Abstract: To save energy and alleviate interference, connected dominating set (CDS) was proposed to serve as a virtual backbone of wireless sensor networks (WSNs). Because sensor nodes may fail due to accidental damages or energy depletion, it is desirable to construct a fault tolerant virtual backbone with high redundancy in both coverage and connectivity. This can be modeled as a $k$ -connected $m$ -fold dominating set (abbreviated as $(k,m)$ -CDS) problem. 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. Constant approximation algorithm is known for $(3,m)$ -CDS in unit disk graph, which models homogeneous WSNs. In this paper, we present the first performance guaranteed approximation algorithm for $(3,m)$ -CDS in a heterogeneous WSN. In fact, our performance ratio is valid for any topology. The performance ratio is at most $\gamma $ , where $\gamma =\alpha +8+2\ln (2\alpha -6)$ for $\alpha \geq 4$ and $\gamma =3\alpha +2\ln 2$ for $\alpha , and $\alpha $ is the performance ratio for the minimum $(2,m)$ -CDS problem. Using currently best known value of $\alpha $ , the performance ratio is $\ln \delta +o(\ln \delta )$ , where $\delta $ is the maximum degree of the graph, which is asymptotically best possible in view of the non-approximability of the problem. Applying our algorithm on a unit disk graph, 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.

Abstract: This paper proposes a new polynomial time constant factor approximation algorithm for a more-a-decade-long open NP-hard problem, the minimum four-connected $m$ -dominating set problem in unit disk graph (UDG) with any positive integer $m \geq 1$ for the first time in the literature. We observe that it is difficult to modify the existing constant factor approximation algorithm for the minimum three-connected $m$ -dominating set problem to solve the minimum four-connected $m$ -dominating set problem in UDG due to the structural limitation of Tutte decomposition, which is the main graph theory tool used by Wang et al. to design their algorithm. To resolve this issue, we first reinvent a new constant factor approximation algorithm for the minimum three-connected $m$ -dominating set problem in UDG and later use this algorithm to design a new constant factor approximation algorithm for the minimum four-connected $m$ -dominating set problem in UDG.

Abstract: Given a connected and weighted graph $$G=(V, E)$$G=(V,E) with each vertex v having a nonnegative weight w(v), the minimum weighted connected vertex cover $$P_{3}$$P3 problem $$(MWCVCP_{3})$$(MWCVCP3) is required to find a subset C of vertices of the graph with minimum total weight, such that each path with length 2 has at least one vertex in C, and moreover, the induced subgraph G[C] is connected. This kind of problem has many applications concerning wireless sensor networks and ad hoc networks. When homogeneous sensors are deployed into a three-dimensional space instead of a plane, the mathematical model for the sensor network is a unit ball graph instead of a unit disk graph. In this paper, we propose a new concept called weak c-local and give the first polynomial time approximation scheme (PTAS) for $$MWCVCP_{3}$$MWCVCP3 in unit ball graphs when the weight is smooth and weak c-local.

Abstract: This paper concerns wireless sensor networks (WSNs) of which each node is equipped with k ≥ 1 directional antennas having beam-width θ ∈ (0, 2π). The sum ϕ k of the beam-widths of the k antennas of each node is in (0, 2π). Each node is initially assigned a transmission range 1 that yields a connected unit disk graph spanning all nodes. The objective of the Antenna Orientation (AO) problem concerning symmetric connectivity is to compute an orientation of the antennas and to find a minimum transmission power range r = O(1) such that the induced symmetric communication graph (SCG) is connected. In this paper, we study the AO problem assuming that each node has two antennas (k = 2) each of which has beam-width θ = π/3 or π/4. We propose two approximation algorithms that orient the antennas to yield symmetric connected communication graphs (SCCGs) where the transmission power ranges are bounded by 4 and 5 when θ = π/3 and π/4, respectively. These bounds are the first results for this problem. We also study the performance of our algorithms through simulation.

