A Conditional Information Inequality and Its Combinatorial Applications

We show that the inequality <inline-formula> <tex-math notation="LaTeX">$H(A | B,X) + H(A | B,Y) \leqslant H(A | B)$ </tex-math></inline-formula> for jointly distributed random variables <inline-formula> <tex-math notation="LaTeX">$A,B,X,Y$ </tex-math></inline-formula>, which does not hold in general case, holds under some natural condition on the support of the probability distribution of <inline-formula> <tex-math notation="LaTeX">$A,B,X,Y$ </tex-math></inline-formula>. This result generalizes a version of the conditional Ingleton inequality: if for some distribution <inline-formula> <tex-math notation="LaTeX">$I(X \mskip 5mu {:} \mskip 5mu Y | A) = H(A | X,Y)=0$ </tex-math></inline-formula>, then <inline-formula> <tex-math notation="LaTeX">$I(A \mskip 5mu {:} \mskip 5mu B) \leqslant I(A \mskip 5mu {:} \mskip 5mu B | X) + I(A \mskip 5mu {:} \mskip 5mu B | Y) + I(X \mskip 5mu {:} \mskip 5mu Y)$ </tex-math></inline-formula>. We present two applications of our result. The first one is the following easy-to-formulate theorem on edge colorings of bipartite graphs: assume that the edges of a bipartite graph are colored in <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> colors so that each two edges sharing a vertex have different colors and for each pair (left vertex <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula>, right vertex <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula>) there is at most one color <inline-formula> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> such both <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$y$ </tex-math></inline-formula> are incident to edges with color <inline-formula> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula>; assume further that the degree of each left vertex is at least <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula> and the degree of each right vertex is at least <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula>. Then <inline-formula> <tex-math notation="LaTeX">$K \geqslant LR$ </tex-math></inline-formula>. The second application is a new method to prove lower bounds for biclique cover of bipartite graphs.

[1]  Klaus Jansen,et al.  Constrained Bipartite Edge Coloring with Applications to Wavelength Routing , 1997, ICALP.

[2]  Stasys Jukna,et al.  On Graph Complexity , 2006, Combinatorics, Probability and Computing.

[3]  Madhav V. Marathe,et al.  Strong edge coloring for channel assignment in wireless radio networks , 2006, Fourth Annual IEEE International Conference on Pervasive Computing and Communications Workshops (PERCOMW'06).

[4]  Jaroslaw Grytczuk,et al.  Nonrepetitive colorings of graphs , 2007, Electron. Notes Discret. Math..

[5]  Nikolai K. Vereshchagin,et al.  Inequalities for Shannon Entropy and Kolmogorov Complexity , 1997, J. Comput. Syst. Sci..

[6]  Terence Chan.  Recent Progresses in Characterising Information Inequalities , 2011, Entropy.

[7]  Zhen Zhang,et al.  A non-Shannon-type conditional inequality of information quantities , 1997, IEEE Trans. Inf. Theory.

[8]  Stasys Jukna,et al.  On covering graphs by complete bipartite subgraphs , 2009, Discret. Math..

[9]  Raymond W. Yeung.  Facets of entropy , 2015, Commun. Inf. Syst..

[10]  Randall Dougherty,et al.  Networks, Matroids, and Non-Shannon Information Inequalities , 2007, IEEE Transactions on Information Theory.

[11]  A. Ingleton,et al.  Conditions for representability and transversality of matroids , 1971 .

[12]  Troy Lee,et al.  Lower Bounds in Communication Complexity , 2009, Found. Trends Theor. Comput. Sci..

[13]  F. Mattt,et al.  Conditional Independences among Four Random Variables Iii: Final Conclusion , 1999 .

[14]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[15]  Zhen Zhang,et al.  On Characterization of Entropy Function via Information Inequalities , 1998, IEEE Trans. Inf. Theory.

[16]  Andrei E. Romashchenko,et al.  Conditional Information Inequalities for Entropic and Almost Entropic Points , 2012, IEEE Transactions on Information Theory.

[17]  Jaroslav Nešetřil,et al.  Complete and pseudocomplete colourings of a graph , 1976 .

[18]  Raymond W. Yeung,et al.  A First Course in Information Theory , 2002 .

[19]  Andrei E. Romashchenko,et al.  On essentially conditional information inequalities , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[20]  Piotr Formanowicz,et al.  A survey of graph coloring - its types, methods and applications , 2012 .

[21]  Frantisek Matús,et al.  Infinitely Many Information Inequalities , 2007, 2007 IEEE International Symposium on Information Theory.

[22]  Nikolai K. Vereshchagin,et al.  A new class of non-Shannon-type inequalities for entropies , 2002, Commun. Inf. Syst..

[23]  J. Orlin.  Contentment in graph theory: Covering graphs with cliques , 1977 .