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Let (X,Y) be a pair of random variables distributed over a finite product set V×W according to a probability distribution P(x,y). The following source coding problem is considered: the encoder knows X, while the decoder knows Y and wants to learn X without error. The minimum zero-error asymptotic rate of transmission is shown to be the complementary graph entropy of an associated graph. Thus, previous results in the literature provide upper and lower bounds for this minimum rate (further, these bounds are tight for the important class of perfect graphs). The algorithmic aspects of instantaneous code design are considered next. It is shown that optimal code design is NP-hard. An optimal code design algorithm is derived. Polynomial-time suboptimal algorithms are also presented, and their average and worst case performance guarantees are established.