem
Takeo Yamada*
Department of Computer Science, The National Defense Academy, Yokosuka, Kanagawa, Japan
(Received 23 September 2007; final version received ll Novefhber 2008)
We fomulate the problem of political districtlng aS amini-max spannlng・forest problem, and present some localsearch-based heuristics to solve the problem approximately・ Through numerical experiments, we evaluate the
performance of the developed algorithms・ We also glVe a Case Study of a prefecture in Japan for the election ofthe Lower House Members of the National Diet・ We observe that 'hyperopIC・ algorithm usually grves satisfactory
solutions, With the resulting districts all connected and usually balanced in size.
Keywords: political districtlng; heuristic algorithm; spannlng forest;mini-max optlmisation
1. 1mtroduction
Gerrymandering (Morris 2006) is a form of districting
in which electoral district or constituency boundaries
are manlpulated fわr an electoral advantage. To prevent
this and to design more fair and reasonable districting,
mathematical methods have been explored (Balinski
and Young 1982) ・ For example, the split-line algorithm
(Smith and Kok 2008) and the Voronoi method
(Balinski, Brams, and Pukelsheim 2004) are based on
geographical configuration of the region and divided
into constituencies with artificial, pleCeWise linear
boundaries.
Cluster analysis (Romesburg 2004) and statistical
physics have been applied to this problem. These
methods introduce such measures as compactness
quotients (Nguyen and Kreinovich 1999; Bottman,Essig, and Whittle 2007) or HamiltoniaT energy (Chou
and Li 2006) to evaluate the approprlaten.esp Of the
resulting districts and try to find the distrlCtlng that
maximises (or mini.mies) the sum of such measuresover all constituencleS. Lush, Gamez, and Kreinovich
(2007) observed that naive cluste空g approach can
lead to a disproportional representatlOn・
In the mathematical programmlng approach cost
is associated with each possible district, and the
problem is usually formulated as a sort of the set
packing/coverL'ng problem (Lawler 1976) to miTimisethe sum or these costs over all possible combination Or
districts・ Some heuristic algorithms have been pro-
posed to solve this 0-1 linear programmlng problem
approximately, using tabu search (Bozkaya, Erkut, and
Laporte 2003) or GRASP (Rios-Mercado and
Fernandez 2009) methods. Exact algorithms have
also been explored to solve this problem to optlmality・
These include techniques such as column generation
and branch-and-price (Mehrotra, Johnson, and
Nemhauser 1998), network optimisation (George,
Lamar, and Wallace 1997), and capacitated transpoト
tation problem (Hojati 1996).
In all these works, the objective function was a sum
or some measure or approprlateneSS OVer all constitu-
encies・ Unfわrtunately, the resulting districts can be
unbalanced; we may have some very large districts
together with some small ones・AIso, in the optlmisation
approach, unless district costs are carefully defined,
we may obtain disconnected districts as a part or an
optimal solution.
The purpose or this article is to take connectedness
and balance or constituencies explicitly Into a∝Ount,
and present a mathematical method to solve this
problem to some polnt Of satisfaction・ To this end,
in Section 2 We fわrmulate the problem as a kind or
mini-max spanning forest problem (MMSFP), and
explain the relation or this fわrmulation to the
MMSFPs studied in earlier works (Yamada,
Takahashi, and Kataoka 1996, 1997). Section 3
presents two kind or heuristic algorithms to solve this
problem approximately・ After a numerical example for
a smalトsized example in Section 4, Section 5 gives
a summary or numerical experiments fわr larger
artificial instances・ Finally, Section 6 describes a case
study of Kanagawa Prefecture, Japan for the election