Exact Algorithms for the Max-Min Dispersion Problem

Akagi, Toshihiro; Araki, Tetsuya; Horiyama, Takashi; Nakano, Shin-ichi; Okamoto, Yoshio; Otachi, Yota; Saitoh, Toshiki; Uehara, Ryuhei; Uno, Takeaki; Wasa, Kiyotaka

doi:10.1007/978-3-319-78455-7_20

Cited by 20 publications

(14 citation statements)

References 19 publications

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“…Recently, in [1], the exact algorithm for the problem was shown by establishing a relationship between the max-min dispersion problem and the maximum independent set problem. They proposed an O(n wk/3 log n) time where w < 2.373.…”

Section: Related Workmentioning

confidence: 99%

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Approximation Algorithms For The Euclidean Dispersion Problems

Mishra,

Das

2021

Preprint

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In this article, we consider the Euclidean dispersion problems. Let P = {p 1 , p 2 , . . . , p n } be a set of n points in R 2 . For each point p ∈ P and S ⊆ P , we define cost γ (p, S) as the sum of Euclidean distance from p to the nearest γ point in S \ {p}. We define cost γ (S) = min p∈S {cost γ (p, S)} for S ⊆ P . In the γ-dispersion problem, a set P of n points in R 2 and a positive integer k ∈ [γ + 1, n] are given. The objective is to find a subset S ⊆ P of size k such that cost γ (S) is maximized. We consider both 2-dispersion and 1-dispersion problem in R 2 . Along with these, we also consider 2-dispersion problem when points are placed on a line. In this paper, we propose a simple polynomial time (2 √ 3 + )-factor approximation algorithm for the 2-dispersion problem, for any > 0, which is an improvement over the best known approximation factor 4 √ 3 [Amano, K. and Nakano, S. I., An approximation algorithm for the 2-dispersion problem, IEICE Transactions on Information and Systems, Vol. 103(3), pp. 506-508, 2020]. Next, we develop a common framework for designing an approximation algorithm for the Euclidean dispersion problem. With this common framework, we improve the approximation factor to 2 √ 3 for the 2-dispersion problem in R 2 . Using the same framework, we propose a polynomial time algorithm, which returns an optimal solution for the 2-dispersion problem when points are placed on a line. Moreover, to show the effectiveness of the framework, we also propose a 2-factor approximation algorithm for the 1-dispersion problem in R 2 .

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Section: Related Workmentioning

confidence: 99%

“…They proposed an O(n wk/3 log n) time where w < 2.373. In [1], Akagi et al also studied two special cases where a set of n points lies on a line and a set of n points lies on a circle separately. They proposed a polynomial time exact algorithm for both special cases.…”

Section: Related Workmentioning

confidence: 99%

Approximation Algorithms For The Euclidean Dispersion Problems

Mishra,

Das

2021

Preprint

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“…Furthermore, they also demonstrated that unless P = NP , the max-min dispersion problem cannot be approximated within a factor of 2 even if the distance function satisfies the triangular inequality. Recently, in [1], the exact algorithm for the max-min dispersion problem was shown by establishing a relationship with the maximum independent set problem. They proposed an O(n wk/3 log n) time algorithm, where w < 2.373.…”

Section: Related Workmentioning

confidence: 99%

“…They proposed an O(n wk/3 log n) time algorithm, where w < 2.373. In [1], Akagi et al also studied two special cases of the max-min dispersion problem where set of points (1) lies on a line, and (2) lies on a circle. They proposed a polynomial time exact algorithm for both the cases.…”

Section: Related Workmentioning

confidence: 99%

Approximation Algorithms For The Dispersion Problems in a Metric Space

Mishra,

Das

2021

Preprint

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In this article, we consider the c-dispersion problem in a metric space (X, d).Let P = {p 1 , p 2 , . . . , p n } be a set of n points in a metric space (X, d). For each point p ∈ P and S ⊆ P , we define cost c (p, S) as the sum of distances from p to the nearest c points in S \ {p}, where c ≥ 1 is a fixed integer. We define cost c (S) = min p∈S {cost c (p, S)} for S ⊆ P . In the c-dispersion problem, a set P of n points in a metric space (X, d) and a positive integer k ∈ [c + 1, n] are given. The objective is to find a subset S ⊆ P of size k such that cost c (S) is maximized.We propose a simple polynomial time greedy algorithm that produces a 2c-factor approximation result for the c-dispersion problem in a metric space. The best known result for the c-dispersion problem in the Euclidean metric space (X, d) is 2c 2 , where P ⊆ R 2 and the distance function is Euclidean distance [ Amano, K. and Nakano, S. I., Away from Rivals, CCCG, pp.68-71, 2018 ]. We also prove that the c-dispersion problem in a metric space is W [1]-hard.

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“…Based on the bounded search tree method we propose an exact fixed-parameter algorithm in O(2 k (n 2 log n + n(log 2 n)(log k))) time, for this problem, where k is the parameter. The proposed exact algorithm is better than the current best exact exponential algorithm [n O( √ k)time algorithm by Akagi et al,(2018)] whenever k < c log 2 n for some constant c. We then present an O(log n)-time 1 2 √ 2 -approximation algorithm for the problem when k = 3 if the points are given in convex position order.…”

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confidence: 97%

Max-Min $k$-Dispersion on a Convex Polygon

Singireddy¹,

Basappa²

2022

Preprint

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In this paper we consider the following k-dispersion problem. Given a set S of n points placed in the plane in a convex position, and an integer k (0 < k < n), the objective is to compute a subset S ⊂ S such that |S | = k and the minimum distance between a pair of points in S is maximized. Based on the bounded search tree method we propose an exact fixed-parameter algorithm in O(2 k (n 2 log n + n(log 2 n)(log k))) time, for this problem, where k is the parameter. The proposed exact algorithm is better than the current best exact exponential algorithm [n O( √ k)time algorithm by Akagi et al.,(2018)] whenever k < c log 2 n for some constant c. We then present an O(log n)-time 1 2 √ 2 -approximation algorithm for the problem when k = 3 if the points are given in convex position order.

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Exact Algorithms for the Max-Min Dispersion Problem

Cited by 20 publications

References 19 publications

Approximation Algorithms For The Euclidean Dispersion Problems

Approximation Algorithms For The Euclidean Dispersion Problems

Approximation Algorithms For The Dispersion Problems in a Metric Space

Max-Min $k$-Dispersion on a Convex Polygon

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