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Constant factor FPT approximation for capacitated k-median
[article]

2018
*
arXiv
*
pre-print

*Capacitated*

*k*-

*median*is one of the few outstanding optimization problems

*for*which the existence of a polynomial time

*constant*

*factor*

*approximation*algorithm remains an open problem. ... In this work we provide an

*FPT*-time

*constant*

*factor*

*approximation*algorithm preserving both cardinality and capacity of the facilities. ... Subsequently, Li broke this integrality gap barrier by giving a

*constant*

*factor*algorithm

*for*the

*capacitated*

*k*-

*median*by opening (1 + ε) ·

*k*facilities [19, 20] . ...

##
###
Constant-Factor FPT Approximation for Capacitated k-Median

2019
*
European Symposium on Algorithms
*

*Capacitated*

*k*-

*median*is one of the few outstanding optimization problems

*for*which the existence of a polynomial time

*constant*

*factor*

*approximation*algorithm remains an open problem. ... In this work we provide an

*FPT*-time

*constant*

*factor*

*approximation*algorithm preserving both cardinality and capacity of the facilities. ... E S A 2 0 1 9 1:8

*Constant*-

*Factor*

*FPT*

*Approximation*

*for*

*Capacitated*

*k*-

*Median*Definition 8. ...

##
###
A constant FPT approximation algorithm for hard-capacitated k-means
[article]

2019
*
arXiv
*
pre-print

As our main contribution, we propose an

arXiv:1901.04628v3
fatcat:smhtswk2yzcijlxw6jol5yhjsm
*FPT*(*k*) algorithm with performance guarantee of 69+ϵ*for*any HCKM instances in this paper. ... To the best our knowledge, no*constant**approximation*algorithm or existence proof of such an algorithm is known. ... [1] propose a*constant**FPT**approximation**for*both uniform and nonuniform*capacitated**k*-*median*, which inspires our work. ...##
###
On the Fixed-Parameter Tractability of Capacitated Clustering

2019
*
International Colloquium on Automata, Languages and Programming
*

*constant*

*factor*exists. ... We show that there exists a (3 + )-

*approximation*algorithm

*for*the

*capacitated*

*k*-

*median*and a (9 + )-

*approximation*algorithm

*for*the

*capacitated*

*k*-means problem in general metric spaces whose running times ... Indeed,

*constant*

*factor*

*approximation*algorithms are known if the capacities [22, 23] or the number of clusters can be violated by a (1 + )

*factor*[4, 13] ,

*for*

*constant*. ...

##
###
FPT Approximation for Constrained Metric k-Median/Means
[article]

2020
*
arXiv
*
pre-print

We give

arXiv:2007.11773v1
fatcat:josqnidnabaytcmgdf2wuav2rm
*FPT*algorithms with*constant**approximation*guarantee*for*a range of constrained*k*-*median*/means problems. ...*For*this special case, our algorithm gives a (2+ε)-*approximation*and (4+ε)-*approximation**for*the constrained versions of*k*-*median*and*k*-means problem respectively in*FPT*time. ... Acknowledgements The authors would like to thank Anup Bhattacharya*for*useful discussions. ...##
###
Capacitated Sum-Of-Radii Clustering: An FPT Approximation

2020
*
European Symposium on Algorithms
*

As a warm-up

doi:10.4230/lipics.esa.2020.62
dblp:conf/esa/0002V20
fatcat:hhphvg2ucfdvblzoyovctnxvnq
*for*this result, we also give a*constant**approximation**for*the uncapacitated sum of radii clustering problem with matroid constraints, thus obtaining the first*FPT**approximation**for*this problem ... While*constant**approximations*are known*for*the uncapacitated version of the problem, there is no work on the*capacitated*version. ... Adapting techniques they develop*for**capacitated**k*-*median*, we can obtain an*FPT**approximation**for**capacitated*sum of radii in metrics of*constant*doubling dimension. ...##
###
Tight FPT Approximation for Socially Fair Clustering
[article]

2021
*
arXiv
*
pre-print

We design (3+ε) and (9 + ε)

arXiv:2106.06755v2
fatcat:2zoqyak7mffaldszq6unlocvjq
*approximation*algorithms*for*the socially fair*k*-*median*and*k*-means problems, respectively, in*FPT*(fixed parameter tractable) time f(*k*,ε) · n^O(1), where f(*k*,ε) = (*k*/ε)^O(*k*... Furthermore, we show that if Gap-ETH holds, then better*approximation*guarantees are not possible in*FPT*time. ... Acknowledgement Thanks to Karthik C.S.*for*pointing us to the work of Pasin Manurangsi [Man20] . ...##
###
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

2020
*
Algorithms
*

Parameterization and

doi:10.3390/a13060146
fatcat:2u2vv3uksfguvj6473t2gsq42a
*approximation*are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. ... Does*CAPACITATED**k*-*MEDIAN*admit an (1 + 2/e)-*approximation*algorithm in*FPT*time with parameter*k*? Do*CAPACITATED*EUCLIDEAN*k*-MEANS/*k*-*MEDIAN*admit an EPAS with parameter*k*or d? ... Second, PIH implies that*k*-CLIQUE is hard to*approximate*to within any*constant**factor*: Lemma 1. Assuming PIH, there is no*constant**factor**FPT**approximation*algorithm*for**k*-CLIQUE. ...##
###
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
[article]

