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Register a new userThe Heterogeneous Capacitated $k$-Center Problem
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Added by
Deeparnab Chakrabarty
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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In this paper we initiate the study of the heterogeneous capacitated $k$-center problem: given a metric space $X = (F cup C, d)$, and a collection of capacities. The goal is to open each capacity at a unique facility location in $F$, and also to assign clients to facilities so that the number of clients assigned to any facility is at most the capacity installed; the objective is then to minimize the maximum distance between a client and its assigned facility. If all the capacities $c_i$s are identical, the problem becomes the well-studied uniform capacitated $k$-center problem for which constant-factor approximations are known. The additional choice of determining which capacity should be installed in which location makes our problem considerably different from this problem, as well the non-uniform generalizations studied thus far in literature. In fact, one of our contributions is in relating the heterogeneous problem to special-cases of the classical Santa Claus problem. Using this connection, and by designing new algorithms for these special cases, we get the following results: (a)A quasi-polynomial time $O(log n/epsilon)$-approximation where every capacity is violated by $1+varepsilon$, (b) A polynomial time $O(1)$-approximation where every capacity is violated by an $O(log n)$ factor. We get improved results for the {em soft-capacities} version where we can place multiple facilities in the same location.
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In this paper, we introduce and study the Non-Uniform k-Center problem (NUkC). Given a finite metric space $(X,d)$ and a collection of balls of radii ${r_1geq cdots ge r_k}$, the NUkC problem is to find a placement of their centers on the metric space and find the minimum dilation $alpha$, such that the union of balls of radius $alphacdot r_i$ around the $i$th center covers all the points in $X$. This problem naturally arises as a min-max vehicle routing problem with fleets of different speeds. The NUkC problem generalizes the classic $k$-center problem when all the $k$ radii are the same (which can be assumed to be $1$ after scaling). It also generalizes the $k$-center with outliers (kCwO) problem when there are $k$ balls of radius $1$ and $ell$ balls of radius $0$. There are $2$-approximation and $3$-approximation algorithms known for these problems respectively; the former is best possible unless P=NP and the latter remains unimproved for 15 years. We first observe that no $O(1)$-approximation is to the optimal dilation is possible unless P=NP, implying that the NUkC problem is more non-trivial than the above two problems. Our main algorithmic result is an $(O(1),O(1))$-bi-criteria approximation result: we give an $O(1)$-approximation to the optimal dilation, however, we may open $Theta(1)$ centers of each radii. Our techniques also allow us to prove a simple (uni-criteria), optimal $2$-approximation to the kCwO problem improving upon the long-standing $3$-factor. Our main technical contribution is a connection between the NUkC problem and the so-called firefighter problems on trees which have been studied recently in the TCS community.
We study the Capacitated k-Median problem, for which all the known constant factor approximation algorithms violate either the number of facilities or the capacities. While the standard LP-relaxation can only be used for algorithms violating one of the two by a factor of at least two, Shi Li [SODA15, SODA16] gave algorithms violating the number of facilities by a factor of 1+{epsilon} exploring properties of extended relaxations. In this paper we develop a constant factor approximation algorithm for Uniform Capacitated k-Median violating only the capacities by a factor of 1+{epsilon}. The algorithm is based on a configuration LP. Unlike in the algorithms violating the number of facilities, we cannot simply open extra few facilities at selected locations. Instead, our algorithm decides about the facility openings in a carefully designed dependent rounding process.
In recent years, the capacitated center problems have attracted a lot of research interest. Given a set of vertices $V$, we want to find a subset of vertices $S$, called centers, such that the maximum cluster radius is minimized. Moreover, each center in $S$ should satisfy some capacity constraint, which could be an upper or lower bound on the number of vertices it can serve. Capacitated $k$-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 $k$ 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.
We first show that a better analysis of the algorithm for The Two-Sage Stochastic Facility Location Problem from Srinivasan cite{sri07} and the algorithm for The Robust Fault Tolerant Facility Location Problem from Byrka et al cite{bgs10} can render improved approximation factors of 2.206 and alpha+4 where alpha is the maximum number an adversary can close, respectively, and which are the best ratios so far. We then present new models for the soft-capacitated facility location problem with uncertainty and design constant factor approximation algorithms to solve them. We devise the stochastic and robust approaches to handle the uncertainty incorporated into the original model. Explicitly, in this paper we propose two new problem, named The 2-Stage Soft-Capacitated Facility Location Problem and The Robust Soft-Capacitated Facility Location Problem respectively, and present constant factor approximation algorithms for them both. Our method uses reductions between facility location problems and linear-cost models, the randomized thresholding technique of Srinivasan cite{sri07} and the filtering and clustering technique of Byrka et al cite{bgs10}.
The $k$-Facility Location problem is a generalization of the classical problems $k$-Median and Facility Location. The goal is to select a subset of at most $k$ facilities that minimizes the total cost of opened facilities and established connections between clients and opened facilities. We consider the hard-capacitated version of the problem, where a single facility may only serve a limited number of clients and creating multiple copies of a facility is not allowed. We construct approximation algorithms slightly violating the capacities based on rounding a fractional solution to the standard LP. It is well known that the standard LP (even in the case of uniform capacities and opening costs) has unbounded integrality gap if we only allow violating capacities by a factor smaller than $2$, or if we only allow violating the number of facilities by a factor smaller than $2$. In this paper, we present the first constant-factor approximation algorithms for the hard-capacitated variants of the problem. For uniform capacities, we obtain a $(2+varepsilon)$-capacity violating algorithm with approximation ratio $O(1/varepsilon^2)$; our result has not yet been improved. Then, for non-uniform capacities, we consider the case of $k$-Median, which is equivalent to $k$-Facility Location with uniform opening cost of the facilities. Here, we obtain a $(3+varepsilon)$-capacity violating algorithm with approximation ratio $O(1/varepsilon)$.
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