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Characterization of gadget reductions via polymorphisms/plurimorphisms

Characterize gadget reductions (and more generally definability of constraints) between valued promise templates in terms of polymorphisms or plurimorphisms, providing algebraic criteria that are both necessary and sufficient.

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Background

Beyond individual constructions, a general algebraic characterization of when gadget reductions exist would unify multiple reduction techniques. In crisp CSPs, definability is linked to polymorphisms; extending this to valued PCSPs requires capturing weighted constraints and promises.

Such a result would directly enhance the reduction theorem’s explanatory power for standard hardness transfers in approximation.

References

There are however many basic questions and theory-building tasks left open already for finite-domain valued PCSPs: to incorporate the trivial reduction as in e.g. \cref{ex:crips-vs-valued}; to characterize gadget reductions (or versions of definability) in terms of polymorphisms or plurimorphisms; to characterize plurimorphism valued minions of templates; to clarify whether plurimorphisms are necessary to determine computational complexity or enough information is provided already by polymorphisms; to develop methods for proving nonexistence of homomorphisms; to revisit the valued CSP dichotomy without fixed threshold and Raghavendra's result on unique games hardness of approximation for all MaxCSPs; among others. An interesting special case for a full complexity classification is the valued non-promise CSPs with fixed threshold.

— The Rise of Plurimorphisms: Algebraic Approach to Approximation (2401.15186 - Barto et al., 26 Jan 2024) in Conclusion