The rate loss of single letter characterization for the "dirty" multiple access channel

For general memoryless systems, the typical information theoretic solution, when exists, has a "single-letter" form. This reflects the fact that optimum performance can be approached by a random code (or a random binning scheme), generated using independent and identically distributed copies of some single-letter distribution. Is that the form of the solution of any (information theoretic) problem? In fact, some counter examples are known, perhaps the most famous being the Korner-Marton "two help one" problem, where the modulo-two sum of two binary sources is to be decoded from their independent encodings. In this paper we provide another counter example, the "doubly-dirty" multiple access channel (MAC). Like the Korner-Marton problem, this example is associated with a multiterminal scenario where side information is distributed among several terminals; each transmitter knows part of the channel interference but the receiver is not aware of any part of it. We give an explicit solution for the capacity region of a binary version of the doubly-dirty MAC, demonstrate how this capacity region can be approached using a linear coding scheme, and prove that the "best known single-letter region" is strictly contained in it. We also state a conjecture regarding a similar rate loss of single letter characterization in the Gaussian case.