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.
There are few examples of local algorithms which approximate linear problems beyond packing and covering LPs. Kuhn al @cite_11 study an LP relaxation of the @math -fold dominating set problem and obtain a local constant-factor approximation for bounded-degree graphs.
@cite_11
2149477108
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.