On Critical Index Coding Problems

The question of under what condition some side information for index coding can be removed without affecting the capacity region is studied, which was originally posed by Tahmasbi, Shahrasbi, and Gohari. To answer this question, the notion of unicycle for the side information graph is introduced and it is shown that any edge that belongs to a unicycle is critical, namely, it cannot be removed without reducing the capacity region. Although this sufficient condition for criticality is not necessary in general, a partial converse is established, which elucidates the connection between the notion of unicycle and the maximal acylic induced subgraph outer bound on the capacity region by Bar-Yossef, Birk, Jayram, and Kol.Comment: 5 pages, accepted to 2015 IEEE Information Theory Workshop (ITW), Jeju Island, Kore

Structural Properties of Index Coding Capacity Using Fractional Graph Theory

The capacity region of the index coding problem is characterized through the notion of confusion graph and its fractional chromatic number. Based on this multiletter characterization, several structural properties of the capacity region are established, some of which are already noted by Tahmasbi, Shahrasbi, and Gohari, but proved here with simple and more direct graph-theoretic arguments. In particular, the capacity region of a given index coding problem is shown to be simple functionals of the capacity regions of smaller subproblems when the interaction between the subproblems is none, one-way, or complete.Comment: 5 pages, to appear in the 2015 IEEE International Symposium on Information Theory (ISIT

Graph Theory versus Minimum Rank for Index Coding

We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic factor from the chromatic number. This is in striking contrast to minrank since prior work has shown that it can outperform the chromatic number by a polynomial factor in some cases. The conclusion is that all known graph theoretic bounds are not much stronger than the chromatic number.Comment: 8 pages, 2 figures. Submitted to ISIT 201