r-Gatherings on a Star

Ahmed, Shareef; Nakano, Shin-ichi; Rahman, Md. Saidur

doi:10.1007/978-3-030-10564-8_3

Cited by 6 publications

(16 citation statements)

References 9 publications

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“…This result is the best possible in the sense of the number of parameters with superpolynomial dependence because at least one parameter (e.g., degree) is necessary according to Theorem 1. Our algorithms have lower parameter dependencies than previous algorithms [2] The proof appears in Sect. 5.…”

Section: Introductionmentioning

confidence: 89%

“…A cluster is single-leg if it contains users from a single leg; otherwise, it is multi-leg. Ahmed et al [2] showed that there is an optimal solution that has a specific singleleg/multi-leg structure as follows.…”

Section: Fpt Algorithm For R-gather Clustering and R-gathering On Spidermentioning

confidence: 99%

“…Our Contribution. In this study, we answer the open question that asks the complexity of r-gathering problem, posed by Ahmed et al [2] by closing the gap between the tractability and intractability of the problems on a spider.…”

Section: Introductionmentioning

confidence: 96%

“…We also consider the r-gather clustering problem [2,3,8], which is a variant of the r-gathering problem in which the facilities are located everywhere.…”

Section: Introductionmentioning

confidence: 99%

“…Such clusters are obtained by solving the r-gather clustering problem. Another typical problem is the sport-team formation problem [2]. Imagine that a town has n football players and m football courts.…”

Section: Introductionmentioning

confidence: 99%

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r-Gathering Problems on Spiders: Hardness, FPT Algorithms, and PTASes

Kumabe

Maehara

2021

WALCOM: Algorithms and Computation

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We consider the min-max r-gathering problem described as follows: We are given a set of users and facilities in a metric space. We open some of the facilities and assign each user to an opened facility such that each facility has at least r users. The goal is to minimize the maximum distance between the users and the assigned facility. We also consider the min-max r-gather clustering problem, which is a special case of the r-gathering problem in which the facilities are located everywhere.In this paper, we study the tractability and the hardness when the underlying metric space is a spider, which answers the open question posed by Ahmed et al. [WALCOM'19]. First, we show that the problems are NP-hard even if the underlying space is a spider. Then, we propose FPT algorithms parameterized by the degree d of the center. This improves the previous algorithms because they are parameterized by both r and d. Finally, we propose PTASes to the problems. These are best possible because there are no FPTASes unless P = NP.

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Section: Introductionmentioning

confidence: 89%

Section: Fpt Algorithm For R-gather Clustering and R-gathering On Spidermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

“…We also consider the r-gather clustering problem [2,3,8], which is a variant of the r-gathering problem in which the facilities are located everywhere.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

r-Gathering Problems on Spiders: Hardness, FPT Algorithms, and PTASes

Kumabe

Maehara

2021

WALCOM: Algorithms and Computation

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One-Dimensional r-Gathering Under Uncertainty

Ahmed

Nakano

Rahman

2019

Algorithmic Aspects in Information and Management

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r-Gatherings on a star and uncertain r-gatherings on a line

Ahmed

Nakano

Rahman

2021

Discrete Math. Algorithm. Appl.

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Let [Formula: see text] be a set of [Formula: see text] customers and [Formula: see text] be a set of [Formula: see text] facilities. An [Formula: see text]-gather clustering of [Formula: see text] is a partition of the customers in clusters such that each cluster contains at least [Formula: see text] customers. The [Formula: see text]-gather clustering problem asks to find an [Formula: see text]-gather clustering which minimizes the maximum distance between a pair of customers in a cluster. An [Formula: see text]-gathering of [Formula: see text] to [Formula: see text] is an assignment of each customer [Formula: see text] to a facility [Formula: see text] such that each facility has zero or at least [Formula: see text] customers. The [Formula: see text]-gathering problem asks to find an [Formula: see text]-gathering that minimizes the maximum distance between a customer and his/her facility. In this work, we consider the [Formula: see text]-gather clustering and [Formula: see text]-gathering problems when the customers and the facilities are lying on a “star”. We show that the [Formula: see text]-gather clustering problem and the [Formula: see text]-gathering problem with customers and facilities on a star with [Formula: see text] rays can be solved in [Formula: see text] and [Formula: see text] time, respectively. Furthermore, we prove the hardness of a variant of the [Formula: see text]-gathering problem, called the min-max-sum [Formula: see text]-gathering problem, even when the customers and the facilities are on a star. We also study the [Formula: see text]-gathering problem when the customers and the facilities are on a line, and each customer location is uncertain. We show that the [Formula: see text]-gathering problem can be solved in [Formula: see text] and [Formula: see text] time when the customers and the facilities are on a line, and the customer locations are given by piecewise uniform functions of at most [Formula: see text] pieces and “well-separated” uniform distribution functions, respectively.

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r-Gatherings on a Star

Cited by 6 publications

References 9 publications

r-Gathering Problems on Spiders: Hardness, FPT Algorithms, and PTASes

r-Gathering Problems on Spiders: Hardness, FPT Algorithms, and PTASes

One-Dimensional r-Gathering Under Uncertainty

r-Gatherings on a star and uncertain r-gatherings on a line

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