Replica Placement on Bounded Treewidth Graphs

Aggarwal, Anshul; Chakaravarthy, Venkatesan T.; Gupta, Neelima; Sabharwal, Yogish; Sharma, Sachin; Thakral, Sonika

doi:10.1007/978-3-319-62127-2_2

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“…Another closely related problem to the CSC problem is the so-called Replica Placement problem. For the graphs of treewidth bounded by t, an O(t) approximation algorithm for this problem is presented in [1]. Finally, PTASes for the Capacitated Dominating Set, and Capacitated Vertex Cover problems on the planar graphs is presented in [6], under the assumption that the demands and capacities of the vertices are upper bounded by a constant.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, since M 1 and M 2 are disjoint, |M | = |M 1 | + |M | ≥ αN + 3(1−α)N Now, usingLemma 12, we conclude that the cost of an optimal solution to I is at least B + αN + 3(1−α)N It is easy to verify that the previous quantity is exactly equal to1 + 1−α 8(3p+1) • (B + N ), recalling that B = 3N • (4p + 1).Now, from Lemma 11, Corollary 1, and Corollary 2, we obtain the following APX-hardness result. For any constant c ≥ 1, there exists a constant c > 0 such that it is NP-hard to obtain a (1 + c , c)-approximation for the uniform capacitated version of MMCC 1.…”

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confidence: 99%

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Capacitated Covering Problems in Geometric Spaces

Bandyapadhyay

Bhowmick

Inamdar

et al. 2019

Discrete Comput Geom

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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...

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%