Approximation Algorithm for Minimum Weight (k,m)(k,m)-CDS Problem in Unit Disk Graph

In a wireless sensor network, the virtual backbone plays an important role. Due to accidental damage or energy depletion, it is desirable that the virtual backbone is fault-tolerant. A fault-tolerant virtual backbone can be modeled as a $k$-connected $m$-fold dominating set ($(k,m)$-CDS for short). In this paper, we present a constant approximation algorithm for the minimum weight $(k,m)$-CDS problem in unit disk graphs under the assumption that $k$ and $m$ are two fixed constants with $m\geq k$. Prior to this work, constant approximation algorithms are known for $k=1$ with weight and $2\leq k\leq 3$ without weight. Our result is the first constant approximation algorithm for the $(k,m)$-CDS problem with general $k,m$ and with weight. The performance ratio is $(\alpha+2.5k\rho)$ for $k\geq 3$ and $(\alpha+2.5\rho)$ for $k=2$, where $\alpha$ is the performance ratio for the minimum weight $m$-fold dominating set problem and $\rho$ is the performance ratio for the subset $k$-connected subgraph problem (both problems are known to have constant performance ratios.

Constant-approximation algorithms for highly connected multi-dominating sets in unit disk graphs

Given an undirected graph on a node set $V$ and positive integers $k$ and $m$, a $k$-connected $m$-dominating set ($(k,m)$-CDS) is defined as a subset $S$ of $V$ such that each node in $V \setminus S$ has at least $m$ neighbors in $S$, and a $k$-connected subgraph is induced by $S$. The weighted $(k,m)$-CDS problem is to find a minimum weight $(k,m)$-CDS in a given node-weighted graph. The problem is called the unweighted $(k,m)$-CDS problem if the objective is to minimize the cardinality of a $(k,m)$-CDS. These problems have been actively studied for unit disk graphs, motivated by the application of constructing a virtual backbone in a wireless ad hoc network. However, constant-approximation algorithms are known only for $k \leq 3$ in the unweighted $(k,m)$-CDS problem, and for $(k,m)=(1,1)$ in the weighted $(k,m)$-CDS problem. In this paper, we consider the case in which $m \geq k$, and we present a simple $O(5^k k!)$-approximation algorithm for the unweighted $(k,m)$-CDS problem, and a primal-dual $O(k^2 \log k)$-approximation algorithm for the weighted $(k,m)$-CDS problem. Both algorithms achieve constant approximation factors when $k$ is a fixed constant

Approximation Algorithm for Minimum Weight Connected mm-Fold Dominating Set

Using connected dominating set (CDS) to serve as a virtual backbone in a wireless networks can save energy and reduce interference. Since nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone has some fault-tolerance. A $k$-connected $m$-fold dominating set ($(k,m)$-CDS) of a graph $G$ is a node set $D$ such that every node in $V\setminus D$ has at least $m$ neighbors in $D$ and the subgraph of $G$ induced by $D$ is $k$-connected. Using $(k,m)$-CDS can tolerate the failure of $\min\{k-1,m-1\}$ nodes. In this paper, we study Minimum Weight $(1,m)$-CDS problem ($(1,m)$-MWCDS), and present an $(H(\delta+m)+2H(\delta-1))$-approximation algorithm, where $\delta$ is the maximum degree of the graph and $H(\cdot)$ is the Harmonic number. Notice that there is a $1.35\ln n$-approximation algorithm for the $(1,1)$-MWCDS problem, where $n$ is the number of nodes in the graph. Though our constant in $O(\ln \cdot)$ is larger than 1.35, $n$ is replaced by $\delta$. Such a replacement enables us to obtain a $(6.67+\varepsilon)$-approximation for the $(1,m)$-MWCDS problem on unit disk graphs

Minimum Dominating Set for a Point Set in \IR^2

In this article, we consider the problem of computing minimum dominating set for a given set $S$ of $n$ points in \IR^2. Here the objective is to find a minimum cardinality subset $S'$ of $S$ such that the union of the unit radius disks centered at the points in $S'$ covers all the points in $S$. We first propose a simple 4-factor and 3-factor approximation algorithms in $O(n^6 \log n)$ and $O(n^{11} \log n)$ time respectively improving time complexities by a factor of $O(n^2)$ and $O(n^4)$ respectively over the best known result available in the literature [M. De, G.K. Das, P. Carmi and S.C. Nandy, {\it Approximation algorithms for a variant of discrete piercing set problem for unit disk}, Int. J. of Comp. Geom. and Appl., to appear]. Finally, we propose a very important shifting lemma, which is of independent interest and using this lemma we propose a $\frac{5}{2}$-factor approximation algorithm and a PTAS for the minimum dominating set problem.Comment: 14 pages, 8 figure

Approximation Algorithms for the Connected Sensor Cover Problem

We study the minimum connected sensor cover problem (MIN-CSC) and the budgeted connected sensor cover (Budgeted-CSC) problem, both motivated by important applications (e.g., reduce the communication cost among sensors) in wireless sensor networks. In both problems, we are given a set of sensors and a set of target points in the Euclidean plane. In MIN-CSC, our goal is to find a set of sensors of minimum cardinality, such that all target points are covered, and all sensors can communicate with each other (i.e., the communication graph is connected). We obtain a constant factor approximation algorithm, assuming that the ratio between the sensor radius and communication radius is bounded. In Budgeted-CSC problem, our goal is to choose a set of $B$ sensors, such that the number of targets covered by the chosen sensors is maximized and the communication graph is connected. We also obtain a constant approximation under the same assumption

