On the Additivity of Optimal Rates for Independent Zero-Error Source and Channel Problems

TL;DR

The paper tackles whether optimal zero-error rates add across independent source or channel subproblems, linking source and channel additivity through the complementary graph entropy $\overline{H}$ and the zero-error capacity $C_0$. It proves an equivalence: additivity of $\overline{H}$ for the AND product is equivalent to additivity for the disjoint union, and shows a network of equivalences among additivity properties of $C_0$, $C$, and $\overline{H}$ under various product operations. It derives new single-letter characterizations for products of perfect graphs and for the product with $C_5$, and provides counterexamples (notably involving the Schläfli graph) showing that additivity can fail in general. The work also connects to capacity-achieving distributions, capacity-relations $C(G,P_X)$, and extends the discussion to partial side information, highlighting both the reach and the limits of current additivity results in zero-error information theory.

Abstract

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 probability distribution decomposes into a product of probability distributions. A crucial step is the additivity property of the optimal rate, which does not always hold in the zero-error regime, unlike in the vanishing error regime. When the additivity holds, the concatenation of optimal codes is optimal. We derive a condition under which the additivity of the complementary graph entropy $\overline{H}$ for the AND product of graphs 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 additivity of $\overline{H}$ does not hold for the product of the Schläfli graph with its complementary graph.

Figures (11)

• Figure 1: The characteristic graph $C_7$ of the noisy-typewriter channel with $7$ letters.
• Figure 2: The source coding problem with decoder side information, called side-information problem.
• Figure 3: Equivalences of additivity properties for the zero-error capacity $C_0$, for the zero-error capacity relative to a distribution $C$, and for the complementary graph entropy $\overline{H}$. The equivalences of our Theorem \ref{['th:caraclin']}, Proposition \ref{['prop:marton']}, Theorem \ref{['th:mainC0']} and Theorem \ref{['th:linC_0sqcup']} are valid under specific conditions on the probability distributions.
• Figure 4: The pentagon graphs $C_5$ with uniform distribution $P_X=\mathop{\mathrm{Unif}}\limits (\lbrace 1, ...,5\rbrace )$ over the vertices.
• Figure 5: Independent side-information problems

Theorems & Definitions (70)

• Definition 1
• Definition 2: Characteristic graph
• Definition 3: Coloring, chromatic number $\chi$
• Definition 4: AND product $\wedge$
• Theorem 1: from alon1996source
• Definition 5
• Theorem 2: from koulgi2003zero
• Lemma 1: from simonyi2003witsenhausen
• Proposition 1: Full support, from witsenhausen1976zero
• Proposition 2