Fast Distributed Algorithms Via Primal-Dual (Extended Abstract)

Abstract

We would like to discuss what seems to be a general methodology to develop fast distributed algorithms for optimization problems on graphs, based on the primal-dual schema. The kind of problems we have in mind are of the following type. We have a synchronous, message-passing network that is to compute a global function of its own topology. Examples of such functions are maximal independent sets, vertex and edge colorings, small dominating sets, vertex covers and so on. Crucially, nodes only know their neighbours and have very little or no global information. In what follows, the only global information allowed will be n, the number of nodes in the network (or an upper bound on it). In this setting the running time of a protocol is given by the number of communication rounds needed to compute the output. By the end of the algorithm each node or edge will have decided its final status: its own color, whether or not to be part of the dominating set etc. In many situations of interest the cost of communication is orders of magnitude larger than local computation cost, and the model provides a rough, but quite useful, quantitative framework to develop and analyze interesting algorithms.