We investigate certain optimization problems for Shannon information
measures, namely, minimization of joint and conditional entropies H(X,Y),
H(X∣Y), H(Y∣X), and maximization of mutual information I(X;Y), over
convex regions. When restricted to the so-called transportation polytopes (sets
of distributions with fixed marginals), very simple proofs of NP-hardness are
obtained for these problems because in that case they are all equivalent, and
their connection to the well-known \textsc{Subset sum} and \textsc{Partition}
problems is revealed. The computational intractability of the more general
problems over arbitrary polytopes is then a simple consequence. Further, a
simple class of polytopes is shown over which the above problems are not
equivalent and their complexity differs sharply, namely, minimization of
H(X,Y) and H(Y∣X) is trivial, while minimization of H(X∣Y) and
maximization of I(X;Y) are strongly NP-hard problems. Finally, two new
(pseudo)metrics on the space of discrete probability distributions are
introduced, based on the so-called variation of information quantity, and
NP-hardness of their computation is shown.Comment: IEEE Information Theory Workshop (ITW) 201