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Optimizing alphabet reduction pairs of arrays

Jean-François Culus, Sophie Toulouse

TL;DR

It is shown 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.

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.

Optimizing alphabet reduction pairs of arrays

TL;DR

It is shown that, when it comes to maximizing the frequency of the words in ARPAs and in CPAs, ARPAs and CPAs are equivalent.

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 : is the size of the symbol set in which the coefficients of the arrays take their values; is the maximum number of distinct symbols allowed in a row of the second array of the pair; is the larger integer for which the two arrays of the pair coincide -- up to the order of their rows -- on any -ary subset of their columns. The first array must contain at least one occurrence of the word . Intuitively, the idea is to cover "as many as possible" occurrences of this word of symbols with "as few as possible" words of at most different symbols. These designs are related to the approximability of "Constraint Satisfaction Problems with bounded constraint arity", known as CSPs. In this context, we are particularly interested in ARPAs in which the frequency of the word 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 -ary subset of their columns. The purpose is, as it were, to cover "as many as possible" occurrences of the word of ones using "as few as possible" -length Boolean words of weight at most . We show that, when it comes to maximizing the frequency of the words in ARPAs and in CPAs, ARPAs and CPAs are equivalent. We prove the optimality of the ARPAs given in [1] for the case . In addition, we provide optimal ARPAs for the cases and . We emphasize the fact that both families of combinatorial designs are related to the approximability of CSPs.
Paper Structure (17 sections, 17 theorems, 89 equations, 8 tables, 1 algorithm)

This paper contains 17 sections, 17 theorems, 89 equations, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation and previous results
  3. Outline and notations
  4. Cover pairs of arrays
  5. From alphabet reduction to cover pairs of arrays
  6. Regular pairs of arrays
  7. From cover to alphabet reduction pairs of arrays
  8. Construction
  9. Proof of Theorem \ref{['thm-reg_joker']}
  10. Characterizing optimal regular CPAs through linear programming
  11. Consequences on numbers $\gamma(q, p, k)$
  12. Concluding remarks and further research
  13. Omitted proofs
  14. Proof of identity (\ref{['eq-qkk-MN-rec']}) (Theorem \ref{['thm-delta_d=k-UB']})
  15. Extended proof of Proposition \ref{['prop-reg_delta']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

For all constant integers $k\geq 2$, $p\geq k$ and $q\geq p$, on any instance $I$ of $\mathsf{k\,CSP\!-\!q}$, the best solutions among those whose components take at most $p$ distinct values are $\gamma(q, p, k)$-differential approximate. Formally, $\mathrm{opt}_p(I)$ satisfies: Moreover, $\mathsf{k\,CSP\!-\!q}$ reduces to $\mathsf{k\,CSP\!-\!p}$ with an expansion of $\gamma(q, p, k)$ on the diff

Theorems & Definitions (42)

  • Definition 1.1: Alphabet reduction pairs of arrays
  • Theorem 1.1: CT18
  • Theorem 1.2: CT18
  • Definition 1.2: Cover pair of arrays
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof : Proof (sketch)
  • Definition 2.2
  • ...and 32 more