New Sufficient Algebraic Conditions for Local Consistency over Homogeneous Structures of Finite Duality

2025-02-04

AI Summary

This work investigates the algebraic characterization of local consistency and polynomial-time solvability for infinite-domain constraint satisfaction problems (CSPs), focusing on the subclass of homogeneous structures with finite duality within the Bodirsky–Pinsker conjecture framework. Method: Combining universal algebra, Maltsev condition analysis, model theory of homogeneous structures, and CSP width theory, we employ a finite/infinite domain comparison strategy to analyze height-1 Maltsev conditions. Contribution/Results: We establish, for the first time in this setting, that a specific height-1 Maltsev condition implies bounded width—thereby guaranteeing polynomial-time solvability—and thus overcome prior algebraic characterizations limited to bounded strict width. Our result yields the first nontrivial sufficient condition for polynomial-time tractability applicable to multiple important infinite-domain templates, including several whose complexity classifications remain open. This significantly advances the complexity classification program for infinite-domain CSPs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The path to the solution of Feder-Vardi dichotomy conjecture by Bulatov and Zhuk led through showing that more and more general algebraic conditions imply polynomial-time algorithms for the finite-domain Constraint Satisfaction Problems (CSPs) whose templates satisfy them. These investigations resulted in the discovery of the appropriate height 1 Maltsev conditions characterizing bounded strict width, bounded width, the applicability of the few-subpowers algorithm, and many others. For problems in the range of the similar Bodirsky-Pinsker conjecture on infinite-domain CSPs, one can only find such a characterization for the notion of bounded strict width, with a proof essentially the same as in the finite case. In this paper, we provide the first non-trivial results showing that certain height 1 Maltsev conditions imply bounded width, and in consequence tractability, for a natural subclass of templates within the Bodirsky-Pinsker conjecture which includes many templates in the literature as well as templates for which no complexity classification is known.

Problem

• Algebraic conditions for CSP tractability
• Height 1 Maltsev conditions analysis
• Bodirsky-Pinsker conjecture subclass study

Innovation

• Height 1 Maltsev conditions
• Bounded width characterization
• Tractability for infinite-domain CSPs