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Optimal Inapproximability of Promise Equations over Finite Groups

Silvia Butti, Alberto Larrauri, Stanislav Živný

TL;DR

The paper proves that beating the random assignment threshold $1/|\mathrm{Im}(\varphi)|$ for promise 3-LIN over finite groups remains NP-hard, even when the instance is almost satisfiable over a restrictive subgroup/quotient via a homomorphism $\varphi$. The authors achieve this via a reduction from Gap Label Cover, combining folding over subgroups, non-Abelian Fourier analysis, and a novel use of penalized characters to handle the trivial Fourier term, plus a robust low/high-degree analysis enabled by a noise model. They situate the result within the algebraic theory of promise CSPs by relating the construction to valued minions and plurimorphisms, showing a homomorphism to the Gap Label Cover world and thus NP-hardness within the Max-PCSP framework. The work extends optimal inapproximability results from Abelian and non-Abelian groups to a broad promise-constraint setting, providing a tight, algebraically grounded understanding of when random assignments are information-theoretically optimal. This synthesis advances the PCP/Promise-PCSP landscape by bridging Fourier-analytic reductions in group-theoretic CSPs with universal-algebraic reductions via minions.

Abstract

A celebrated result of Hastad established that, for any constant $\varepsilon>0$, it is NP-hard to find an assignment satisfying a $(1/|G|+\varepsilon)$-fraction of the constraints of a given 3-LIN instance over an Abelian group $G$ even if one is promised that an assignment satisfying a $(1-\varepsilon)$-fraction of the constraints exists. Engebretsen, Holmerin, and Russell showed the same result for 3-LIN instances over any finite (not necessarily Abelian) group. In other words, for almost-satisfiable instances of 3-LIN the random assignment achieves an optimal approximation guarantee. We prove that the random assignment algorithm is still best possible under a stronger promise that the 3-LIN instance is almost satisfiable over an arbitrarily more restrictive group.

Optimal Inapproximability of Promise Equations over Finite Groups

TL;DR

The paper proves that beating the random assignment threshold for promise 3-LIN over finite groups remains NP-hard, even when the instance is almost satisfiable over a restrictive subgroup/quotient via a homomorphism . The authors achieve this via a reduction from Gap Label Cover, combining folding over subgroups, non-Abelian Fourier analysis, and a novel use of penalized characters to handle the trivial Fourier term, plus a robust low/high-degree analysis enabled by a noise model. They situate the result within the algebraic theory of promise CSPs by relating the construction to valued minions and plurimorphisms, showing a homomorphism to the Gap Label Cover world and thus NP-hardness within the Max-PCSP framework. The work extends optimal inapproximability results from Abelian and non-Abelian groups to a broad promise-constraint setting, providing a tight, algebraically grounded understanding of when random assignments are information-theoretically optimal. This synthesis advances the PCP/Promise-PCSP landscape by bridging Fourier-analytic reductions in group-theoretic CSPs with universal-algebraic reductions via minions.

Abstract

A celebrated result of Hastad established that, for any constant , it is NP-hard to find an assignment satisfying a -fraction of the constraints of a given 3-LIN instance over an Abelian group even if one is promised that an assignment satisfying a -fraction of the constraints exists. Engebretsen, Holmerin, and Russell showed the same result for 3-LIN instances over any finite (not necessarily Abelian) group. In other words, for almost-satisfiable instances of 3-LIN the random assignment achieves an optimal approximation guarantee. We prove that the random assignment algorithm is still best possible under a stronger promise that the 3-LIN instance is almost satisfiable over an arbitrarily more restrictive group.

Paper Structure

This paper contains 23 sections, 36 theorems, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Paper structure
  4. Background
  5. Groups and group homomorphisms
  6. Label Cover
  7. Linear Algebra
  8. Fourier Analysis over non-Abelian Groups
  9. Fourier Analysis over Direct Products
  10. Overview of Results
  11. Reduction
  12. Proof Outline
  13. Proof of \ref{['th:main']}
  14. Completeness
  15. Soundness
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\mathscr{G}_1$ and $\mathscr{G}_2$ be finite groups, $\mathscr{H}_1$ a subgroup of $\mathscr{G}_1$, and $\varphi$ a group homomorphism from $\mathscr{H}_1$ to $\mathscr{H}_2=\textnormal{Im}(\varphi)$, which is a subgroup of $\mathscr{G}_2$. For arbitrary $\epsilon,\delta>0$, given a system of l

Figures (1)

  • Figure 1: The sampling procedure for $\Phi_\Sigma$.

Theorems & Definitions (53)

  • Theorem : Main theorem, informal statement
  • Theorem 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Lemma 2.5
  • Lemma 2.6: Terras_1999
  • Definition 2.7
  • Lemma 2.8: Terras_1999
  • ...and 43 more