Forming Plurality at Minimum Cost

Lin, Wei Yin; Wu, Yen Wei; Wang, Hung Lung; Chao, Kun-Mao

doi:10.1007/978-3-319-15612-5_8

Cited by 3 publications

(8 citation statements)

References 12 publications

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“…In this section, we show how to compute a plurality point for V with respect to the L 2 norm in O (n log n) time, if it exists. We start by proving several properties of the plurality point in higher dimensions, which generalize similar properties as Lin et al [18] proved in R 2 . These properties imply that if a plurality point exists, it is unique (unless all points are collinear).…”

Section: Plurality Points In the L Normmentioning

confidence: 53%

“…Verifying the condition in Proposition 2.1 directly is not efficient. Hence, we will prove alternative conditions for a point p to be a plurality point in R d , which generalize the conditions Lin et al [18] stated for the planar case. First, we define some concepts introduced by Lin et al…”

Section: Properties Of Plurality Points In the L 2 Normmentioning

confidence: 57%

“…Let L * (p) be the set of all unbalanced lines in L(p) and let R * (p) be the corresponding set of rays. We now define the so-called alternating property, as introduced by Plott [20] and later by Lin et al [18]. This property is restricted to the 2-dimensional setting, where we can order the rays in R * (p) around p. In this setting, the point p is said to have the alternating property if the following holds: the circular sequence of nonzero labels of the rays in R * (p), which we obtain when we visit the rays in R * (p) in clockwise order around p, alternates between labels +1 and −1 (see Figure 1).…”

Section: Properties Of Plurality Points In the L 2 Normmentioning

confidence: 99%

“…In fact, the set V of voters must, in a certain sense, be highly symmetric to admit a plurality point. For example, when the number of voters is even, any line containing a plurality point p must contain the same number of voters on either side of p. This led Lin et al [18] to study the minimumcost plurality problem. Here each voter is assigned a cost, and the goal is to find a minimum-cost subset W ⊂ V of voters such that if we ignore the voters in W -that is, if we consider V \ Wthen a plurality point exists.…”

Section: Introductionmentioning

confidence: 99%

“…1 We then turn our attention to the minimum-cost plurality problem. We solve the open problem of Lin et al [18] by presenting an algorithm that solves the problem in O (n 4 ) time. Note that this even improves on the O (n 5 log n) running time for the planar case.…”

Section: Introductionmentioning

confidence: 99%

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Faster Algorithms for Computing Plurality Points

Berg

Gudmundsson

Mehr

2018

ACM Trans. Algorithms

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Let V be a set of n points in R d , which we call voters. A point p ∈ R d is preferred over another point p ′ ∈ R d by a voter υ ∈ V if dist(υ, p ) < dist(υ, p ′). A point p is called a plurality point if it is preferred by at least as many voters as any other point p ′. We present an algorithm that decides in O ( n log n ) time whether V admits a plurality point in the L 2 norm and, if so, finds the (unique) plurality point. We also give efficient algorithms to compute a minimum-cost subset W ⊂ V such that V \ W admits a plurality point, and to compute a so-called minimum-radius plurality ball. Finally, we consider the problem in the personalized L 1 norm, where each point υ ∈ V has a preference vector 〈 w 1 (υ),…, w d (υ)〉 and the distance from υ to any point p ∈ R d is given by ∑ i =1 d w i (υ)· | x i (υ)− x i ( p )|. For this case we can compute in O ( n d −1 ) time the set of all plurality points of V . When all preference vectors are equal, the running time improves to O ( n ).

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Section: Plurality Points In the L Normmentioning

confidence: 53%

Section: Properties Of Plurality Points In the L 2 Normmentioning

confidence: 57%

Section: Properties Of Plurality Points In the L 2 Normmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Faster Algorithms for Computing Plurality Points

Berg

Gudmundsson

Mehr

2018

ACM Trans. Algorithms

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Plurality in Spatial Voting Games with Constant $$\beta $$

Filtser,

Filtser

2024

Discrete Comput Geom

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Condorcet Relaxation In Spatial Voting

Filtser

2021

AAAI

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Consider a set of voters V, represented by a multiset in a metric space (X,d). The voters have to reach a decision - a point in X. A choice p∈ X is called a β-plurality point for V, if for any other choice q∈ X it holds that |{v∈ V ∣ β⋅ d(p,v)≤ d(q,v)}| ≥|V|/2 . In other words, at least half of the voters ``prefer'' over q, when an extra factor of β is taken in favor of p. For β=1, this is equivalent to Condorcet winner, which rarely exists. The concept of β-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [SoCG 2020] as a relaxation of the Condorcet criterion. Denote by β*(X,d) the value sup{ β ∣ every finite multiset V in X admits a β-plurality point}}. The parameter β* determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane β*(ℝ2,\|⋅\|2)=√3/2 , and more generally, for d-dimensional Euclidean space, 1/√d ≤ β*(ℝd,\|⋅\|2)≤√3/2 . In this paper, we show that 0.557≤ β*(ℝd,\|⋅\|2) for any dimension d (notice that 1/√d

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Forming Plurality at Minimum Cost

Cited by 3 publications

References 12 publications

Faster Algorithms for Computing Plurality Points

Faster Algorithms for Computing Plurality Points

Plurality in Spatial Voting Games with Constant $$\beta $$

Condorcet Relaxation In Spatial Voting

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