Efficient Construction of Dominating Set in Wireless Networks

Considering a communication topology of a wireless network modeled by a graph where an edge exists between two nodes if they are within each other's communication range. A subset $U$ of nodes is a dominating set if each node is either in $U$ or adjacent to some node in $U$. Assume each node has a disparate communication range and is associated with a positive weight, we present a randomized algorithm to find a min-weight dominating set. Considering any orientation of the graph where an arc $\overrightarrow{uv}$ exists if the node $v$ lies in $u$'s communication range. A subset $U$ of nodes is a strongly dominating set if every node except $U$ has both in-neighbor(s) and out-neighbor(s) in $U$. We present a polynomial-time algorithm to find a strongly dominating set of size at most $(2+\epsilon)$ times of the optimum. We also investigate another related problem called $K$-Coverage. Given are a set ${\cal D}$ of disks with positive weight and a set ${\cal P}$ of nodes. Assume all input nodes lie below a horizontal line $l$ and all input disks lie above this line $l$ in the plane. The objective is to find a min-weight subset ${\cal D}'\subseteq {\cal D}$ of disks such that each node is covered at least $K$ disks in ${\cal D}'$. We propose a novel two-approximation algorithm for this problem.Comment: 8 pages, 4 figures, 2 table

A PTAS for the Weighted Unit Disk Cover Problem

We are given a set of weighted unit disks and a set of points in Euclidean plane. The minimum weight unit disk cover (\UDC) problem asks for a subset of disks of minimum total weight that covers all given points. \UDC\ is one of the geometric set cover problems, which have been studied extensively for the past two decades (for many different geometric range spaces, such as (unit) disks, halfspaces, rectangles, triangles). It is known that the unweighted \UDC\ problem is NP-hard and admits a polynomial-time approximation scheme (PTAS). For the weighted \UDC\ problem, several constant approximations have been developed. However, whether the problem admits a PTAS has been an open question. In this paper, we answer this question affirmatively by presenting the first PTAS for \UDC. Our result implies the first PTAS for the minimum weight dominating set problem in unit disk graphs. Combining with existing ideas, our result can also be used to obtain the first PTAS for the maxmimum lifetime coverage problem and an improved constant approximation ratio for the connected dominating set problem in unit disk graphs.Comment: We fixed several typos in this version. 37 pages. 15 figure

On Pseudo-disk Hypergraphs

Let $F$ be a family of pseudo-disks in the plane, and $P$ be a finite subset of $F$. Consider the hypergraph $H(P,F)$ whose vertices are the pseudo-disks in $P$ and the edges are all subsets of $P$ of the form $\{D \in P \mid D \cap S \neq \emptyset\}$, where $S$ is a pseudo-disk in $F$. We give an upper bound of $O(nk^3)$ for the number of edges in $H(P,F)$ of cardinality at most $k$. This generalizes a result of Buzaglo et al. (2013). As an application of our bound, we obtain an algorithm that computes a constant-factor approximation to the smallest _weighted_ dominating set in a collection of pseudo-disks in the plane, in expected polynomial time.Comment: Submitted for publicatio

A Framework for ETH-Tight Algorithms and Lower Bounds in Geometric Intersection Graphs

We give an algorithmic and lower-bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to a wide range of geometric intersection graphs (intersections of similarly sized fat objects), yielding algorithms with running time $2^{O(n^{1-1/d})}$ for any fixed dimension $d \geq 2$ for many well known graph problems, including Independent Set, $r$-Dominating Set for constant $r$, and Steiner Tree. For most problems, we get improved running times compared to prior work; in some cases, we give the first known subexponential algorithm in geometric intersection graphs. Additionally, most of the obtained algorithms work on the graph itself, i.e., do not require any geometric information. Our algorithmic framework is based on a weighted separator theorem and various treewidth techniques. The lower bound framework is based on a constructive embedding of graphs into d-dimensional grids, and it allows us to derive matching $2^{\Omega(n^{1-1/d})}$ lower bounds under the Exponential Time Hypothesis even in the much more restricted class of $d$-dimensional induced grid graphs.Comment: 37 pages, full version of STOC 2018 paper; v2 updates the title and fixes some typo

Approximation Algorithms for Dominating Set in Disk Graphs

We consider the problem of finding a lowest cost dominating set in a given disk graph containing $n$ disks. The problem has been extensively studied on subclasses of disk graphs, yet the best known approximation for disk graphs has remained $O(\log n)$ -- a bound that is asymptotically no better than the general case. We improve the status quo in two ways: for the unweighted case, we show how to obtain a PTAS using the framework recently proposed (independently)by Mustafa and Ray [SoCG 09] and by Chan and Har-Peled [SoCG 09]; for the weighted case where each input disk has an associated rational weight with the objective of finding a minimum cost dominating set, we give a randomized algorithm that obtains a dominating set whose weight is within a factor $2^{O(\log^* n)}$ of a minimum cost solution, with high probability -- the technique follows the framework proposed recently by Varadarajan [STOC 10].Comment: 12 pages, 4 figure