Optimal Inapproximability of Promise Equations over Finite Groups

2024-11-03

arXiv.org

AI Summary

This paper investigates the approximation hardness of 3-LIN over an arbitrary finite group (G) under subgroup commitments: specifically, whether random assignment remains optimal—achieving the best possible polynomial-time approximation ratio—when constraints are restricted to proper subgroups of (G). Method: Building on the PCP theorem, group representation theory, algebraic coding, and Fourier analysis over finite groups, we construct a tight reduction that generalizes Håstad’s and Engebretsen et al.’s NP-hardness-of-approximation results to the subgroup-commitment model for general finite groups. Contribution/Results: We rigorously prove that, for any finite group (G), no polynomial-time algorithm can satisfy a 3-LIN equation over (G) with probability exceeding (1/|G|); this bound is precisely matched by the performance of random assignment. Consequently, random assignment is information-theoretically optimal in this setting. This yields the first universal characterization of approximation thresholds for group-constrained CSPs, establishing tight hardness for subgroup-committed 3-LIN across all finite groups.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Large-scale security measurements

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.

Problem

• Optimal inapproximability of promise equations over finite groups
• Hardness of approximating 3-LIN instances under strong promises
• Random assignment optimality for almost-satisfiable 3-LIN instances

Innovation

• Optimal inapproximability for promise equations
• Random assignment best for almost-satisfiable instances
• Extends results to restrictive group constraints