Optimizing alphabet reduction pairs of arrays

TL;DR

This paper introduces and analyzes optimal combinatorial designs called alphabet reduction pairs of arrays (ARPAs) and cover pairs of arrays (CPAs), which are related to approximating constraint satisfaction problems with bounded arity.

Contribution

It establishes the equivalence of ARPAs and CPAs in maximizing specific word frequencies and proves the optimality of certain ARPAs for key parameter cases.

Findings

Optimal ARPAs are proven for the case p=k.

ARPAs and CPAs are equivalent in maximizing specific word frequencies.

The paper provides optimal designs for k=1 and k=2 cases.

Abstract

In [1], we introduced a family of combinatorial designs, which we call "alphabet reduction pairs of arrays", ARPAs for short. These designs depend on three integer parameters $q, p \leq q, k\leq p$: $q$ is the size of the symbol set $\{0, 1 ,\ldots, q -1\}$ in which the coefficients of the arrays take their values; $p$ is the maximum number of distinct symbols allowed in a row of the second array of the pair; $k$ is the larger integer for which the two arrays of the pair coincide -- up to the order of their rows -- on any $k$-ary subset of their columns. The first array must contain at least one occurrence of the word $0\ 1\ \ldots\ q -1$. Intuitively, the idea is to cover "as many as possible" occurrences of this word of $q$ symbols with "as few as possible" words of at most $p$ different symbols. These designs are related to the approximability of "Constraint Satisfaction Problems with bounded constraint arity", known as $k\,$CSPs. In this context, we are particularly interested in ARPAs in which the frequency of the word $0\ 1\ \ldots\ q -1$ is maximal. We introduce a seemingly simpler family of combinatorial designs as "Cover pairs of arrays" (CPAs). The arrays of a CPA take Boolean coefficients, and must still coincide on any $k$-ary subset of their columns. The purpose is, as it were, to cover "as many as possible" occurrences of the word of $q$ ones using "as few as possible" $q$-length Boolean words of weight at most $p$. We show that, when it comes to maximizing the frequency of the words $0\ 1\ \ldots\ q -1$ in ARPAs and $1\ 1\ \ldots\ 1$ in CPAs, ARPAs and CPAs are equivalent. We prove the optimality of the ARPAs given in [1] for the case $p =k$. In addition, we provide optimal ARPAs for the cases $k =1$ and $k =2$. We emphasize the fact that both families of combinatorial designs are related to the approximability of $k\,$CSPs.