2,036 research outputs found
Centrality of Trees for Capacitated k-Center
There is a large discrepancy in our understanding of uncapacitated and
capacitated versions of network location problems. This is perhaps best
illustrated by the classical k-center problem: there is a simple tight
2-approximation algorithm for the uncapacitated version whereas the first
constant factor approximation algorithm for the general version with capacities
was only recently obtained by using an intricate rounding algorithm that
achieves an approximation guarantee in the hundreds.
Our paper aims to bridge this discrepancy. For the capacitated k-center
problem, we give a simple algorithm with a clean analysis that allows us to
prove an approximation guarantee of 9. It uses the standard LP relaxation and
comes close to settling the integrality gap (after necessary preprocessing),
which is narrowed down to either 7, 8 or 9. The algorithm proceeds by first
reducing to special tree instances, and then solves such instances optimally.
Our concept of tree instances is quite versatile, and applies to natural
variants of the capacitated k-center problem for which we also obtain improved
algorithms. Finally, we give evidence to show that more powerful preprocessing
could lead to better algorithms, by giving an approximation algorithm that
beats the integrality gap for instances where all non-zero capacities are
uniform.Comment: 21 pages, 2 figure
Capacitated Center Problems with Two-Sided Bounds and Outliers
In recent years, the capacitated center problems have attracted a lot of
research interest. Given a set of vertices , we want to find a subset of
vertices , called centers, such that the maximum cluster radius is
minimized. Moreover, each center in should satisfy some capacity
constraint, which could be an upper or lower bound on the number of vertices it
can serve. Capacitated -center problems with one-sided bounds (upper or
lower) have been well studied in previous work, and a constant factor
approximation was obtained.
We are the first to study the capacitated center problem with both capacity
lower and upper bounds (with or without outliers). We assume each vertex has a
uniform lower bound and a non-uniform upper bound. For the case of opening
exactly centers, we note that a generalization of a recent LP approach can
achieve constant factor approximation algorithms for our problems. Our main
contribution is a simple combinatorial algorithm for the case where there is no
cardinality constraint on the number of open centers. Our combinatorial
algorithm is simpler and achieves better constant approximation factor compared
to the LP approach
An Improved Approximation Algorithm for the Hard Uniform Capacitated k-median Problem
In the -median problem, given a set of locations, the goal is to select a
subset of at most centers so as to minimize the total cost of connecting
each location to its nearest center. We study the uniform hard capacitated
version of the -median problem, in which each selected center can only serve
a limited number of locations.
Inspired by the algorithm of Charikar, Guha, Tardos and Shmoys, we give a
-approximation algorithm for this problem with increasing the
capacities by a factor of , which improves
the previous best -approximation algorithm proposed by Byrka,
Fleszar, Rybicki and Spoerhase violating the capacities by factor
.Comment: 19 pages, 1 figur
Constant Factor Approximation for Capacitated k-Center with Outliers
The -center problem is a classic facility location problem, where given an
edge-weighted graph one is to find a subset of vertices ,
such that each vertex in is "close" to some vertex in . The
approximation status of this basic problem is well understood, as a simple
2-approximation algorithm is known to be tight. Consequently different
extensions were studied.
In the capacitated version of the problem each vertex is assigned a capacity,
which is a strict upper bound on the number of clients a facility can serve,
when located at this vertex. A constant factor approximation for the
capacitated -center was obtained last year by Cygan, Hajiaghayi and Khuller
[FOCS'12], which was recently improved to a 9-approximation by An, Bhaskara and
Svensson [arXiv'13].
In a different generalization of the problem some clients (denoted as
outliers) may be disregarded. Here we are additionally given an integer and
the goal is to serve exactly clients, which the algorithm is free to
choose. In 2001 Charikar et al. [SODA'01] presented a 3-approximation for the
-center problem with outliers.
In this paper we consider a common generalization of the two extensions
previously studied separately, i.e. we work with the capacitated -center
with outliers. We present the first constant factor approximation algorithm
with approximation ratio of 25 even for the case of non-uniform hard
capacities.Comment: 15 pages, 3 figures, accepted to STACS 201
On the Cost of Essentially Fair Clusterings
Clustering is a fundamental tool in data mining. It partitions points into
groups (clusters) and may be used to make decisions for each point based on its
group. However, this process may harm protected (minority) classes if the
clustering algorithm does not adequately represent them in desirable clusters
-- especially if the data is already biased.
At NIPS 2017, Chierichetti et al. proposed a model for fair clustering
requiring the representation in each cluster to (approximately) preserve the
global fraction of each protected class. Restricting to two protected classes,
they developed both a 4-approximation for the fair -center problem and a
-approximation for the fair -median problem, where is a parameter
for the fairness model. For multiple protected classes, the best known result
is a 14-approximation for fair -center.
We extend and improve the known results. Firstly, we give a 5-approximation
for the fair -center problem with multiple protected classes. Secondly, we
propose a relaxed fairness notion under which we can give bicriteria
constant-factor approximations for all of the classical clustering objectives
-center, -supplier, -median, -means and facility location. The
latter approximations are achieved by a framework that takes an arbitrary
existing unfair (integral) solution and a fair (fractional) LP solution and
combines them into an essentially fair clustering with a weakly supervised
rounding scheme. In this way, a fair clustering can be established belatedly,
in a situation where the centers are already fixed
Constant-Factor FPT Approximation for Capacitated k-Median
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 a series of recent papers algorithms producing solutions violating either the number of facilities or the capacity by a multiplicative factor were obtained. However, to produce solutions without violations appears to be hard and potentially requires different algorithmic techniques. Notably, if parameterized by the number of facilities k, the problem is also W[2] hard, making the existence of an exact FPT algorithm unlikely. In this work we provide an FPT-time constant factor approximation algorithm preserving both cardinality and capacity of the facilities. The algorithm runs in time 2^O(k log k) n^O(1) and achieves an approximation ratio of 7+epsilon
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