Ineffectiveness for Search and Undecidability of PCSP Meta-Problems

It is an open question whether the search and decision versions of promise CSPs are equivalent. Most known algorithms for PCSPs solve only their \emph{decision} variant, and it is unknown whether they can be adapted to solve \emph{search} as well. The main approaches, called BLP, AIP and BLP+AIP, handle a PCSP by finding a solution to a relaxation of some integer program. We prove that rounding those solutions to a proper search certificate can be as hard as any problem in the class TFNP. In other words, these algorithms are ineffective for search. Building on the algebraic approach to PCSPs, we find sufficient conditions that imply ineffectiveness for search. Our tools are tailored to algorithms that are characterized by minions in a suitable way, and can also be used to prove undecidability results for meta-problems. This way, we show that the families of templates solvable via BLP, AIP, and BLP+AIP are undecidable. Using the same techniques we also analyze several algebraic conditions that are known to guarantee the tractability of finite-template CSPs. We prove that several meta-problems related to cyclic polymorphims and WNUs are undecidable for PCSPs. In particular, there is no algorithm deciding whether a finite PCSP template (1) admits cyclic a polymorphism, (2) admits a WNU.