5 research outputs found
Approximation Algorithm for Minimum Weight Connected -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 -connected -fold dominating set
(-CDS) of a graph is a node set such that every node in
has at least neighbors in and the subgraph of
induced by is -connected. Using -CDS can tolerate the failure of
nodes. In this paper, we study Minimum Weight -CDS
problem (-MWCDS), and present an
-approximation algorithm, where is the
maximum degree of the graph and is the Harmonic number. Notice that
there is a -approximation algorithm for the -MWCDS problem,
where is the number of nodes in the graph. Though our constant in is larger than 1.35, is replaced by . Such a replacement
enables us to obtain a -approximation for the -MWCDS
problem on unit disk graphs
Approximation Algorithm for Minimum Weight -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 -connected -fold dominating set (-CDS for short). In this paper,
we present a constant approximation algorithm for the minimum weight
-CDS problem in unit disk graphs under the assumption that and
are two fixed constants with . Prior to this work, constant
approximation algorithms are known for with weight and
without weight. Our result is the first constant approximation algorithm for
the -CDS problem with general and with weight. The performance
ratio is for and for ,
where is the performance ratio for the minimum weight -fold
dominating set problem and is the performance ratio for the subset
-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 and positive integers and
, a -connected -dominating set (-CDS) is defined as a subset
of such that each node in has at least neighbors in
, and a -connected subgraph is induced by . The weighted -CDS
problem is to find a minimum weight -CDS in a given node-weighted graph.
The problem is called the unweighted -CDS problem if the objective is to
minimize the cardinality of a -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 in the unweighted -CDS problem,
and for in the weighted -CDS problem. In this paper, we
consider the case in which , and we present a simple -approximation algorithm for the unweighted -CDS problem, and a
primal-dual -approximation algorithm for the weighted
-CDS problem. Both algorithms achieve constant approximation factors
when is a fixed constant
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
Adaptive Algorithm for Finding Connected Dominating Sets in Uncertain Graphs
The problem of finding a minimum-weight connected dominating set (CDS) of a
given undirected graph has been studied actively, motivated by operations of
wireless ad hoc networks. In this paper, we formulate a new stochastic variant
of the problem. In this problem, each node in the graph has a hidden random
state, which represents whether the node is active or inactive, and we seek a
CDS of the graph that consists of the active nodes. We consider an adaptive
algorithm for this problem, which repeat choosing nodes and observing the
states of the nodes around the chosen nodes until a CDS is found. Our
algorithms have a theoretical performance guarantee that the sum of the weights
of the nodes chosen by the algorithm is at most
times that of any adaptive algorithm in expectation, where is an
approximation factor for the node-weighted polymatroid Steiner tree problem and
is the minimum probability of possible scenarios on the node states.Comment: This is the accepted version of a paper to be published by IEEE/ACM
Transactions on Networkin