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 $h:V\to \{0,1,\ldots,d-1\}$ such that $h(e)\in R$ for every $e\in E$? This can be cast as a promise constraint satisfaction problem (PCSP) of the form $\operatorname{PCSP}(1-in-3,\mathbf{B})$, where $\mathbf{B}$ defines the relation $R$, and is an example of $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$, where $\mathbf{A}$ (and thus wlog also $\mathbf{B}$) is symmetric. The computational complexity of such problems is understood for $\mathbf{A}$ and $\mathbf{B}$ on Boolean domains by the work of Ficak, Kozik, Ol\v{s}\'{a}k, and Stankiewicz [ICALP'19]. As our first result, we establish a dichotomy for $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$, where $\mathbf{A}$ is Boolean and symmetric and $\mathbf{B}$ is functional (on a domain of any size); i.e, all but one element of any tuple in a relation in $\mathbf{B}$ determine the last element. This includes PCSPs of the form $\operatorname{PCSP}(q-in-r,\mathbf{B})$, where $\mathbf{B}$ is functional, thus making progress towards a classification of $\operatorname{PCSP}(1-in-3,\mathbf{B})$, which were studied by Barto, Battistelli, and Berg [STACS'21] for $\mathbf{B}$ on three-element domains. As our second result, we show that for $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$, where $\mathbf{A}$ contains a single Boolean symmetric relation and $\mathbf{B}$ 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]

On the complexity of symmetric vs. functional PCSPs

The complexity of the promise constraint satisfaction problem PCSP(A, B) is largely unknown, even for symmetric A and B, except for the case when A and B are Boolean. First, we establish a dichotomy for PCSP(A, B) where A, B are symmetric, B is functional (i.e. any r − 1 elements of an r-ary tuple uniquely determines the last one), and (A, B) satisfies technical conditions we introduce called dependency and additivity. This result implies a dichotomy for PCSP(A, B) with A, B symmetric and B functional if (i) A is Boolean, or (ii) A is a hypergraph of a small uniformity, or (iii) A has a relation RA of arity at least 3 such that the hypergraph diameter of (A, RA) is at most 1. Second, we show that for PCSP(A, B), where A and B contain a single relation, A satisfies a technical condition called balancedness, and B is arbitrary, the combined basic linear programming relaxation (BLP) and the affine integer programming relaxation (AIP) is no more powerful than the (in general strictly weaker) AIP relaxation. Balanced A include symmetric A or, more generally, A preserved by a transitive permutation group

Approximate Graph Colouring and the Hollow Shadow

We show that approximate graph colouring is not solved by constantly many levels of the lift-and-project hierarchy for the combined basic linear programming and affine integer programming relaxation. The proof involves a construction of tensors whose fixed-dimensional projections are equal up to reflection and satisfy a sparsity condition, which may be of independent interest.Comment: Generalises and subsumes results from Section 6 in arXiv:2203.02478; builds on and generalises results in arXiv:2210.0829

The complexity of promise SAT on non-Boolean domains

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin, Guruswami, and Håstad [FOCS'14/SICOMP'17] proved a result known as "(2+ε)-SAT is NP-hard". They showed that the problem of distinguishing k-CNF formulas that are g-satisfiable (i.e. some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if g/k < 1/2 and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus we give a dichotomy for a natural fragment of promise constraint satisfaction problems (PCSPs) on arbitrary finite domains

The complexity of promise SAT on non-Boolean domains

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin et al. [SICOMP’17] proved a result known as “(2+ɛ)-SAT is NP-hard.” They showed that the problem of distinguishing k-CNF formulas that are g-satisfiable (i.e., some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if g/k < 1/2 and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus, we give a dichotomy for a natural fragment of promise constraint satisfaction problems (PCSPs) on arbitrary finite domains. The hardness side is proved using the algebraic approach via a new general NP-hardness criterion on polymorphisms, which is based on a gap version of the Layered Label Cover problem. We show that previously used criteria are insufficient—the problem hence gives an interesting benchmark of algebraic techniques for proving hardness of approximation in problems such as PCSPs