2020
*
arXiv
*
pre-print

Parameterization and

arXiv:2006.04411v1
fatcat:hjgu7f3s7zbydkcnioq3qlzgza
*approximation*are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. ... Does*CAPACITATED**k*-*MEDIAN*admit an (1 + 2/e)-*approximation*algorithm in*FPT*time with parameter*k*? Do*CAPACITATED*EUCLIDEAN*k*-MEANS/*k*-*MEDIAN*admit an EPAS with parameter*k*or d? ... Second, PIH implies that*k*-CLIQUE is hard to*approximate*to within any*constant**factor*: Lemma 1. Assuming PIH, there is no*constant**factor**FPT**approximation*algorithm*for**k*-CLIQUE. ...##
###
On the Fixed-Parameter Tractability of Capacitated Clustering
[article]

2022
*
arXiv
*
pre-print

*constant*

*factor*exists. ... We show that there exists a (3+ϵ)-

*approximation*algorithm

*for*the

*capacitated*

*k*-

*median*and a (9+ϵ)-

*approximation*algorithm

*for*the

*capacitated*

*k*-means problem in general metric spaces whose running times ... Indeed,

*constant*

*factor*

*approximation*algorithms are known if the capacities [22, 23] or the number of clusters can be violated by a (1 + )

*factor*[4, 13],

*for*

*constant*. ...

##
###
A Parameterized Approximation Algorithm for the Chromatic k-Median Problem

2021
*
IEEE Access
*

In this paper, we give an

doi:10.1109/access.2021.3060422
fatcat:yrnbhxxuifca7gq4cmmcj3ejg4
*FPT*(*k*)-time*approximation*algorithm*for*chromatic*k*-*median*. ... The algorithm achieves a (3 + )-*approximation*and runs in ( We propose a different random sampling algorithm*for*opening facilities, which is the crucial step in getting the*constant**factor*parameterized ... It remains an open problem that whether a*constant**factor**approximation**for*chromatic*k*-*median*can be achieved. ...##
###
Parameterized Inapproximability for Steiner Orientation by Gap Amplification

2020
*
International Colloquium on Automata, Languages and Programming
*

We show that

doi:10.4230/lipics.icalp.2020.104
dblp:conf/icalp/000120
fatcat:x3myqmzqlnfgdeasbbojyrnqpm
*k*-Steiner Orientation is unlikely to admit an*approximation*algorithm with any*constant**factor*, even within*FPT*running time. ... The problem is known to be W[1]-hard when parameterized by*k*and hard to*approximate*up to some*constant**for**FPT*algorithms assuming Gap-ETH. ...*For**Capacitated**k*-*Median*, a*constant**factor**FPT**approximation*has been obtained [1, 14] , whereas the best-known polynomial-time*approximation**factor*is O(log*k*). ...##
###
On Coresets for Fair Clustering in Metric and Euclidean Spaces and Their Applications
[article]

2020
*
arXiv
*
pre-print

In particular, we obtain the first true

arXiv:2007.10137v1
fatcat:4aytetpfg5agvglviignswgisa
*constant*-*approximation*algorithm*for*metric fair clustering, whose running time is fixed-parameter tractable (*FPT*). ... This leads to improved*constant*-*approximations**for*these problems in general metrics and near-linear time (1+ϵ)-*approximations*in the Euclidean metric. ... The authors are thankful to Vincent Cohen-Addad*for*sharing the full version of [35] . A Proof of Lemma 4.14 We consider a minimum cost feasible flow φ in G Y*for*Y = 1. ...##
###
Parameterized inapproximability for Steiner Orientation by Gap Amplification
[article]

2020
*
arXiv
*
pre-print

We show that

arXiv:1907.06529v3
fatcat:awyzxap73na2zkbmo2rcinup4q
*k*-Steiner Orientation is unlikely to admit an*approximation*algorithm with any*constant**factor*, even within*FPT*running time. ... The problem is known to be W[1]-hard when parameterized by*k*and hard to*approximate*up to some*constant**for**FPT*algorithms assuming Gap-ETH. ...*For**Capacitated**k*-*Median*, a*constant**factor**FPT**approximation*has been obtained [1] , whereas the best-known polynomial-time*approximation**factor*is O(log*k*). ...##
###
Lossy Kernelization of Same-Size Clustering
[article]

2021
*
arXiv
*
pre-print

In this work, we study the

arXiv:2107.07383v1
fatcat:unbmeiysaza6rcxpesfxz7g5lu
*k*-*median*clustering problem with an additional equal-size constraint on the clusters, from the perspective of parameterized preprocessing. ... Our main result is the first lossy (2-*approximate*) polynomial kernel*for*this problem, parameterized by the cost of clustering. ... It includes tight algorithmic and complexity results*for**k*-means and*k*-*median*[13] and*constant**factor**FPT*-*approximation**for**capacitated*clustering [14] . ...
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