
The Complexity of Gerrymandering Over Graphs: Paths and Trees
Roughly speaking, gerrymandering is the systematic manipulation of the b...
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Grid Recognition: Classical and Parameterized Computational Perspectives
Grid graphs, and, more generally, k× r grid graphs, form one of the most...
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Minimizing Margin of Victory for Fair Political and Educational Districting
In many practical scenarios, a population is divided into disjoint group...
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On Parameterized Complexity of Liquid Democracy
In liquid democracy, each voter either votes herself or delegates her vo...
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Multivariate Analysis of Scheduling Fair Competitions
A fair competition, based on the concept of envyfreeness, is a nonelim...
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FineGrained View on Bribery for Group Identification
Given a set of agents qualifying or disqualifying each other, group iden...
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Fair redistricting is hard
Gerrymandering is a longstanding issue within the U.S. political system...
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A Refined Complexity Analysis of Fair Districting over Graphs
We study the NPhard Fair Connected Districting problem: Partition a vertexcolored graph into k connected components (subsequently referred to as districts) so that in each district the most frequent color occurs at most a given number of times more often than the second most frequent color. Fair Connected Districting is motivated by various realworld scenarios where agents of different types, which are onetoone represented by nodes in a network, have to be partitioned into disjoint districts. Herein, one strives for "fair districts" without any type being in a dominating majority in any of the districts. This is to e.g. prevent segregation or political domination of some political party. Our work builds on a model recently proposed by Stoica et al. [AAMAS 2020], thereby also strengthening and extending computational hardness results from there. More specifically, with Fair Connected Districting we identify a natural, already hard special case of their Fair Connected Regrouping problem. We conduct a finegrained analysis of the (parameterized) computational complexity of Fair Connected Districting, proving that it is polynomialtime solvable on paths, cycles, stars, caterpillars, and cliques, but already becomes NPhard on trees. Motivated by the latter negative result, we perform a parameterized complexity analysis with respect to various graph parameters, including treewidth, and problemspecific parameters, including the numbers of colors and districts. We obtain a rich and diverse, close to complete picture of the corresponding parameterized complexity landscape (that is, a classification along the complexity classes FPT, XP, W[1]hardness, and paraNPhardness). Doing so, we draw a fine line between tractability and intractability and identify structural properties of the underlying graph that make Fair Connected Districting computationally hard.
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