Two algorithms for determining a minimum independent dominating set

The problem of determining a minimum independent dominating set is fundamental to both the theory and applications of graphs. Computationally it belongs to the class of hard combinatorial optimization problems known as NP-hard. In this paper, we develop a backtracking algorithm and a dynamic programming algorithm to determine a minimum independent dominating set. Computational experience with the backtracking algorithm on more than 1000 randomly generated graphs ranging from 100 to 200 vertices and from 10% to 60% densities has shown that the algorithm is effective.     

Approximating minimum coloring and maximum independent set in dotted interval graphs

Dotted interval graphs are introduced by Aumann et al. as a generalization of interval graphs. We study two optimization problems in dotted interval graphs that find application in high-throughput genotyping. We present improved approximations for minimum coloring and the first approximation for maximum independent set in dotted interval graphs.     

Constructing minimum extended weakly-connected dominating sets for clustering in ad hoc networks

Motivated by cooperative communication in ad hoc networks, Wu et al. proposed extended dominating set (EDS) where each node in an ad hoc network is covered by either a dominating neighbor or several 2-hop dominating neighbors, and defined two types of dominating sets: extended strongly connected dominating set (ECDS) and extended weakly connected dominating set (EWCDS), according to the success of a broadcast process. An EWCDS is an effective method for clustering. In this paper, we extend the dominative capabilities of nodes such that each forward node dominates not only itself and its regular neighbors fully, but also its quasi-neighbors partly. Based on this extension, three novel algorithms to find EWCDSs in ad hoc networks are proposed. The correctness and performance of our algorithms are confirmed through theoretical analysis and comprehensive simulations.     

Greed is good: Approximating independent sets in sparse and bounded-degree graphs

Theminimum-degree greedy algorithm, or Greedy for short, is a simple and well-studied method for finding independent sets in graphs. We show that it achieves a performance ratio of (Δ+2)/3 for approximating independent sets in graphs with degree bounded by Δ. The analysis yields a precise characterization of the size of the independent sets found by the algorithm as a function of the independence number, as well as a generalization of Turán’s bound. We also analyze the algorithm when run in combination with a known preprocessing technique, and obtain an improved performance ratio on graphs with average degree , improving on the previous best of Hochbaum. Finally, we present an efficient parallel and distributed algorithm attaining the performance guarantees of Greedy. Gordon Gekko [29]. A preliminary version of this paper appeared at the 26th ACM Symposium on Theory of Computing, 1994. This work was done while both authors were at the Japan Advanced Institute of Science and Technology, Hokuriku.     

On positive influence dominating sets in social networks

In this paper, we investigate the positive influence dominating set (PIDS) which has applications in social networks. We prove that PIDS is APX-hard and propose a greedy algorithm with an approximation ratio of H(δ) where H is the harmonic function and δ is the maximum vertex degree of the graph representing a social network.     

New approximations for minimum-weighted dominating sets and minimum-weighted connected dominating sets on unit disk graphs

Given a node-weighted graph, the minimum-weighted dominating set (MWDS) problem is to find a minimum-weighted vertex subset such that, for any vertex, it is contained in this subset or it has a neighbor contained in this set. And the minimum-weighted connected dominating set (MWCDS) problem is to find a MWDS such that the graph induced by this subset is connected. In this paper, we study these two problems on a unit disk graph. A (4 +ε)-approximation algorithm for an MWDS based on a dynamic programming algorithm for a Min-Weight Chromatic Disk Cover is presented. Meanwhile, we also propose a (1 +ε)-approximation algorithm for the connecting part by showing a polynomial-time approximation scheme for a Node-Weighted Steiner Tree problem when the given terminal set is c-local and thus obtain a (5 +ε)-approximation algorithm for an MWCDS.     

Two algorithms for minimum 2-connected r-hop dominating set

For a graph G=(V,E), a subset D⊆V is an r-hop dominating set if every vertex u∈V−D is at most r-hops away from D. It is a 2-connected r-hop dominating set if the subgraph of G induced by D is 2-connected. In this paper, we present two approximation algorithms to compute minimum 2-connected r-hop dominating set. The first one is a greedy algorithm using ear decomposition of 2-connected graphs. This algorithm is applicable to any 2-connected general graph. The second one is a three-phase algorithm which is only applicable to unit disk graphs. For both algorithms, performance ratios are given.     

Approximating minimum cocolorings

A cocoloring of a graph G is a partition of the vertex set of G such that each set of the partition is either a clique or an independent set in G. Some special cases of the minimum cocoloring problem are of particular interest.We provide polynomial-time algorithms to approximate a minimum cocoloring on graphs, partially ordered sets and sequences. In particular, we obtain an efficient algorithm to approximate within a factor of 1.71 a minimum partition of a partially ordered set into chains and antichains, and a minimum partition of a sequence into increasing and decreasing subsequences.     

Clustering wireless ad hoc networks with weakly connected dominating set

The increasing popular personal communications and mobile computing require a wireless network infrastructure that supports self-configuration and self-management. Efficient clustering protocol for constructing virtual backbone is becoming one of the most important issues in wireless ad hoc networks. The weakly connected dominating set   (WCDS) is very suitable for cluster formation. As finding the minimum WCDS in an arbitrary graph is a NP-Hard problem, we propose an area-based distributed algorithm for WCDS construction in wireless ad hoc networks with time and message complexity O(n)$O(n)$. This Area algorithm is divided into three phases: area partition, WCDS construction for each area and adjustment along the area borders. We confirm the effectiveness of our algorithm through analysis and comprehensive simulation study. The number of nodes in the WCDS constructed by this Area algorithm is up to around 50% less than that constructed by the previous well-known algorithm.     

