Network Coding for Computing: Cut-Set Bounds

TL;DR

This paper extends the concept of min-cut bounds to network computing, establishing capacity limits for computing functions over networks and analyzing their tightness for various network classes and functions.

Contribution

It generalizes the min-cut bound to network computing, proving its tightness for certain networks and functions, and provides bounds for computing capacity related to Steiner tree packing.

Findings

Min-cut remains an upper bound on computing capacity.

The bound is tight for multi-edge tree networks and linear functions.

Computing capacity can be arbitrarily smaller than the min-cut bound in some cases.

Abstract

The following \textit{network computing} problem is considered. Source nodes in a directed acyclic network generate independent messages and a single receiver node computes a target function $f$ of the messages. The objective is to maximize the average number of times $f$ can be computed per network usage, i.e., the ``computing capacity''. The \textit{network coding} problem for a single-receiver network is a special case of the network computing problem in which all of the source messages must be reproduced at the receiver. For network coding with a single receiver, routing is known to achieve the capacity by achieving the network \textit{min-cut} upper bound. We extend the definition of min-cut to the network computing problem and show that the min-cut is still an upper bound on the maximum achievable rate and is tight for computing (using coding) any target function in multi-edge tree networks and for computing linear target functions in any network. We also study the bound's tightness for different classes of target functions. In particular, we give a lower bound on the computing capacity in terms of the Steiner tree packing number and a different bound for symmetric functions. We also show that for certain networks and target functions, the computing capacity can be less than an arbitrarily small fraction of the min-cut bound.