Optimizing Alphabet Reduction Pairs of Arrays.

An Alphabet Reduction Pair of Arrays (ARPA) on a set Σ _q of q symbols consists of two arrays of q columns each. The first array contains a target word composed of all symbols from Σ _q . For some p< q , each row of the second array contains at most p distinct symbols from Σ _q . For some k≤ p , the two arrays, when restricted to any subset of k columns, yield the same multiset of rows. ARPAs were introduced in the study of Constraint Satisfaction Problems with bounded constraint arity ( k -CSPs) to transfer positive differential approximation results from instances over alphabets of size p≥ k to instances over alphabets of larger size q . In this context, the goal is to maximize the frequency of the target word in the first array. This work focuses on ARPAs that achieve this maximum frequency, which we refer to as optimal ARPAs. We show that the frequency of the target word in an optimal ARPA can be determined by solving a linear program with Θ (q) continuous variables and Θ (k) constraints. In addition, we prove the optimality of previously known ARPAs for the case p=k and provide new optimal constructions for the cases k=1 and k=2 . These results are obtained by relating ARPAs to simpler objects called Cover Pairs of Arrays (CPAs), which partially encode the structure of ARPAs in Boolean terms. This connection highlights the relevance of CPAs in the study of the approximability of k -CSPs.