A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We study the applicability of local algorithms to max-min LPs where the objective is to maximise mink Sigmav CkvXv subject to Sigmav alphaivXv les 1 far each i and Xv ges 0 far each v. Here ckv ges 0, and the support sets Vi = v : alphaiv> 0 , Vk = v : ckv > 0 , Iv = i: alphaiv > 0 and Kv = k : Ckv > 0 have bounded size. In the distributed setting, each agent v is responsible for choosing the value of Xv, and the communication network is a hypergraph H where the sets Vk and Vi constitute the hyperedges. We present inapproximability results for a wide range of structural assumptions; for example, even if |Vi| and |Vk| are bounded by some constants larger than 2, there is no local approximation scheme. To contrast the negative results, we present a local approximation algorithm which achieves good approximation ratios if we can bound the relative growth of the vertex neighbourhoods in H.
We study fractional scheduling problems in sensor networks, in particular, sleep scheduling (generalisation of fractional domatic partition) and activity scheduling (generalisation of fractional graph colouring). The problems are hard to solve in general even in a centralised setting; however, we show that there are practically relevant families of graphs where these problems admit a local distributed approximation algorithm; in a local algorithm each node utilises information from its constant-size neighbourhood only. Our algorithm does not need the spatial coordinates of the nodes; it suffices that a subset of nodes is designated as markers during network deployment. Our algorithm can be applied in any marked graph satisfying certain bounds on the marker density; if the bounds are met, guaranteed near-optimal solutions can be found in constant time, space and communication per node.We also show that auxiliary information is necessary--no local algorithm can achieve a satisfactory approximation guarantee on unmarked graphs. Finding a small dominating set is one of the most fundamental problems of traditional graph theory. In this paper, we present a new fully distributed approximation algorithm based on LP relaxation techniques. For an arbitrary parameter k and maximum degree Δ, our algorithm computes a dominating set of expected size O(kΔ2 k log Δ|DSOPT|) in O(k2) rounds where each node has to send O(k2Δ) messages of size O(logΔ). This is the first algorithm which achieves a non-trivial approximation ratio in a constant number of rounds. In this paper, we review a recently developed class of algorithms that solve global problems in unit distance wireless networks by means of local algorithms. A local algorithm is one in which any node of a network only has information on nodes at distance at most k from itself, for a constant k. For example, given a unit distance wireless network N, we want to obtain a planar subnetwork of N by means of an algorithm in which all nodes can communicate only with their neighbors in N, perform some operations, and then halt. We review algorithms for obtaining planar subnetworks, approximations to minimum weight spanning trees, Delaunay triangulations, and relative neighbor graphs. Given a unit distance wireless network N, we present new local algorithms to solve the following problems:1.Calculate small dominating sets (not necessarily connected) of N. 2.Extract a bounded degree planar subgraph H of N and obtain a proper edge coloring of H with at most 12 colors. The second of these algorithms can be used in the channel assignment problem. This paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion. The computational model is such that each node of the graph is occupied by a processor which has its own ID. Processors are restricted to collecting data from others which are at a distance at most t away from them in t time units, but are otherwise computationally unbounded. This model focuses on the issue of locality in distributed processing, namely, to what extent a global solution to a computational problem can be obtained from locally available data.Three results are proved within this model: • A 3-coloring of an n-cycle requires time @math . This bound is tight, by previous work of Cole and Vishkin. • Any algorithm for coloring the d-regular tree of radius r which runs for time at most @math requires at least @math colors. • In an n-vertex graph of largest degree @math , an @math -coloring may be found in time @math . The purpose of this paper is a study of computation that can be done locally in a distributed network, where "locally" means within time (or distance) independent of the size of the network. Locally checkable labeling (LCL) problems are considered, where the legality of a labeling can be checked locally (e.g., coloring). The results include the following: There are nontrivial LCL problems that have local algorithms. There is a variant of the dining philosophers problem that can be solved locally. Randomization cannot make an LCL problem local; i.e., if a problem has a local randomized algorithm then it has a local deterministic algorithm. It is undecidable, in general, whether a given LCL has a local algorithm. However, it is decidable whether a given LCL has an algorithm that operates in a given time @math . Any LCL problem that has a local algorithm has one that is order-invariant (the algorithm depends only on the order of the processor IDs).  In this paper, we study distributed approximation algorithms for fault-tolerant clustering in wireless ad hoc and sensor networks. A k-fold dominating set of a graph G = (V,E) is a subset S of V such that every node v V S has at least k neighbors in S. We study the problem in two network models. In general graphs, for arbitrary parameter t, we propose a distributed algorithm that runs in time O(t^2) and achieves an approximation ratio of O(t ^2 t log ), where n and denote the number of nodes in the network and the maximal degree, respectively. When the network is modeled as a unit disk graph, we give a probabilistic algorithm that runs in time O(log log n) and achieves an O(1) approximation in expectation. Both algorithms require only small messages of size O(log n) bits.