The Complexity of Embedding Orders into Small Products of Chains

Embedding a partially ordered set into a product of chains is a classical way to encode it. Such encodings have been used in various fields such as object oriented programming or distributed computing. The embedding associates with each element a sequence of integers which is used to perform comparisons between elements. A critical measure is the space required by the encoding, and several authors have investigated ways to minimize it, which comes to embedding partially ordered sets into small products of chains. The minimum size of such an encoding has been called the encoding dimension [24], and the string dimension [22] for a slightly different definition of embeddings. This paper investigates some new properties of the encoding dimension. We clear up the links with the string dimension and we answer the computational complexity questions raised in [22] and [24]: both these parameters are NP-hard to compute.