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Register a new userAn approximation algorithm for Uniform Capacitated k-Median problem with 1 + {epsilon} capacity violation
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Bartosz Rybicki
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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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.
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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 $O(1/varepsilon)$.
This study considers the (soft) capacitated vertex cover problem in a dynamic setting. This problem generalizes the dynamic model of the vertex cover problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted graph $G=(V,E)$, which allows edge insertions and edge deletions, the goal is to design a data structure that maintains an approximate minimum vertex cover while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex $v$ in the cover, the number of $v$s incident edges covered by the copy is up to a given capacity of $v$. We extend Bhattacharya et al.s work [SODA15 and ICALP15] to obtain a deterministic primal-dual algorithm for maintaining a constant-factor approximate minimum capacitated vertex cover with $O(log n / epsilon)$ amortized update time, where $n$ is the number of vertices in the graph. The algorithm can be extended to (1) a more general model in which each edge is associated with a nonuniform and unsplittable demand, and (2) the more general capacitated set cover problem.
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.
We present a packing-based approximation algorithm for the $k$-Set Cover problem. We introduce a new local search-based $k$-set packing heuristic, and call it Restricted $k$-Set Packing. We analyze its tight approximation ratio via a complicated combinatorial argument. Equipped with the Restricted $k$-Set Packing algorithm, our $k$-Set Cover algorithm is composed of the $k$-Set Packing heuristic cite{schrijver} for $kgeq 7$, Restricted $k$-Set Packing for $k=6,5,4$ and the semi-local $(2,1)$-improvement cite{furer} for 3-Set Cover. We show that our algorithm obtains a tight approximation ratio of $H_k-0.6402+Theta(frac{1}{k})$, where $H_k$ is the $k$-th harmonic number. For small $k$, our results are 1.8667 for $k=6$, 1.7333 for $k=5$ and 1.5208 for $k=4$. Our algorithm improves the currently best approximation ratio for the $k$-Set Cover problem of any $kgeq 4$.
Let $G=(V, E)$ be a given edge-weighted graph and let its {em realization} $mathcal{G}$ be a random subgraph of $G$ that includes each edge $e in E$ independently with probability $p$. In the {em stochastic matching} problem, the goal is to pick a sparse subgraph $Q$ of $G$ without knowing the realization $mathcal{G}$, such that the maximum weight matching among the realized edges of $Q$ (i.e. graph $Q cap mathcal{G}$) in expectation approximates the maximum weight matching of the whole realization $mathcal{G}$. In this paper, we prove that for any desirably small $epsilon in (0, 1)$, every graph $G$ has a subgraph $Q$ that guarantees a $(1-epsilon)$-approximation and has maximum degree only $O_{epsilon, p}(1)$. That is, the maximum degree of $Q$ depends only on $epsilon$ and $p$ (both of which are known to be necessary) and not for example on the number of nodes in $G$, the edge-weights, etc. The stochastic matching problem has been studied extensively on both weighted and unweighted graphs. Previously, only existence of (close to) half-approximate subgraphs was known for weighted graphs [Yamaguchi and Maehara, SODA18; Behnezhad et al., SODA19]. Our result substantially improves over these works, matches the state-of-the-art for unweighted graphs [Behnezhad et al., STOC20], and essentially settles the approximation factor.
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