On the Maximum Rate of Networked Computation in a Capacitated Network

Given a capacitated communication network $\mathcal{N}$ and a function f that needs to be computed on $\mathcal{N},$ we study the problem of generating a computation and communication schedule in $\mathcal{N}$ to maximize the rate of computation of f. Shah et. al.[IEEE Journal of Selected Areas in Communication, 2013] studied this problem when the computation schema $\mathcal{G}$ for f is a tree. We define the notion of a schedule when $\mathcal{G}$ is a general DAG and show that finding an optimal schedule is equivalent to finding the solution of a packing LP. We prove that approximating the maximum rate is MAX SNP-hard by looking at the packing LP. For this packing LP we prove that solving the separation oracle of its dual is equivalent to solving the LP. The separation oracle of the dual reduces to the problem of finding minimum cost embedding given $\mathcal{N},\mathcal{G},$ which we prove to be MAX SNP-hard even when $\mathcal{G}$ has bounded degree and bounded edge weights and $\mathcal{N}$ has just three vertices. We present a polynomial time algorithm to compute the maximum rate of function computation when $\mathcal{N}$ has two vertices by reducing the problem to a version of submodular function minimization problem. For the general $\mathcal{N}$ we study restricted class of schedules and its equivalent packing LP. We observe that for this packing LP also the separation oracle of its dual reduces to finding minimum cost embedding. A version of this minimum cost embedding problem has been studied in literature. We present a quadratic integer program for the minimum cost embedding problem and its linear programming relaxation based on earthmover metric. We also present some approximate algorithms for special classes of $\mathcal{G}.$