  An interference alignment perspective is used to identify the simplest
instances (minimum possible number of edges in the alignment graph, no more
than 2 interfering messages at any destination) of index coding problems where
non-Shannon information inequalities are necessary for capacity
characterization. In particular, this includes the first known example of a
multiple unicast (one destination per message) index coding problem where
non-Shannon information inequalities are shown to be necessary. The simplest
multiple unicast example has 7 edges in the alignment graph and 11 messages.
The simplest multiple groupcast (multiple destinations per message) example has
6 edges in the alignment graph, 6 messages, and 10 receivers. For both the
simplest multiple unicast and multiple groupcast instances, the best outer
bound based on only Shannon inequalities is $\frac{2}{5}$, which is tightened
to $\frac{11}{28}$ by the use of the Zhang-Yeung non-Shannon type information
inequality, and the linear capacity is shown to be $\frac{5}{13}$ using the
Ingleton inequality. Conversely, identifying the minimal challenging aspects of
the index coding problem allows an expansion of the class of solved index
coding problems up to (but not including) these instances.
