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Ideal Membership Problem for Boolean Minority and Dual Discriminator

Arpitha P. Bharathi, Monaldo Mastrolilli

TL;DR

This paper proves that the IMP_d for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time, and proves the identification of the tractability for the Boolean IMP$_d.

Abstract

We consider the polynomial Ideal Membership Problem (IMP) for ideals encoding combinatorial problems that are instances of CSPs over a finite language. In this paper, the input polynomial $f$ has degree at most $d=O(1)$ (we call this problem IMP$_d$). We bridge the gap in \cite{MonaldoMastrolilli2019} by proving that the IMP$_d$ for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This completes the identification of the tractability for the Boolean IMP$_d$. We also prove that the proof of membership for the IMP$_d$ for problems constrained by the dual discriminator polymorphism over any finite domain can be found in polynomial time. Our results can be used in applications such as Nullstellensatz and Sum-of-Squares proofs.

Ideal Membership Problem for Boolean Minority and Dual Discriminator

TL;DR

This paper proves that the IMP_d for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time, and proves the identification of the tractability for the Boolean IMP$_d.

Abstract

We consider the polynomial Ideal Membership Problem (IMP) for ideals encoding combinatorial problems that are instances of CSPs over a finite language. In this paper, the input polynomial has degree at most (we call this problem IMP). We bridge the gap in \cite{MonaldoMastrolilli2019} by proving that the IMP for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This completes the identification of the tractability for the Boolean IMP. We also prove that the proof of membership for the IMP for problems constrained by the dual discriminator polymorphism over any finite domain can be found in polynomial time. Our results can be used in applications such as Nullstellensatz and Sum-of-Squares proofs.

Paper Structure

This paper contains 20 sections, 19 theorems, 26 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ideals and varieties
  4. CSP and polymorphisms
  5. The ideal-CSP correspondence
  6. Gröbner bases
  7. The ideal membership problem IMP(Gamma)
  8. Boolean Minority
  9. Overiew of the proof of \ref{['thm:MainTheorem']}.
  10. Gröbner bases in lex order
  11. Computing a truncated Gröbner basis
  12. Expansion of a product of Boolean functions
  13. Our conversion algorithm
  14. Dual discriminator
  15. Permutation constraints
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\Gamma$ be a constraint language over the Boolean domain that is closed under the minority polymorphism. For each instance $\mathcal{C}$ of $\textsc{CSP}(\Gamma)$, the $d$-truncated reduced Gröbner basis in the graded lexicographic monomial ordering of the combinatorial ideal $\emph{I}_\mathcal

Theorems & Definitions (49)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Cox, Th.15, p.196
  • Definition 2.5
  • Corollary 2.6: Cox, Cor.3, p.190
  • ...and 39 more