arXiv subjects

Jiao Zhou

Publications and source records attributed to Jiao Zhou.

4 recordsLinked to original sources

arXiv2023

A New Approximation Algorithm for Minimum-Weight $(1,m)$--Connected Dominating Set

Consider a graph with nonnegative node weight. A vertex subset is called a CDS (connected dominating set) if every other node has at least one neighbor in the subset and the subset induces a connected subgraph. Furthermore, if every other node has at least $m$ neighbors in the subset, then the node subset is called a $(1,m)$CDS. The minimum-weight $(1,m)$CDS problem aims at finding a $(1,m)$CDS with minimum total node weight. In this paper, we present a new polynomial-time approximation algorithm for this problem with approximation ratio $2H(δ_{\max}+m-1)$, where $δ_{\max}$ is the maximum degree of the given graph and $H(\cdot)$ is the Harmonic function, i.e., $H(k)=\sum_{i=1}^k \frac{1}{i}$.

cs.DS↗

arXiv2017

Approximation Algorithm for Fault-Tolerant Virtual Backbone in Wireless Sensor Networks

To save energy and alleviate interferences in a wireless sensor network, the usage of virtual backbone was proposed. Because of accidental damages or energy depletion, it is desirable to construct a fault tolerant virtual backbone, which can be modeled as a $k$-connected $m$-fold dominating set (abbreviated as $(k,m)$-CDS) in a graph. A node set $C\subseteq V(G)$ is a $(k,m)$-CDS of graph $G$ if every node in $V(G)\backslash C$ is adjacent with at least $m$ nodes in $C$ and the subgraph of $G$ induced by $C$ is $k$-connected. In this paper, we present an approximation algorithm for the minimum $(3,m)$-CDS problem with $m\geq3$. The performance ratio is at most $γ$, where $γ=α+8+2\ln(2α-6)$ for $α\geq4$ and $γ=3α+2\ln2$ for $α<4$, and $α$ is the performance ratio for the minimum $(2,m)$-CDS problem. Using currently best known value of $α$, the performance ratio is $\lnδ+o(\lnδ)$, where $δ$ is the maximum degree of the graph, which is asymptotically best possible in view of the non-approximability of the problem. This is the first performance-guaranteed algorithm for the minimum $(3,m)$-CDS problem on a general graph. Furthermore, applying our algorithm on a unit disk graph which models a homogeneous wireless sensor network, the performance ratio is less than 27, improving previous ratio 62.3 by a large amount for the $(3,m)$-CDS problem on a unit disk graph.

cs.DM↗

arXiv2017

Approximation Algorithm for Minimum Weight Connected $m$-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 $(H(δ+m)+2H(δ-1))$-approximation algorithm, where $δ$ 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 $δ$. Such a replacement enables us to obtain a $(6.67+\varepsilon)$-approximation for the $(1,m)$-MWCDS problem on unit disk graphs.

cs.DM↗

arXiv2016

Performance Guaranteed Approximation Algorithm for Minimum $k$-Connected $m$-Fold Dominating Set

To achieve an efficient routing in a wireless sensor network, connected dominating set (CDS) is used as virtual backbone. A fault-tolerant virtual backbone can be modeled as a $(k,m)$-CDS. For a connected graph $G=(V,E)$ and two fixed integers $k$ and $m$, a node set $C\subseteq V$ is a $(k,m)$-CDS of $G$ if every node in $V\setminus C$ has at least $m$ neighbors in $C$, and the subgraph of $G$ induced by $C$ is $k$-connected. Previous to this work, approximation algorithms with guaranteed performance ratio in a general graph were know only for $k\leq 3$. This paper makes a significant progress by presenting a $(2k-1)α_0$ approximation algorithm for general $k$ and $m$ with $m\geq k$, where $α_0$ is the performance ratio for the minimum CDS problem. Using currently best known ratio for $α_0$, our algorithm has performance ratio $O(\lnΔ)$, where $Δ$ is the maximum degree of the graph.

cs.DM↗