Capacitated Domination Problems on Planar Graphs

Becker, Amariah

doi:10.1007/978-3-319-89441-6_1

Cited by 5 publications

(5 citation statements)

References 23 publications

(15 reference statements)

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“…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%

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%

Capacitated Covering Problems in Geometric Spaces

Bandyapadhyay

Bhowmick

Inamdar

et al. 2019

Discrete Comput Geom

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“…Liedloff et al [52] presented a solution for the CapDS problem in O * (1.8463 n ) time by benefiting from dynamic programming over subsets. Later on, Becker [53] suggested a polynomial-time approximation method for CapDS, particularly for planar and unweighted graphs, where the maximum capacity and maximum demand are both bounded. Li et al [54], on the other hand, proposed a local search algorithm to solve CapDS.…”

Section: Related Workmentioning

confidence: 99%

ACapDS: An Energy-Efficient and Fault-Tolerant Distributed Capacitated Dominating Set Algorithm for Industrial IoT

Arapoglu,

Cabuk,

Dagdeviren

et al. 2024

IEEE Access

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Design and analysis of energy-efficient and fault-tolerant dominating set (DS) algorithms are vital tasks for Industrial Internet of Things (IIoT) scenarios as operational efficiency is a key objective in most industries. Such algorithms must maintain topology control, clustering, routing, and data aggregation. An IIoT deployment is said to pose the self-stabilization property if it can always recover to a steady state in a bounded time without any interruption whenever it is started at an unstable state. Self-stabilization is a fruitful principle for building fault-tolerant IIoT deployments. This paper proposes a novel distributed faulttolerant capacitated DS (capDS) algorithm for IIoT systems. The algorithm is the first self-stabilizing capDS approach to the best of our knowledge. Proofs concerning the asynchronous behaviors of the algorithm, as well as the self-stabilization feature with regard to convergence and closure properties, are provided. Besides, our theoretical analysis showed that the algorithm's approximation ratio is 6 for IIoT setups implemented as unit disk graphs. Measurements made using our testbed of 40 IRIS motes and the extensive TOSSIM simulations revealed that the proposed algorithm is up to 55% better in terms of coefficient of variation, requires up to 61% fewer moves, causes up to 13% less data traffic, and consumes up to 14% less energy when compared to a randomized approach and a minimum ID priority-based approach.INDEX TERMS Industrial Internet of Things, IIoT, minimum capacitated dominating set, distributed algorithms, self-stabilizing algorithms, fault tolerance.

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“…Liedloff et al [11] further improved the time complexity of the algorithm of [10] to O(1.8463 |V | ) by utilizing dynamic programming over subsets. Becker [12] developed a polynomial-time approximation scheme for solving the problem of capacitated domination on planar graphs under bounded maximum capacity and bounded maximum demand.…”

Section: Related Workmentioning

confidence: 99%

“…A number of approximation schemes [7][8][9][10][11][12], heuristics [13,14], metaheuristics [15] and metaheuristics [16] have been proposed in the literature for solving CAPMDS problem and its variants. We have discussed these approaches in the next section.…”

Section: Introductionmentioning

confidence: 99%