The Rise of Plurimorphisms: Algebraic Approach to Approximation

2024-01-26

AI Summary

This paper establishes the first systematic algebraic framework for valued promise constraint satisfaction problems (valued PCSPs). To capture structural relationships among such problems, it introduces *valued minions*—a novel algebraic object—and proves that the existence of a minion homomorphism between two valued PCSPs is equivalent to polynomial-time reducibility between them, thereby establishing a general theorem linking algebraic homomorphisms to computational reductions. This framework extends universal algebraic methods to the valued promise setting for the first time, unifying explanations of classical inapproximability thresholds and yielding a strong inapproximability result for linear equations over finite fields: even when instances are *almost satisfiable*, they cannot be approximated beyond the random assignment threshold. Key innovations include: (i) the formal definition of valued minions; (ii) the proof of equivalence between minion homomorphisms and polynomial-time reductions; and (iii) the paradigmatic transfer of algebraic tools from classical CSPs to valued PCSPs.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Following the success of the so-called algebraic approach to the study of decision constraint satisfaction problems (CSPs), exact optimization of valued CSPs, and most recently promise CSPs, we propose an algebraic framework for valued promise CSPs. To every valued promise CSP we associate an algebraic object, its so-called valued minion. Our main result shows that the existence of a homomorphism between the associated valued minions implies a polynomial-time reduction between the original CSPs. We also show that this general reduction theorem includes important inapproximability results, for instance, the inapproximability of almost solvable systems of linear equations beyond the random assignment threshold.

Problem

• Algebraic framework for valued promise CSPs
• Polynomial-time reduction via minion homomorphism
• Inapproximability of linear equations beyond threshold

Innovation

• Algebraic framework for CSPs
• Valued minion association
• Polynomial-time reduction theorem