  We provide a novel upper-bound on Witsenhausen's rate, the rate required in
the zero-error analogue of the Slepian-Wolf problem; our bound is given in
terms of a new information-theoretic functional defined on a certain graph. We
then use the functional to give a single letter lower-bound on the error
exponent for the Slepian-Wolf problem under the vanishing error probability
criterion, where the decoder has full (i.e. unencoded) side information. Our
exponent stems from our new encoding scheme which makes use of source
distribution only through the positions of the zeros in the `channel' matrix
connecting the source with the side information, and in this sense is
`semi-universal'. We demonstrate that our error exponent can beat the
`expurgated' source-coding exponent of Csisz\'{a}r and K\"{o}rner,
achievability of which requires the use of a non-universal maximum-likelihood
decoder. An extension of our scheme to the lossy case (i.e. Wyner-Ziv) is
given. For the case when the side information is a deterministic function of
the source, the exponent of our improved scheme agrees with the sphere-packing
bound exactly (thus determining the reliability function). An application of
our functional to zero-error channel capacity is also given.
