Revisiting valued CSP dichotomy and optimal approximation under UGC

Revisit and adapt the valued CSP dichotomy (without fixed threshold) and Raghavendra’s optimal approximation result for MaxCSPs to the valued PCSP framework, determining whether their conclusions can be derived or reformulated via valued minion homomorphisms.

Background

The crisp CSP dichotomy and Raghavendra’s UGC-based optimality are cornerstones of CSP approximability. Extending or aligning these with the valued PCSP algebraic approach would connect the two major strands (algebraic and analytical) more tightly.

Such a revisitation could yield unified classifications or reveal inherent limitations of the current homomorphism framework.

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