一种参考能量的最小连通支配集近似算法

在无线传感器网络中,能量效率问题至关重要,构造精简的虚拟骨干网可以节约有限资源,这等同于在图论中求解最小连通支配集(MCDS)问题.由此,提出一种构造MCDS的启发式算法.首先根据均值公式为顶点建立次序表,其次构造极大独立集(MIS),再次连接MIS节点,最后优化.仿真实验表明:该算法能够在短时间内找到规模较小的连通支配集(CDS),并且有效地均衡了各节点能量,延长了网络生命周期.     

连通支配集一种集中式近似算法

简单无向图的最小连通支配集问题是NP完全问题,目前还没有成熟解法。提出了一种用有序表构建独立集求解连通支配集的算法,算法从图中度最大的顶点开始将顶点加入到有序表中,并在加入过程中构建独立集,同时加入其他节点连接独立集使其成为连通集。当图中所有节点处理完成,有序表中标记为独立集的节点和连接节点就形成了一个连通支配集。实验表明算法生成的支配集较小,运行时间复杂度比较低。     

连通支配集一种集中式近似算法

简单无向图的最小连通支配集问题是NP完全问题,目前还没有成熟解法。提出了一种用有序袁构建独立集求解连通支配集的算法,算法从图中度最大的顶点开始将顶点加入到有序表中,并在加入过程中构建独立集,同时加入其他节点连接独立集使其成为连通集当图中所有节点处理完成,有序表中标记为独立集的节点和连接节点就形成了一个连通支配集。实验表明算法生成的支配集较小,运行时间复杂度比较低。     

A strengthened analysis of a local algorithm for the minimum dominating set problem in planar graphs

In recent years growing interest in local distributed algorithms has widely been observed. This results from their high resistance to errors and damage, as well as from their good performance, which is independent of the size of the network. A local deterministic distributed algorithm finding an approximation of a Minimum Dominating Set in planar graphs has been presented by Lenzen et al., and they proved that the algorithm returns a 130-approximation of the Minimum Dominating Set. In this article we will show that the algorithm is two times more effective than was previously assumed, and we prove that the algorithm by Lenzen et al. outputs a 52-approximation to a Minimum Dominating Set. Therefore the gap between the lower bound and the approximation ratio of the best yet local deterministic distributed algorithm is reduced by half.     

Quorum and connected dominating sets based location service in wireless ad hoc, sensor and actuator networks

Location service provides position of mobile destination to source node to enable geo-routing. In existing quorum-based location service protocols, destination node registers its location along a ‘column’ while source node makes a query along a ‘row’. Grid and quorum-based location service is based on division of network into square grids, and selecting ‘leader’ location server node in each grid. Location updates, leader reelection and information transfer are performed whenever destination and leader nodes are moving to a different grid. We propose here to apply connected dominating sets (DS) as an alternative to grids. We also improved basic quorum, and applied on DS-quorum (DS based quorum) better criterion for triggering local information exchanges and global location updates, by meeting two criteria: certain distance movement and certain number of observed link changes with (DS) nodes. Backbones created by DS nodes (using 1-hop neighborhood information) are small size, do not have a parameter like grid size, and preserve network connectivity without the help of other nodes. Location updates and destination searches are restricted to backbone nodes. Both methods use ‘hello’ messages to learn neighbors. While this suffices to construct DS, grid leader (re)election requires additional messages. Simulation results show that using DS as backbone for quorum construction is superior to using grid as backbone or no backbone at all. The proposed DS-quorum location service can achieve higher (or similar) success rate with much less communication overhead than grid-based approaches.     

Large independent sets in random regular graphs

We present algorithmic lower bounds on the size _sd$s_d$ of the largest independent sets of vertices in random d$d$-regular graphs, for each fixed d≥3$d\ge 3$. For instance, for d=3$d=3$ we prove that, for graphs on n$n$ vertices, _sd≥0.43475n$s_d\ge 0.43475n$ with probability approaching one as n$n$ tends to infinity.     

Hardness and approximation of minimum maximal matchings

In this paper, we consider the minimum maximal matching problem in some classes of graphs such as regular graphs. We show that the minimum maximal matching problem is NP-hard even in regular bipartite graphs, and a polynomial time exact algorithm is given for almost complete regular bipartite graphs. From the approximation point of view, it is well known that any maximal matching guarantees the approximation ratio of 2 but surprisingly very few improvements have been obtained. In this paper we give improved approximation ratios for several classes of graphs. For example any algorithm is shown to guarantee an approximation ratio of (2-o(1)) in graphs with high average degree. We also propose an algorithm guaranteeing for any graph of maximum degree Δ an approximation ratio of (2?1/Δ), which slightly improves the best known results. In addition, we analyse a natural linear-time greedy algorithm guaranteeing a ratio of (2?23/18k) in k-regular graphs admitting a perfect matching.     

On mimicking networks representing minimum terminal cuts

Given a capacitated undirected graph G=(V,E)$G=(V,E)$ with a set of terminals K⊂V$K\subset V$, a mimicking network   is a smaller graph H=(_VH,_EH)$H=(V_H,E_H)$ which contains the set of terminals K   and for every bipartition [U,K−U]$[U,K-U]$ of the terminals, the cost of the minimum cut separating U   from K−U$K-U$ in G is exactly equal to the cost of the minimum cut separating U   from K−U$K-U$ in H.