Conditional Information Inequalities and Combinatorial Applications

Abstract

We show that the inequality $H(A\mid B,X)+H(A\mid B,Y)\le H(A\mid B)$ for jointly distributed random variables $A,B,X,Y$, which does not hold in general case, holds under some natural condition on the support of the probability distribution of $A,B,X,Y$. This result generalizes a version of the conditional Ingleton inequality: if for some distribution $I(X:Y\mid A)=H(A\mid X,Y)=0$, then $I(A:B)\le I(A:B\mid X)+I(A:B\mid Y)+I(X:Y)$. We present two applications of our result. The first one is the following easy-to-formulate combinatorial theorem: assume that the edges of a bipartite graph are partitioned into $K$ matchings such that for each pair (left vertex $x$, right vertex $y$) there is at most one matching in the partition involving both $x$ and $y$; assume further that the degree of each left vertex is at least $L$ and the degree of each right vertex is at least $R$. Then $K\ge LR$. The second application is a new method to prove lower bounds for biclique coverings of bipartite graphs.

References in corpus (1)

• Conditional Information Inequalities for Entropic and Almost Entropic Points