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Local PTAS for Independent Set and Vertex Cover in Location Aware Unit Disk Graphs
 In Proceedings of the 4th IEEE/ACM International Conference on Distributed Computing in Sensor Systems (DCOSS
, 2008
"... Abstract. We present the first local approximation schemes for maximum independent set and minimum vertex cover in unit disk graphs. In the graph model we assume that each node knows its geographic coordinates in the plane (location aware nodes). Our algorithms are local in the sense that the status ..."
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Abstract. We present the first local approximation schemes for maximum independent set and minimum vertex cover in unit disk graphs. In the graph model we assume that each node knows its geographic coordinates in the plane (location aware nodes). Our algorithms are local in the sense that the status of each node v (whether or not v is in the computed set) depends only on the vertices which are a constant number of hops away from v. This constant is independent of the size of the network. We give upper bounds for the constant depending on the desired approximation ratio. We show that the processing time which is necessary in order to compute the status of a single vertex is bounded by a polynomial in the number of vertices which are at most a constant number of vertices away from it. Our algorithms give the best possible approximation ratios for this setting. The technique which we use to obtain the algorithm for vertex cover can also be employed for constructing the first global PTAS for this problem in unit disk graph which does not need the embedding of the graph as part of the input. 1.
3D Local Algorithm for Dominating Sets of Unit Disk Graphs
"... A dominating set of a graph G = (V, E) is a subset V ′ ∈ V of the nodes such that for all nodes v ∈ V, either v ∈ V ′ or a neighbor u of v is in V ′. Several routing protocols in ad hoc networks use a dominating set of the nodes for routing. None of the existing algorithms has a constant approximat ..."
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A dominating set of a graph G = (V, E) is a subset V ′ ∈ V of the nodes such that for all nodes v ∈ V, either v ∈ V ′ or a neighbor u of v is in V ′. Several routing protocols in ad hoc networks use a dominating set of the nodes for routing. None of the existing algorithms has a constant approximation factor in a constant number of rounds in 3D. In this paper, we use the nodes ’ geometric locations to propose the first local, constant time algorithm that constructs a Dominating Set and Connected Dominating Set of a Unit Disk Graph (UDG) in a 3D environment. The approximation ratios of our algorithms are given. 1
Local approximation algorithms for a class of 0/1 maxmin linear programs
 Manuscript
, 2008
"... Abstract — We study the applicability of distributed, local algorithms P to 0/1 maxmin LPs where the objective is to maximise mink v ckvxv subject to P v aivxv ≤ 1 for each i and xv ≥ 0 for each v. Here ckv ∈ {0, 1}, aiv ∈ {0, 1}, and the support sets Vi = {v: aiv> 0} and Vk = {v: ckv> 0} have boun ..."
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Abstract — We study the applicability of distributed, local algorithms P to 0/1 maxmin LPs where the objective is to maximise mink v ckvxv subject to P v aivxv ≤ 1 for each i and xv ≥ 0 for each v. Here ckv ∈ {0, 1}, aiv ∈ {0, 1}, and the support sets Vi = {v: aiv> 0} and Vk = {v: ckv> 0} have bounded size; in particular, we study the case Vk  ≤ 2. Each agent v is responsible for choosing the value of xv based on information within its constantsize neighbourhood; the communication network is the hypergraph where the sets Vk and Vi constitute the hyperedges. We present a local approximation algorithm which achieves an approximation ratio arbitrarily close to the theoretical lower bound presented in prior work. I.
Impact of locality on location aware unit disk graphs
, 2007
"... Abstract. A network algorithm is local if the status of a vertex depends only on the vertices which are at most a constant (independent of the size of the network) number of hops away from it. Due to their importance for studies on wireless networks, recent years have seen a surge of activity on the ..."
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Abstract. A network algorithm is local if the status of a vertex depends only on the vertices which are at most a constant (independent of the size of the network) number of hops away from it. Due to their importance for studies on wireless networks, recent years have seen a surge of activity on the design of local algorithms for the solution of a variety of network tasks. Nevertheless, there are only a few lower bounds known for approximation factors of local algorithms and none for local algorithms in the setting of location aware nodes. In this paper we investigate the impact of very low locality (i.e., number of hops) on the design of algorithms in location aware UDGs. We prove the first ever lower bounds for local algorithms of a given locality for minimum dominating and connected dominating set, maximum independent set and minimum vertex cover in the location aware setting. Then we study the prospects of algorithms with very low localities. Despite of this restriction we propose local constant ratio approximation algorithms for solving these problems in Unit Disk Graphs. We compare the bounds obtained by designing even tighter upper bounds on Unit Line Graphs (a special class of UDGs whereby all vertices lie on the same line) and contrast them by proving lower bounds for arbitrary locality on these graphs. 1.
Deterministic Dominating Set Construction in Networks with Bounded Degree
, 2011
"... This paper considers the problem of calculating dominating sets in networks with bounded degree. In these networks, the maximal degree of any node is bounded by Δ, which is usually significantly smaller than n, the total number of nodes in the system. Such networks arise in various settings of wirel ..."
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This paper considers the problem of calculating dominating sets in networks with bounded degree. In these networks, the maximal degree of any node is bounded by Δ, which is usually significantly smaller than n, the total number of nodes in the system. Such networks arise in various settings of wireless and peertopeer communication. A trivial approach of choosing all nodes into the dominating set yields an algorithm with the approximation ratio of Δ + 1. We show that any deterministic algorithm with nontrivial approximation ratio requires Ω(log ∗ n) rounds, meaning effectively that no o(Δ)approximation deterministic algorithm with a running time independent of the size of the system may ever exist. On the positive side, we show two deterministic algorithms that achieve log Δ and 2 log Δapproximation in O(Δ 3 +log ∗ n) and O(Δ 2 log Δ +log ∗ n) time, respectively. These algorithms rely on coloring rather than node IDs to break symmetry.