The Rise of Plurimorphisms: Algebraic Approach to Approximation
Libor Barto, Silvia Butti, Alexandr Kazda, Caterina Viola, Stanislav Živný
TL;DR
The paper develops an algebraic framework for valued promise CSPs ($PCSP$s) by introducing valued minions and plurimorphisms to capture solution structure. It proves a central reduction theorem: the existence of a valued minion homomorphism implies a polynomial-time reduction between $PCSP$s, via a valued Minor Condition (VMC) problem and canonical payoff formulations. This framework unifies reductions and connects to classic inapproximability results (e.g., Hastad-type results for almost solvable linear systems) by showing how valuations and symmetries govern tractability boundaries. The work also furnishes concrete instances (gadget reductions, Gap Label Cover) and outlines a program for extending the theory toward PCP-like hardness, Unique Games, and broader infinite-structure settings.
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.