Alphabet Sizes of Auxiliary Variables in Canonical Inner Bounds

TL;DR

This paper derives the alphabet size of auxiliary variables in canonical inner bounds, improving estimates for special cases and generalizing to multiterminal setups through geometric and combinatorial analysis.

Contribution

It introduces a generalized method for determining minimal auxiliary alphabet sizes in multiterminal information theory problems.

Findings

Derived alphabet size bounds for auxiliary variables.

Extended estimates from special cases to general multiterminal scenarios.

Provided a geometric approach using hyperplane tangency and Caratheodory's theorem.

Abstract

Alphabet size of auxiliary random variables in our canonical description is derived. Our analysis improves upon estimates known in special cases, and generalizes to an arbitrary multiterminal setup. The salient steps include decomposition of constituent rate polytopes into orthants, translation of a hyperplane till it becomes tangent to the achievable region at an extreme point, and derivation of minimum auxiliary alphabet sizes based on Caratheodory's theorem.