Abstract: We give algorithms with running time 2^{O({\sqrt{k}\log{k}})} n^{O(1)} for the following problems. Given an n-vertex unit disk graph G and an integer k, decide whether G contains (i) a path on exactly/at least k vertices, (ii) a cycle on exactly k vertices, (iii) a cycle on at least k vertices, (iv) a feedback vertex set of size at most k, and (v) a set of k pairwise vertex disjoint cycles. For the first three problems, no subexponential time parameterized algorithms were previously known. For the remaining two problems, our algorithms significantly outperform the previously best known parameterized algorithms that run in time 2^{O(k^{0.75}\log{k})} n^{O(1)}. Our algorithms are based on a new kind of tree decompositions of unit disk graphs where the separators can have size up to k^{O(1)} and there exists a solution that crosses every separator at most O(\sqrt{k}) times. The running times of our algorithms are optimal up to the log{k} factor in the exponent, assuming the Exponential Time Hypothesis.

Abstract: Numerous approximation algorithms for problems on unit disk graphs have been proposed in the literature, exhibiting a sharp trade-off between running times and approximation ratios. We introduce a variation of the known shifting strategy that allows us to obtain linear-time constant-factor approximation algorithms for such problems. To illustrate the applicability of the proposed variation, we obtain results for three well-known optimization problems. Among such results, the proposed method yields linear-time (4 + 𝜀)-approximations for the maximum-weight independent set and the minimum dominating set of unit disk graphs, thus bringing significant performance improvements when compared to previous algorithms that achieve the same approximation ratios. Finally, we use axis-aligned rectangles to illustrate that the same method may be used to derive linear-time approximations for problems on other geometric intersection graph classes.

Abstract: We consider a network localization problem by modeling this as a unit disk graph where nodes are randomly placed with uniform distribution in an area.The connectivity between nodes is defined when the distances fall within a unit range. Under a condition that certain nodes know their locations (anchor nodes), this paper proposes a heuristic approach to find a realization for the rest of the network by applying a tree search algorithm in a depth- first search manner. Our contribution is to put together a priori information and constraints such as graph properties in order to speed up the search. An evaluation function is formed and used to prune down the search space. This evaluation function is used to select the order of the unknown nodes to iterate. This paper also extends the idea further by accommodating a variety of other properties of graphs into the evaluation function. The results show that node degrees, node distances and shortest paths to anchor nodes drastically reduce the number of iterations required for realizing a feasible localization instance both in noise-free and noisy environments. Finally, some preliminary complexity analysis is also given.

Abstract: In this paper, we study the antenna orientation problem concerning symmetric connectivity in directional wireless sensor networks. We are given a set of nodes each of which is equipped with one directional antenna with beam-width $$\theta = 2\pi /3$$ and is initially assigned a transmission range 1 that yields a connected unit disk graph spanning all nodes. The objective of the problem is to compute an orientation of the antennas and to find a minimum transmission power range $$r=O(1)$$ such that the induced symmetric communication graph is connected. We propose two algorithms that orient the antennas to yield symmetric connected communication graphs where the transmission power ranges are bounded by 6 and 5, which are currently the best results for this problem. We also study the performance of our algorithms through simulations.

Abstract: Geographic routing has emerged as one of the most suitable strategies for routing in Wireless Sensor Networks (WSNs). Greedy forwarding (GF) is an efficient form of geographic routing in which a packet is progressively pushed closest to the destination in each hop. The presence of communication voids may lead GF to fail at dead-ends. The dead-end situation is usually handled by using methods like flooding, face routing, and routing along network boundaries. Existing schemes for handling void problems are either too inefficient or are rely on some unrealistic assumptions like unit disk graph model of connectivity. In this chapter, a scheme called Rolling Circle Algorithm (RCA) is presented for recovering from dead-ends by routing packets along the boundary of the WSN until GF can resume again. The proposed technique is based on a variant of alpha-shapes method of detecting boundaries of a point cloud. The value of the parameter alpha used to detect boundary nodes adapts according to the local topology and location information of 1-hop neighbors of the forwarding node.

Abstract: Finding the Minimum-size Connected Dominating Set (MCDS), i.e., the communication backbone with the minimum number of nodes is a key problem in wireless networks, which is crucial for designing efficient routing algorithms and for network energy efficiency etc. This paper proposes a new greedy algorithm to approach the MCDS problem. The key idea is to separate nodes in CDS into core nodes and supporting nodes. The core nodes dominate the supporting nodes in CDS, while the supporting nodes dominate other nodes that are not in CDS. The proposed algorithm is verified by simulation, the simulation results show that the CDS constructed by our algorithm has smaller size than the state of the art algorithms in [10].