# Boolean symmetric vs. functional PCSP dichotomy

Given a 3-uniform hypergraph (V,E) that is promised to admit a {0,1}-colouring such that every edge contains exactly one 1, can one find a d-colouring h:V→{0,1,…,d-1} such that h(e)∈ R for every e∈ E? This can be cast as a promise constraint satisfaction problem (PCSP) of the form PCSP(1-in-3,𝐁), where 𝐁 defines the relation R, and is an example of PCSP(𝐀,𝐁), where 𝐀 (and thus wlog also 𝐁) is symmetric. The computational complexity of such problems is understood for 𝐀 and 𝐁 on Boolean domains by the work of Ficak, Kozik, Olšák, and Stankiewicz [ICALP'19]. As our first result, we establish a dichotomy for PCSP(𝐀,𝐁), where 𝐀 is Boolean and symmetric and 𝐁 is functional (on a domain of any size); i.e, all but one element of any tuple in a relation in 𝐁 determine the last element. This includes PCSPs of the form PCSP(q-in-r,𝐁), where 𝐁 is functional, thus making progress towards a classification of PCSP(1-in-3,𝐁), which were studied by Barto, Battistelli, and Berg [STACS'21] for 𝐁 on three-element domains. As our second result, we show that for PCSP(𝐀,𝐁), where 𝐀 contains a single Boolean symmetric relation and 𝐁 is arbitrary (and thus not necessarily functional), the combined basic linear programmin relaxation (BLP) and the affine integer programming relaxation (AIP) of Brakensiek et al. [SICOMP'20] is no more powerful than the (in general strictly weaker) AIP relaxation of Brakensiek and Guruswami [SICOMP'21].

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