Linear-Time Approximation Algorithms for Unit Disk Graphs

Fonseca, Guilherme D. da; Sá, Vinícius Gusmão Pereira de; Figueiredo, Celina M. H. de

doi:10.1007/978-3-319-18263-6_12

Cited by 3 publications

(7 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We show that the problem admits a simple constant-factor approximation algorithm for any graphs in which the minimum-weight dominating set problem admits a constant-factor approximation. This class includes unit disk graphs [24,6]. Below, we let β denote the approximation factor for the minimum-weight dominating set problem.…”

Section: Algorithm For the Connected Dominating Set Problem With T = Vmentioning

confidence: 99%

Computing a Tree Having a Small Vertex Cover

Fukunaga

Maehara

2016

Combinatorial Optimization and Applications

View full text Add to dashboard Cite

We consider a new Steiner tree problem, called vertex-cover-weighted Steiner tree problem. This problem defines the weight of a Steiner tree as the minimum weight of vertex covers in the tree, and seeks a minimum-weight Steiner tree in a given vertex-weighted undirected graph. Since it is included by the Steiner tree activation problem, the problem admits an O(log n)approximation algorithm in general graphs with n vertices. This approximation factor is tight up to a constant because it is NP-hard to achieve an o(log n)-approximation for the vertex-coverweighted Steiner tree problem in general graphs even if the given vertex weights are uniform and a spanning tree is required instead of a Steiner tree. In this paper, we present constantfactor approximation algorithms for the problem in unit disk graphs and in graphs excluding a fixed minor. For the latter graph class, our algorithm can be also applied for the Steiner tree activation problem.consists of at least n/2 vertices. Our problem is to compute a tree that minimizes the number (or, more generally, the weight) of devices required to monitor all of the traffic.More formally, our problem is defined as follows. Let G = (V, E) be an undirected graph associated with nonnegative vertex weights w ∈ R V + . Throughout this paper, we will denote |V | by n. Let T ⊆ V be a set of vertices called terminals. The problem seeks a pair comprising a tree F and a vertex set U ⊆ V (F ) such that (i) F is a Steiner tree with regard to the terminal set T (i.e., T ⊆ V (F )), and (ii) U is a vertex cover of F (i.e., each edge in F is incident to at least one vertex in U ). The objective is to find such a pair (F, U ) that minimizes the weight w(U ) := v∈U w(v) of the vertex cover. We call this the vertex-cover-weighted (VC-weighted) Steiner tree problem. We call the special case in which V = T the vertex-cover-weighted (VC-weighted) spanning tree problem. The aim of this paper is to investigate these fundamental problems.Besides the motivation from the communication networks, there is another reason for the importance of the VC-weighted Steiner tree problem. The VC-weighted Steiner tree problem is a special case of the Steiner tree activation problem, which was formulated by Panigrahi [18]. In the Steiner tree activation problem, we are given a set W of nonnegative real numbers, and each edge uv in the graph is associated with an activation function f uv : W × W → { , ⊥}, where indicates that an edge uv is activated, and ⊥ indicates that it is not. The activation function is assumed to be monotone (i.e., if f uv (i, j) = , i ≤ i , and j ≤ j , then f uv (i , j ) = ). A solution for the problem is defined as a |V |-dimensional vector x ∈ W V . We say that a solution x activates an edge uv if f uv (x(u), x(v)) = . The problem seeks a solution x that minimizes x(V ) := v∈V x(v) subject to the constraint that the edges activated by x include a Steiner tree. To see that the Steiner tree activation problem includes the VC-weighted Steiner tree problem, define W as {0} ∪ {w(v) : v ∈ V }, a...

show abstract

Section: Algorithm For the Connected Dominating Set Problem With T = Vmentioning

confidence: 99%

Computing a Tree Having a Small Vertex Cover

Fukunaga

Maehara

2016

Combinatorial Optimization and Applications

View full text Add to dashboard Cite

show abstract

“…There are polynomialtime approximation schemes (PTASs) for several optimization problems on unit disk graphs [10,14,17,18,22,23], including maximum (weight) independent set. However, the high complexities of the PTASs motivated the recent study of faster constantapproximation algorithms, notably for minimum dominating set [5,12,13,15] and maximum (weight) independent set [13,14].…”

Section: Introductionmentioning

confidence: 99%

“…A different strip decomposition yields an O(n 3 )-time 2-approximation algorithm for the weighted version [14]. The use of coresets give an O(n)-time (4 + ε)-approximation algorithm for the weighted version [13].…”

Section: Introductionmentioning

confidence: 99%

Efficient independent set approximation in unit disk graphs

Das

Fonseca

Jallu

2020

Discrete Applied Mathematics

Self Cite

View full text Add to dashboard Cite

We consider the maximum (weight) independent set problem in unit disk graphs. The high complexity of the existing polynomial-time approximation schemes motivated the development of faster constant-approximation algorithms. In this article, we present a 2.16-approximation algorithm that runs in O(n log 2 n) time and a 2approximation algorithm that runs in O(n 2 log n) time for the unweighted version of the problem. In the weighted version, the running times increase by an O(log n) factor. Our algorithms are based on a classic strip decomposition, but we improve over previous algorithms by efficiently using geometric data structures. We also propose a PTAS for the unweighted version.

show abstract

“…Recently, there has been renewed interest in making shifting strategy algorithms practical, as the PTASes obtained by the shifting strategy are too slow to be applied in practice. In recent work by Fonseca et al [6], the technique of shifting coresets is introduced, giving linear time approximations for various problems on unit disc graphs. They observe that within-window algorithms used in the shifting technique often iterate over m poly( ) candidate solutions, where m is the number of points inside the window and is the size of the window.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we revisit the shifting strategy in the context of the streaming model and develop a streaming-friendly shifting strategy. When combined with the shifting coreset method introduced by Fonseca et al [6], we obtain streaming algorithms for various graph properties of unit disc graphs. As a further application, we present novel approximation algorithms and lower bounds for the unit disc cover (UDC) problem in the streaming model, for which currently no algorithms are known.…”

mentioning

confidence: 99%

Approximation Schemes for Covering and Packing in the Streaming Model

Liaw,

Liu,

Reiss

2017

Preprint

View full text Add to dashboard Cite

The shifting strategy, introduced by Hochbaum and Maass [10], and independently by Baker [1], is a unified framework for devising polynomial approximation schemes to NP-Hard problems. This strategy has been used to great success within the computational geometry community in a plethora of different applications; most notably covering, packing, and clustering problems [2,5,7,8,9]. In this paper, we revisit the shifting strategy in the context of the streaming model and develop a streaming-friendly shifting strategy. When combined with the shifting coreset method introduced by Fonseca et al. [6], we obtain streaming algorithms for various graph properties of unit disc graphs. As a further application, we present novel approximation algorithms and lower bounds for the unit disc cover (UDC) problem in the streaming model, for which currently no algorithms are known.

show abstract

Linear-Time Approximation Algorithms for Unit Disk Graphs

Cited by 3 publications

References 10 publications

Computing a Tree Having a Small Vertex Cover

Computing a Tree Having a Small Vertex Cover

Efficient independent set approximation in unit disk graphs

Approximation Schemes for Covering and Packing in the Streaming Model

Contact Info

Product

Resources

About