On the Linearization of Optimal Rates for Independent Zero-Error Source and Channel Problems
<|reference_start|>On the Linearization of Optimal Rates for Independent Zero-Error Source and Channel Problems: Zero-error coding encompasses a variety of source and channel problems where the probability of error must be exactly zero. This condition is stricter than that of the vanishing error regime, where the error probability goes to zero as the code blocklength goes to infinity. In general, zero-error coding is an open combinatorial question. We investigate two unsolved zero-error problems: the source coding problem with side information and the channel coding problem. We focus our attention on families of independent problems for which the distribution decomposes into a product of distributions, corresponding to solved zero-error problems. A crucial step is the linearization property of the optimal rate which does not always hold in the zero-error regime, unlike in the vanishing error regime. We derive a condition under which the linearization properties of the complementary graph entropy $\overline{H}$ for the AND product of graph and for the disjoint union of graphs are equivalent. Then we establish the connection with a recent result obtained by Wigderson and Zuiddam and by Schrijver, for the zero-error capacity $C_0$. As a consequence, we provide new single-letter characterizations of $\overline{H}$ and $C_0$, for example when the graph is a product of perfect graphs, which is not perfect in general, and for the class of graphs obtained by the product of a perfect graph $G$ with the pentagon graph $C_5$. By building on Haemers result for $C_0$, we also show that the linearization of $\overline{H}$ does not hold for the product of the Schl\"{a}fli graph with its complementary graph.<|reference_end|>
@article{charpenay2024on,
    title={On the Linearization of Optimal Rates for Independent Zero-Error Source
  and Channel Problems},
    author={Nicolas Charpenay, Ma"el Le Treust, Aline Roumy},
    journal={arXiv preprint arXiv:2407.02281},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.02281},
    primaryClass={cs.IT math.IT}
}