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#ComputationalComplexity

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David Bremner

Given a decision problem Π, I need a name for the polytope constructed by taking the convex hull of the characteristic vectors (or binary inputs of fixed length) corresponding to "yes" instances. If nobody has a better suggestion I might use "the characteristic polytope of Π". OTOH, for $reasons I'd much rather use something more standard / widespread.

#ComputationalComplexity#Polytope
Joshua Grochow

Apparently I missed that Zhuk posted a *simplified* proof of the CSP Dichotomy Conjecture back in January: arxiv.org/abs/2404.01080

I'd really love to understand all of this!

arXiv.orgA simplified proof of the CSP Dichotomy Conjecture and XY-symmetric operationsWe develop a new theory of strong subalgebras and linear congruences that are defined globally. Using this theory we provide a new proof of the correctness of Zhuk's algorithm for all tractable CSPs on a finite domain, and therefore a new simplified proof of the CSP Dichotomy Conjecture. Additionally, using the new theory we prove that composing a weak near-unanimity operation of an odd arity $n$ we can derive an $n$-ary operation that is symmetric on all two-element sets. Thus, CSP over a constraint language $Γ$ on a finite domain is tractable if and only if there exist infinitely many polymorphisms of $Γ$ that are symmetric on all two-element sets.
#ComputationalComplexity#complexity#math
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