Xianyue Li

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 +@e)-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 +@e)-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 +@e)-approximation algorithm for an MWCDS.

A Better Approximation Algorithm for Computing Connected Dominating Sets in Unit Ball Graphs

A Virtual Backbone (VB) of a wireless network is a subset of nodes such that only VB nodes are responsible for routing-related tasks. Since a smaller VB causes less overhead, size is the primary quality factor of VB. Frequently, Unit Disk Graphs (UDGs) are used to model 2D homogeneous wireless networks, and the problem of finding minimum VBs in the networks is abstracted as Minimum Connected Dominating Set (MCDS) problem in UDGs. In some applications, the altitude of nodes can be hugely different and UDG cannot abstract the networks accurately. Then, Unit Ball Graph (UBG) can replace UDG. In this paper, we study how to construct quality CDSs in UBGs in distributed environments. We first give an improved upper bound of the number of independent nodes in a UBG, and use this result to analyze the Performance Ratio (PR) of our new centralized algorithm C-CDS-UBG, which computes CDSs in UBGs. Next, we propose a distributed algorithm D-CDS-UBG originated from C-CDS-UBG and analyze its message and time complexities. Our theoretical analysis shows that the PR of D-CDS-UBG is 14.937, which is better than current best, 22. Our simulations also show that D-CDS-UBG outperforms the competitor, on average.

A New Constant Factor Approximation for Computing 3-Connected m-Dominating Sets in Homogeneous Wireless Networks

In this paper, we study the problem of constructing quality fault-tolerant Connected Dominating Sets (CDSs)in homogeneous wireless networks, which can be defined as minimum k-Connected m-Dominating Set ((k;m)-CDS) problem in Unit Disk Graphs (UDGs). We found that every existing approximation algorithm for this problem is incomplete for k >= 3 in a sense that it does not generate a feasible solution in some UDGs. Based on these observations, we propose a new polynomial time approximation algorithm for computing (3;m)-CDSs. We also show that our algorithm is correct and its approximation ratio is a constant.

On construction of quality fault-tolerant virtual backbone in wireless networks

In this paper, we study the problem of computing quality fault-tolerant virtual backbone in homogeneous wireless network, which is defined as the k-connected m-dominating set problem in a unit disk graph. This problem is NP-hard, and thus many efforts have been made to find a constant factor approximation algorithm for it, but never succeeded so far with arbitrary k ≥ 3 and m ≥ 1 pair. We propose a new strategy for computing a smaller-size 3-connected m-dominating set in a unit disk graph with any m ≥ 1. We show the approximation ratio of our algorithm is constant and its running time is polynomial. We also conduct a simulation to examine the average performance of our algorithm. Our result implies that while there exists a constant factor approximation algorithm for the k-connected m-dominating set problem with arbitrary k ≤ 3 and m ≥ 1 pair, the k-connected m-dominating set problem is still open with k > 3.

Construction of Minimum Connected Dominating Set in 3-Dimensional Wireless Network

Connected Dominating Set (CDS) has been a well known approach for constructing a virtual backbone to alleviate the broadcasting storm in wireless networks. Previous literature modeled the wireless network in a 2-dimensional plane and looked for the approximated Minimum CDS (MCDS) distributed or centralized to construct the virtual backbone of the wireless network. However, in some real situations, the wireless network should be modeled as a 3-dimensional space instead of 2-dimensional plane. We propose our approximation algorithm for MCDS construction in 3-dimensional wireless network in this paper. It achieves better upper bound (13 + ln 10)opt+ 1 than the only known result 22opt. This algorithm helps bringing the research for MCDS construction in 3-dimensional wireless network to a new stage.

A PTAS for Node-Weighted Steiner Tree in Unit Disk Graphs

The node-weighted Steiner tree problem is a variation of classical Steiner minimum tree problem. Given a graph G = (V ,E ) with node weight function C :V ***R + and a subset X of V , the node-weighted Steiner tree problem is to find a Steiner tree for the set X such that its total weight is minimum. In this paper, we study this problem in unit disk graphs and present a (1+*** )-approximation algorithm for any *** > 0, when the given set of vertices is c -local. As an application, we use node-weighted Steiner tree to solve the node-weighted connected dominating set problem in unit disk graphs and obtain a (5 + *** )-approximation algorithm.

Two Constant Approximation Algorithms for Node-Weighted Steiner Tree 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 nfor any 0 < a< 1 unless NP? DTIME(nO(logn)), where nis the number of nodes in s. In this paper, we show that for unit disk graph, the problem is still NP-hard, however it has polynomial time constant approximation. We will present a 4-approximation and a 2.5ρ-approximation where ρis 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 polynomial time (9.875+?)-approximation algorithm for minimum weight connected dominating set in unit disk graphs.

Recyclable Connected Dominating Set for Large Scale Dynamic Wireless Networks

Many people studied the Minimum Connected Dominating Set (MCDS) problem to introduce Virtual Backbone (VB) to wireless networks. However, many existing algorithms assume a static wireless network, and when its topology is changed, compute a new CDS all over again. Since wireless networks are highly dynamic due to many reasons, their approaches can be inefficient in practice. Motivated by this observation, we propose Recyclable CDS Algorithm (RCDSA), an efficient VB maintenance algorithm which can handle the activeness of wireless networks. The RCDSA is built on an approximation algorithm CDS-BD-C1 by Kim et. al. [1]. When a node is added to or deleted from current graph, RCDSA recycles current CDS to get a new one. We prove RCDSAs performance ratio is equal to CDS-BD-C1s. In simulation, we compare RCDSA with CDS-BD-C1. Our results show that the average size of CDS by RCDSA is similar with that by CDS-BD-C1 but RCDSA is at least three times faster than CDS-BD-C1 due to its simplicity. Furthermore, at any case, a new CDS by RCDSA highly resembles to its old version than the one by CDS-BD-C1, which means that using RCDSA, a wireless network labors less to maintain its VB when its topology is dynamically changing.

Approximation algorithms on 0---1 linear knapsack problem with a single continuous variable

The 0–1 linear knapsack problem with a single continuous variable (KPC) is a natural generalization of the standard 0–1 linear knapsack problem (KP). In KPC, the capacity of the knapsack is not fixed, but can be adjusted by a continuous variable. This paper studies the approximation algorithm on KPC. Firstly, assuming that the weight of each item is at most the original capacity of the knapsack, we give a 2-approximation algorithm on KPC by generalizing the 2-approximation algorithm on KP. Then, without the above assumption, we give another 2-approximation algorithm on KPC for general cases by extending the first algorithm.

Partial inverse maximum spanning tree in which weight can only be decreased under \(l_p\)-norm

The maximum or minimum spanning tree problem is a classical combinatorial optimization problem. In this paper, we consider the partial inverse maximum spanning tree problem in which the weight function can only be decreased. Given a graph, an acyclic edge set, and an edge weight function, the goal of this problem is to decrease weights as little as possible such that there exists with respect to function containing the given edge set. If the given edge set has at least two edges, we show that this problem is APX-Hard. If the given edge set contains only one edge, we present a polynomial time algorithm.