Bi-Factor Approximation Algorithms for Hard Capacitated <i>k</i>-Median Problems

Byrka, Jarosław; Fleszar, Krzysztof; Rybicki, Bartosz; Spoerhase, Joachim

doi:10.1137/1.9781611973730.49

Cited by 42 publications

(58 citation statements)

References 30 publications

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“…As a special case of CkFLP, their result applies on hard nonuniform CkM with all facility costs being zero. In the same manner, the CkFLP result (1/ 2 ) of Byrka et al [4] is applicable on hard uniform CkM. The result violates capacities by a factor of 2 + .…”

Section: Introductionmentioning

confidence: 70%

“…Several techniques have been used to obtain results for the problem. One of the most widely used technique to ap-proximate the problem is LP Rounding ( [4,5,7,8,9,11,12,13,14]). Charikar et al [7] gave a 20/3 factor approximation algorithm for UkM, which was further improved to 3.25 factor by Charikar and Li in [8].…”

Section: Introductionmentioning

confidence: 99%

“…Capacity violation: Charikar et al [7] gave a 16 factor approximation algorithm for hard uniform CkM violating capacities by a factor of 3 in case of splittable demands and 4 in case of un-splittable demands. In 2015, Byrka et al [4] gave an O(1/ ) approximation algorithm violating capacities by a factor of (3 + ) for hard non-uniform CkM. Demirci et al [9] improved the approximation ratio to O(1/ 5 ) with capacity violation of (1 + ) for the same version of the problem.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Improved Local Search Based Approximation Algorithm for Hard Uniform Capacitated k-Median Problem

Grover

Gupta

Pancholi

2018

IJCAI

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In this paper, we study the hard uniform capacitated k-median problem. We give (5 +) factor approximation for the problem using local search technique, violating cardinality by a factor of 3. Though better results are known for the problem using LP techniques, local search algorithms are well known to be simpler. There is a trade-off viz-a-viz approximation factor and cardinality violation between our result and the result of Korupolu et al. [10] which is the only result known for the problem using local search. They gave (1 + α) approximation factor with (5 + 5/α) factor loss in cardinality. In a sense, our result is an improvement as they violate the cardinality by more than a factor of 6 to achieve 5 factor in approximation. Though in their result, the approximation factor can be made arbitrarily small, cardinality loss is at least 5 and small approximation factor is obtained at a big loss in cardinality. Thus, we improve upon their result with respect to cardinality. Povzetek: Obravnavan je NP problem optimiranja iskanja k median in predlagana izvirna rešitev, ki dosega boljše rezultate v določenih primerjavah.

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Section: Introductionmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Improved Local Search Based Approximation Algorithm for Hard Uniform Capacitated k-Median Problem

Grover

Gupta

Pancholi

2018

IJCAI

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“…We now examine the quantity Considering each cluster in both S and O to have at least ⌈2λ⌉k points, we can extend the result to account for λ. To do this, we apply the replacements leading to inequality (7) ⌈2λ⌉ times to the sum 2λ (o i ,s j )∈Σ s ∈S…”

Section: Approximation Guaranteesmentioning

confidence: 99%

Reconciliation k-median: Clustering with Non-polarized Representatives

Ordozgoiti

Gionis

2019

The World Wide Web Conference

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We propose a new variant of the k-median problem, where the objective function models not only the cost of assigning data points to cluster representatives, but also a penalty term for disagreement among the representatives. We motivate this novel problem by applications where we are interested in clustering data while avoiding selecting representatives that are too far from each other. For example, we may want to summarize a set of news sources, but avoid selecting ideologically-extreme articles in order to reduce polarization.To solve the proposed k-median formulation we adopt the localsearch algorithm of Arya et al. [2], We show that the algorithm provides a provable approximation guarantee, which becomes constant under a mild assumption on the minimum number of points for each cluster. We experimentally evaluate our problem formulation and proposed algorithm on datasets inspired by the motivating applications. In particular, we experiment with data extracted from Twitter, the US Congress voting records, and popular news sources. The results show that our objective can lead to choosing less polarized groups of representatives without significant loss in representation fidelity.

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“…Later, Chuzhoy and Rabani [13] gave a 40-approximation with capacity violation 50, for the more general non-uniform capacitated k-median, where different facilities can have different capacities. Recently, Byrka et al [6] improved the capacity violation constant 3 of [9] for uniform CKM to 2 + and achieved approximation ratio of O(1/ 2 ). This factor was improved to O(1/ ) by Li [21].…”

Section: Introductionmentioning

confidence: 99%

On Uniform Capacitated k -Median Beyond the Natural LP Relaxation

LiShi

2017

ACM Trans. Algorithms

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In this paper, we study the uniform capacitated k-median problem. In the problem, we are given a set F of potential facility locations, a set C of clients, a metric d over F ∪ C, an upper bound k on the number of facilities we can open and an upper bound u on the number of clients each facility can serve. We need to open a subset S ⊆ F of k facilities and connect clients in C to facilities in S so that each facility is connected by at most u clients. The goal is to minimize the total connection cost over all clients. Obtaining a constant approximation algorithm for this problem is a notorious open problem; most previous works gave constant approximations by either violating the capacity constraints or the cardinality constraint. Notably, all these algorithms are based on the natural LP-relaxation for the problem. The LP-relaxation has unbounded integrality gap, even when we are allowed to violate the capacity constraints or the cardinality constraint by a factor of 2 − .Our result is an exp(O(1/ 2 ))-approximation algorithm for the problem that violates the cardinality constraint by a factor of 1 + . That is, we find a solution that opens at most (1 + )k facilities whose cost is at most exp(O(1/ 2 )) times the optimum solution when at most k facilities can be open. This is already beyond the capability of the natural LP relaxation, as it has unbounded integrality gap even if we are allowed to open (2 − )k facilities. Indeed, our result is based on a novel LP for this problem. We hope that this LP is the first step towards a constant approximation for capacitated k-median.The version as we described is the hard-capacitated version of the problem, as we can only open one facility at each location. This is as opposed to the soft-capacitated version, in which we are allowed to open more than one facilities at each location. The hard-capacitated version is more general, since one can convert a soft-capacitated instance to a hard-capacitated instance by making enough copies of each facility location. We give a simple proof that in the uniform capacitated case, the soft-capacitated version and the hard-capacitated version are actually equivalent, up to a small constant loss in the approximation ratio. Moreover, we show that the given potential facility locations do not matter: we can assume F = C.

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Bi-Factor Approximation Algorithms for Hard Capacitated k-Median Problems

Cited by 42 publications

References 30 publications

Improved Local Search Based Approximation Algorithm for Hard Uniform Capacitated k-Median Problem

Improved Local Search Based Approximation Algorithm for Hard Uniform Capacitated k-Median Problem

Reconciliation k-median: Clustering with Non-polarized Representatives

On Uniform Capacitated k -Median Beyond the Natural LP Relaxation

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