Book inequalities
Abstract
Information theoretical inequalities have strong ties with polymatroids and their representability. A polymatroid is entropic if its rank function is given by the Shannon entropy of the subsets of some discrete random variables. The book is a special iterated adhesive extension of a polymatroid with the property that entropic polymatroids have $n$-page book extensions over an arbitrary spine. We prove that every polymatroid has an $n$-page book extension over a single element and over an all-but-one-element spine. Consequently, for polymatroids on four elements, only book extensions over a two-element spine should be considered. F. Matúš proved that the Zhang-Yeung inequalities characterize polymatroids on four elements which have such a 2-page book extension. The $n$-page book inequalities, defined in this paper, are conjectured to characterize polymatroids on four elements which have $n$-page book extensions over a two-element spine. We prove that the condition is necessary; consequently every book inequality is an information inequality on four random variables. Using computer-aided multiobjective optimization, the sufficiency of the condition is verified up to 9-page book extensions.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2013
- DOI:
- arXiv:
- arXiv:1312.6490
- Bibcode:
- 2013arXiv1312.6490C
- Keywords:
-
- Computer Science - Information Theory;
- 05B35;
- 26A12;
- 52B12;
- 90C29;
- 94A17