Gerrymandering on Graphs: Computational Complexity and Parameterized Algorithms

Gupta, Sushmita; Jain, Pallavi; Panolan, Fahad; Roy, Sanjukta; Saurabh, Saket

doi:10.1007/978-3-030-85947-3_10

Cited by 10 publications

(12 citation statements)

References 25 publications

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“…We prove that Gerrymandering is W [2]-hard on trees (even when the depth is 2) with respect to the number of districts k, suggesting that no FPT algorithm exists. This contrasts sharply with the polynomial time algorithm for stars (trees of depth 1), and answers an open question of Gupta et al [9]. To better understand the difference in complexity between trees and paths, we also study the problem in trees with only ℓ leaves.…”

Section: Introductionmentioning

confidence: 78%

“…In contrast, we do not allow ties for the total number of districts won and require the preferred candidate p to win a strict plurality in YES-instances. Previous work [9,10] uses tie-breaking rules both within districts and for the overall number of districts. We note that the constructions in our hardness proofs ensure that our results apply regardless of the choice of tie-breaking rules.…”

Section: Preliminariesmentioning

confidence: 99%

“…Internally, we use a modified version of the FPT algorithm for paths from Gupta et al [9]. Their algorithm creates an instance (H, s, t, k, k * ) of k-Labeled Path which, given a partially edge-labeled directed graph H with vertices s and t, asks if there exists an st-path with exactly k internal vertices such that no edge label is used more than k * times.…”

Section: Weighted Gerrymanderingmentioning

confidence: 99%

“…In 2021, Gupta et al showed that Gerrymandering is fixed parameter tractable (FPT) on paths with respect to the number of districts k (independent of the number of candidates) [9]. They gave an O(2.619 k (n + m) O (1) ) algorithm for Weighted Gerrymandering, a generalization of Gerrymandering which allows vertices to split their votes between multiple candidates.…”

Section: Introductionmentioning

confidence: 99%

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Gerrymandering Trees: Parameterized Hardness

Fraser¹,

Lavallee²,

Sullivan³

2022

Preprint

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In a representative democracy, elections involve partitioning geographical space into districts which each elect a single representative. Legislation is then determined by votes from these representatives, and thus political parties are incentivized to win as many districts as possible (ideally a plurality). Gerrymandering is the process by which these districts' boundaries are manipulated to give favor to a certain candidate or party. Cohen-Zemach et al. (AAMAS 2018) proposed Gerrymandering as a formalization of this problem on graphs (as opposed to Euclidean space) where districts partition vertices into connected subgraphs. More recently, Gupta et al. (SAGT 2021) studied its parameterized complexity and gave an FPT algorithm for paths with respect to the number of districts k. We prove that Gerrymandering is W[2]-hard on trees (even when the depth is two) with respect to k, answering an open question of Gupta et al. Moreover, we prove that Gerrymandering remains W[2]-hard in trees with ℓ leaves with respect to the combined parameter k + ℓ. To complement this result, we provide an algorithm to solve Gerrymandering that is FPT in k when ℓ is a fixed constant.

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Section: Introductionmentioning

confidence: 78%

Section: Preliminariesmentioning

confidence: 99%

Section: Weighted Gerrymanderingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gerrymandering Trees: Parameterized Hardness

Fraser¹,

Lavallee²,

Sullivan³

2022

Preprint

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“…More recently, however, different variants of Gerrymandering have been considered from an algorithmic perspective [28,13]. Notably, the study of Gerrymandering over graphs, which is pretty much analogous to our setting, very recently gained significantly increased interest by various research groups [4,11,18,21]. A similar model for graph-based redistribution scenarios and political districting has been studied under the name "network-based vertex dissolution" [5].…”

Section: Related Workmentioning

confidence: 99%

A Refined Complexity Analysis of Fair Districting over Graphs

Boehmer¹,

Koana²,

Niedermeier³

2021

Preprint

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We study the NP-hard Fair Connected Districting problem: Partition a vertex-colored 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 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.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 fine-grained analysis of the (parameterized) computational complexity of Fair Connected Districting, proving that it is polynomial-time solvable on paths, cycles, stars, caterpillars, and cliques, 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]-hardness, and para-NP-hardness). 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. * This work was initiated at the research retreat of the Algorithmics and Computational Complexity group held in September 2020 in Zinnowitz, Germany.† Supported by the Deutsche Forschungsgemeinschaft (DFG), project MaMu, NI 369/19. ‡ Supported by the Deutsche Forschungsgemeinschaft (DFG), project FPTinP, NI 369/16.

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