On the Symmetry Reduction of Information Inequalities

Information inequalities can be used to derive the fundamental limits of information systems. Many information inequalities and problem-specific constraints are linear equalities or inequalities of joint entropies, and thus, outer bounding the fundamental limits can be viewed as and in principle computed through linear programming. However, for many practical engineering problems, the resultant linear program (LP) is very large, rendering such a computational approach almost completely inapplicable in practice. It was shown recently that symmetry can be used to effectively reduce the scale of the LP; however, the precise amount of reduction was not well understood. In this paper, we provide a method to pinpoint this reduction by counting the number of orbits induced by the symmetry on the set of the LP variables and the LP constraints, respectively. The Pólya counting theorem is a powerful tool for such counting task, which requires identifying the cycle index function. We propose a generic three-layer decomposition of the group structures for the quantities in typical information systems to facilitate such a calculation. Three problems are studied using this approach: extremal pairwise cyclically symmetric entropy inequalities, the regenerating code problem, and the caching problem, for which explicit formulas are provided for the cycle indices of the induced permutations.