Dilworth Rate: A Generalization of Witsenhausen’s Zero-Error Rate for Directed Graphs

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, such as Sperner capacity and dichromatic number. We also look at the natural variant of the above problem, where the decoder should decode the first message with zero-error, not only decide whether its earlier decoding was correct. In this case, the Witsenhausen rate of an appropriately defined undirected graph turns out to be the relevant parameter.