In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B ⊆ B of balls and assign each point in P to some ball in B that contains it, such that the number of points assigned to any ball is at most U . The objective function that we would like to minimize is the cardinality of B .We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for these problems. In fact, in the Euclidean setting, only (1 + ) factor expansion is sufficient for any > 0, with the approximation factor being a polynomial in 1/ . We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems. complexity theoretic assumptions [14]. The same is true for the MUC, as demonstrated by the following reduction from Set Cover. We take a ball of radius 1 corresponding to each set, and a point corresponding to each element. If an element is in a set, then the distance between the center of the corresponding ball and the point is 1. We consider the metric space induced by the centers and the points. It is easy to see that any solution for this instance of MUC directly gives a solution for the input instance of the general Set Cover, implying that for MUC, it is not possible to get any approximation guarantee better than the O(log n) bound for Set Cover.The MUC in fixed dimensional Euclidean spaces has been extensively studied. One interesting variant is when the allowed set B of balls consists of all unit balls. Hochbaum and Maass [19] gave a polynomial time approximation scheme (PTAS) for this using a grid shifting strategy. When B is an arbitrary finite set of balls, the problem seems to be much harder. An O(1) approximation algorithm in the 2-dimensional Euclidean plane was given by Brönnimann and Goodrich [9]. More recently, a PTAS was obatined by Mustafa and Ray [26]. In dimensions 3 and higher, the best known approximation guarantee is still O(log n). Motivated by this, Har-Peled and Lee [18] gave a PTAS for a bi-criteria version where the algorithm is allowed to expand the input balls by a (1 + ) fac...
Voronoi game is a geometric model of competitive facility location problem played between two players. Users are generally modeled as points uniformly distributed on a given underlying space. Each player chooses a set of points in the underlying space to place their facilities. Each user avails service from its nearest facility. Service zone of a facility consists of the set of users which are closer to it than any other facility. Payoff of each player is defined by the quantity of users served by all of its facilities. The objective of each player is to maximize their respective payoff. In this paper we consider the two players Voronoi game where the underlying space is a road network modeled by a graph. In this framework we consider the problem of finding k optimal facility locations of Player 2 given any placement of m facilities by Player 1. Our main result is a dynamic programming based polynomial time algorithm for this problem on tree network. On the other hand, we show that the problem is strongly N P-complete for graphs. This proves that finding a winning strategy of P2 is N P-complete. Consequently, we design an 1 − 1 e factor approximation algorithm, where e ≈ 2.718.
The clustering problem, in its many variants, has numerous applications in operations research and computer science (e.g., in applications in bioinformatics, image processing, social network analysis, etc.). As sizes of data sets have grown rapidly, researchers have focused on designing algorithms for clustering problems in models of computation suited for large-scale computation such as MapReduce, Pregel, and streaming models. The k-machine model (Klauck et al., SODA 2015) is a simple, message-passing model for large-scale distributed graph processing. This paper considers three of the most prominent examples of clustering problems: the uncapacitated facility location problem, the p-median problem, and the pcenter problem and presents O(1)-factor approximation algorithms for these problems running inÕ(n/k) rounds in the k-machine model. These algorithms are optimal up to polylogarithmic factors because this paper also showsΩ(n/k) lower bounds for obtaining polynomial-factor approximation algorithms for these problems. These are the first results for clustering problems in the k-machine model.We assume that the metric provided as input for these clustering problems in only implicitly provided, as an edge-weighted graph and in a nutshell, our main technical contribution is to show that constantfactor approximation algorithms for all three clustering problems can be obtained by learning only a small portion of the input metric.1 problems in the recently proposed k-machine model [21], a synchronous, message-passing model for largescale distributed computation. This model cleanly abstracts essential features of systems such as Pregel [24] and Giraph (see http://giraph.apache.org/) that have been designed for large-scale graph processing 1 , allowing researchers to prove precise upper and lower bounds. One of the main features of the k-machine model is that the input, consisting of n items, is randomly partitioned across k machines. Of particular interest are settings in which n is much larger than k. Communication occurs via bandwidth-restricted communication links between every pair of machines and thus the underlying communication network is a size-k clique. For all three problems, we present constant-factor approximation algorithms that run iñ O(n/k) rounds in the k-machine model. We also show that these algorithms have optimal round complexity, to within polylogarithmic factors, by providing complementaryΩ(n/k) lower bounds for polynomial-factor approximation algorithms 2 . These are the first results on clustering problems in the k-machine model.
We consider the Minimum Dominating Set (MDS) problem on the intersection graphs of geometric objects. Even for simple and widely-used geometric objects such as rectangles, no sublogarithmic approximation is known for the problem and (perhaps surprisingly) the problem is NP-hard even when all the rectangles are "anchored" at a diagonal line with slope -1 (Pandit, CCCG 2017). In this paper, we first show that for any > 0, there exists a (2 + )-approximation algorithm for the MDS problem on "diagonal-anchored" rectangles, providing the first O(1)approximation for the problem on a non-trivial subclass of rectangles. It is not hard to see that the MDS problem on "diagonal-anchored" rectangles is the same as the MDS problem on "diagonal-anchored" L-frames: the union of a vertical and a horizontal line segment that share an endpoint. As such, we also obtain a (2 + )-approximation for the problem with "diagonalanchored" L-frames. On the other hand, we show that the problem is APX-hard in case the input L-frames intersect the diagonal, or the horizontal segments of the L-frames intersect a vertical line. However, as we show, the problem is linear-time solvable in case the L-frames intersect a vertical as well as a horizontal line. Finally, we consider the MDS problem in the so-called "edge intersection model" and obtain a number of results, answering two questions posed by Mehrabi (WAOA 2017).
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