Zero-error source coding with maximum distortion criterion

Let finite source and reproduction alphabets X and Y and a distortion measure d: X×Y→[0,∞) be given. We study the minimum asymptotic rate required to describe a source distributed over X within a (given) distortion threshold D at every sample. The problem is hence a min-max problem, and the distortion measure is extended to vectors as follows: for xⁿ∈Xⁿ, yⁿ∈Yⁿ, d(xⁿ, yⁿ)=max_id(x_i, y_i). In the graph-theoretic formulation we introduce, a code for the problem is a dominating set of an equivalent distortion graph. We introduce a linear programming lower bound for the minimum dominating set size of an arbitrary graph, and show that this bound is also the minimum asymptotic rate required for the corresponding source. Turning then to the optimality of scalar coding, we show that scalar codes are asymptotically optimal if the underlying graph is either an interval graph or a tree.