A refined complexity analysis of fair districting over graphs

We study the NP-hard Fair Connected Districting problem recently proposed by Stoica et al. [AAMAS 2020]: Partition a vertex-colored graph into k  connected components (subsequently referred to as districts) so that in every district the most frequent color occurs at most a given number of times more...

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Published in:Autonomous agents and multi-agent systems Vol. 37; no. 1; p. 13
Main Authors: Boehmer, Niclas, Koana, Tomohiro, Niedermeier, Rolf
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2023
Springer Nature B.V
Subjects:
Apportionment
Artificial Intelligence
Color
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Developing countries
Elections
Ethnicity
LDCs
Minority & ethnic groups
Parameterization
Parameters
Political parties
Polynomials
School districts
Software Engineering/Programming and Operating Systems
Trees (mathematics)
User Interfaces and Human Computer Interaction
Voters
Social Choice on Social Networks
Parameterized Algorithmics
Electoral Districting
Fairness
ISSN:1387-2532, 1573-7454
Online Access:Get full text
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Summary:We study the NP-hard Fair Connected Districting problem recently proposed by Stoica et al. [AAMAS 2020]: Partition a vertex-colored graph into k  connected components (subsequently referred to as districts) so that in every 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 real-world scenarios where agents of different types, which are one-to-one 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. We conduct a fine-grained analysis of the (parameterized) computational complexity of Fair Connected Districting . In particular, we prove that it is polynomial-time solvable on paths, cycles, stars, and caterpillars, but already becomes NP-hard on trees. Motivated by the latter negative result, we perform a parameterized complexity analysis with respect to various graph parameters including treewidth, and problem-specific 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]-hard, and para-NP-hard).
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ISSN:1387-2532
1573-7454
DOI:10.1007/s10458-022-09594-2