On the Maximum Rate of Networked Computation in a Capacitated Network

We are given a capacitated communication network and several infinite sequences of source data each of which is available at some node in the network. A function of the source data is to be computed in the network and made available at a sink node that is also on the network. The schema to compute the function is given as a directed acyclic graph (DAG). We want to generate a computation and communication schedule in the network to maximize the rate of computation of the function for an arbitrary function (represented by DAG). We first analyze the complexity of finding the rate maximizing schedule for the general DAG. We show that finding an optimal schedule is equivalent to solving a packing linear program (LP). We then prove that finding the maximum rate is MAX SNP-hard (by analyzing this packing LP) even when the DAG has bounded degree, bounded edge weights and the network has three vertices. We then consider special cases arising in practical situations. First, a polynomial time algorithm for the network with two vertices is presented. This algorithm is a reduction to a version of a submodular function minimization problem. Next, for the general network we describe a restricted class of schedules and its equivalent packing LP. By relating this LP to minimum cost embedding problem, we present approximation algorithms for special classes of DAGs.