Comments on Cut-Set Bounds on Network Function Computation

TL;DR

This paper critiques existing cut-set bounds on network function computation, identifies their limitations, and proposes a new, valid bound applicable to general functions and network structures, confirming its tightness in specific cases.

Contribution

The paper introduces a new valid cut-set bound for network function computation, correcting previous invalid bounds and extending applicability to general functions and network topologies.

Findings

Existing bounds are invalid for general cases.

The new bound is tight for certain known cases.

Some previous results are validated using the new bound.

Abstract

A function computation problem in directed acyclic networks has been considered in the literature, where a sink node wants to compute a target function with the inputs generated at multiple source nodes. The network links are error-free but capacity-limited, and the intermediate network nodes perform network coding. The target function is required to be computed with zero error. The computing rate of a network code is measured by the average number of times that the target function can be computed for one use of the network, i.e., each link in the network is used at most once. In the papers [1], [2], two cut-set bounds were proposed on the computing rate. However, we show in this paper that these bounds are not valid for general network function computation problems. We analyze the arguments that lead to the invalidity of these bounds and fix the issue with a new cut-set bound, where a new equivalence relation associated with the inputs of the target function is used. Our bound is qualified for general target functions and network topologies. We also show that our bound is tight for some special cases where the computing capacity is known. Moreover, some results in [11], [12] were proved using the invalid upper bound in [1] and hence their correctness needs further justification. We also justify their validity in the paper.