A generalization of Witsenhausen's zero-error rate for directed graphs

TL;DR

This paper introduces a new graph parameter that generalizes Witsenhausen's zero-error rate, analyzing its properties and relationships to other graph parameters within a communication scenario involving noisy and noiseless channels.

Contribution

It defines a novel digraph parameter extending Witsenhausen's zero-error rate and explores its properties and connections to existing graph parameters.

Findings

New digraph parameter generalizes Witsenhausen's zero-error rate

Analyzed the parameter for specific directed graphs

Explored relations to Sperner capacity and dichromatic number

Abstract

We investigate a communication setup where a source output is sent through a free noisy channel first and an additional codeword is sent through a noiseless but expensive channel later. With the help of the second message the decoder should be able to decide with zero-error whether its decoding of the first message was error-free. This scenario leads to the definition of a digraph parameter that generalizes Witsenhausen's zero-error rate for directed graphs. We investigate this new parameter for some specific directed graphs and explore its relations to other digraph parameters like Sperner capacity and dichromatic number. When the original problem is modified to require zero-error decoding of the whole message then we arrive back to the Witsenhausen rate of an appropriately defined undirected graph.