Set In Unit Disk Graphs Research Articles

Parallel algorithms for minimum general partial dominating set and maximum budgeted dominating set in unit disk graph

In a minimum general partial dominating set problem (MinGPDS), given a graph G=(V,E), a profit function p:V→R+ and a threshold K, the goal is to find a minimum subset of vertices D⊆V such that the total profit of those vertices dominated by D is at least K (a vertex is dominated by D if it is either in D or has at least one neighbor in D). In a maximum general budgeted dominating set problem (MaxGBDS), given a budget B, the goal is to find a vertex set D with at most B vertices such that the total profit of those vertices dominated by D is as large as possible. We present the first parallel algorithms for MinGPDS and MaxGBDS in unit disk graphs. They both run in O(log⁡n) rounds on O(n) machines, and achieve constant approximation ratios.

Distributed construction of connected dominating set in unit disk graphs

Let G=(V,E) be a unit disk graph corresponding to a given set P of n points in R2. We propose a distributed approximation algorithm to find a minimum connected dominating set of G. The maintenance of the connected dominating set constructed is fully localized. Our algorithm produces a connected dominating set of size (104opt+52), where opt is the size of a minimum connected dominating set. The time and message complexities of our algorithm are O(Δ) and O(n) respectively, where Δ and n are the maximum node degree and number of nodes in G respectively. Our distributed approximation algorithm outperforms existing algorithms in terms of its time and message complexities.We also propose a scheduling scheme that obtains O(Δ) conflict-free time slots to deal with interference.

A New Constant Factor Approximation to Construct Highly Fault-Tolerant Connected Dominating Set in Unit Disk Graph

This paper proposes a new polynomial time constant factor approximation algorithm for a more-a-decade-long open NP-hard problem, the minimum four-connected $m$ -dominating set problem in unit disk graph (UDG) with any positive integer $m \geq 1$ for the first time in the literature. We observe that it is difficult to modify the existing constant factor approximation algorithm for the minimum three-connected $m$ -dominating set problem to solve the minimum four-connected $m$ -dominating set problem in UDG due to the structural limitation of Tutte decomposition, which is the main graph theory tool used by Wang et al. to design their algorithm. To resolve this issue, we first reinvent a new constant factor approximation algorithm for the minimum three-connected $m$ -dominating set problem in UDG and later use this algorithm to design a new constant factor approximation algorithm for the minimum four-connected $m$ -dominating set problem in UDG.

A new bound on maximum independent set and minimum connected dominating set in unit disk graphs

It is a conjecture that in any unit disk graph $$G$$G, $$\alpha (G) \le 3 \cdot \gamma _c(G) + 3$$?(G)≤3·?c(G)+3 where $$\alpha (G)$$?(G) is the size of the maximum independent set in $$G$$G and $$\gamma _c(G)$$?c(G) is the size of minimum connected dominating set in $$G$$G. In this paper, we show that in any unit disk graph $$G$$G, $$\alpha (G) \le 3.399 \cdot \gamma _c(G) + 4.874$$?(G)≤3.399·?c(G)+4.874. Currently, this is the best-known bound.

Node-weighted Steiner tree approximation in unit disk graphs

Given a graph G=(V,E) with node weight w:V→R+ and a subset S⊆V, find a minimum total weight tree interconnecting all nodes in S. This is the node-weighted Steiner tree problem which will be studied in this paper. In general, this problem is NP-hard and cannot be approximated by a polynomial time algorithm with performance ratio aln n for any 0<a<1 unless NP⊆DTIME(nO(log n)), where n is the number of nodes in s. In this paper, we are the first to show that even though for unit disk graphs, the problem is still NP-hard and it has a polynomial time constant approximation. We present a 2.5ρ-approximation where ρ is the best known performance ratio for polynomial time approximation of classical Steiner minimum tree problem in graphs. As a corollary, we obtain that there is a polynomial time (9.875+e)-approximation algorithm for minimum weight connected dominating set in unit disk graphs, and also there is a polynomial time (4.875+e)-approximation algorithm for minimum weight connected vertex cover in unit disk graphs.

A PTAS for minimum connected dominating set in 3-dimensional Wireless sensor networks

When homogeneous sensors are deployed into a three-dimensional space instead of a plane, the mathematical model for the sensor network is a unit ball graph instead of a unit disk graph. It is known that for the minimum connected dominating set in unit disk graph, there is a polynomial time approximation scheme (PTAS). However, that construction cannot be extended to obtain a PTAS for unit ball graph. In this paper, we will introduce a new construction, which gives not only a PTAS for the minimum connected dominating set in unit ball graph, but also improves running time of PTAS for unit disk graph.

A [formula omitted]-approximation algorithm for minimum weighted dominating set in unit disk graph

We study the minimum weight dominating set problem in weighted unit disk graph, and give a polynomial time algorithm with approximation ratio 5 + ϵ , improving the previous best result of 6 + ϵ in [Yaochun Huang, Xiaofeng Gao, Zhao Zhang, Weili Wu, A better constant-factor approximation for weighted dominating set in unit disk graph, J. Comb. Optim. (ISSN: 1382-6905) (2008) 1573–2886. (Print) (Online)]. Combining the common technique used in the above mentioned reference, we can compute a minimum weight connected dominating set with approximation ratio 9 + ϵ , beating the previous best result of 10 + ϵ in the same work.

A better constant-factor approximation for weighted dominating set in unit disk graph

This paper presents a (10+e)-approximation algorithm to compute minimum-weight connected dominating set (MWCDS) in unit disk graph. MWCDS is to select a vertex subset with minimum weight for a given unit disk graph, such that each vertex of the graph is contained in this subset or has a neighbor in this subset. Besides, the subgraph induced by this vertex subset is connected. Our algorithm is composed of two phases: the first phase computes a dominating set, which has approximation ratio 6+e (e is an arbitrary positive number), while the second phase connects the dominating sets computed in the first phase, which has approximation ratio 4.