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

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]

Linearly ordered colourings of hypergraphs

A linearly ordered (LO) $k$-colouring of an $r$-uniform hypergraph assigns an integer from $\{1, \ldots, k \}$ to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for $r=3$, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied LO colourings on $3$-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results. First, given a 3-uniform hypergraph that admits an LO $2$-colouring, one can find in polynomial time an LO $k$-colouring with $k=O(\sqrt[3]{n \log \log n / \log n})$. Second, given an $r$-uniform hypergraph that admits an LO $2$-colouring, we establish NP-hardness of finding an LO $k$-colouring for every constant uniformity $r\geq k+2$. In fact, we determine relationships between polymorphism minions for all uniformities $r\geq 3$, which reveals a key difference between $r<k+2$ and $r\geq k+2$ and which may be of independent interest. Using the algebraic approach to PCSPs, we actually show a more general result establishing NP-hardness of finding an LO $k$-colouring for LO $\ell$-colourable $r$-uniform hypergraphs for $2 \leq \ell \leq k$ and $r \geq k - \ell + 4$.Comment: Full version (with stronger both tractability and intractability results) of an ICALP 2022 pape

1-in-3 vs. not-all-equal: dichotomy of a broken promise

The 1-in-3 and the Not-All-Equal satisfiability problems for Boolean CNF formulas are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs. Not-All-Equal problem can be solved in polynomial time. In the present work, we investigate this constraint satisfaction problem in a regime where the promise is weakened from either side by a rainbow-free structure, and establish a complexity dichotomy for the resulting class of computational problems

Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs

A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO $3$-colourable, and the case that it is not even LO $4$-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opr\v{s}al, Wrochna, and \v{Z}ivn\'y (2023)

Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs

A linearly ordered (LO) k-colouring of a hypergraph is a colouring of its vertices with colours 1, … , k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the "linearly ordered chromatic number" of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO 3-colourable, and the case that it is not even LO 4-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023)

A logarithmic approximation of linearly-ordered colourings

A linearly ordered (LO) -colouring of a hypergraph assigns to each vertex a colour from the set {0, 1, . . . , − 1} in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO -colouring of an LO 2-colourable 3-uniform hypergraph for any constant ≥ 2 [STACS’21] but even the case = 3 is still open. Nakajima and Zivn ˇ y gave polynomial-time algorithms for ´ finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with ∗ ( √ ) colours [ICALP’22] and an LO colouring with ∗ ( 3 √ ) colours [ACM ToCT’23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with ∗ ( 5 √ ) colours. We present two simple polynomial-time algorithms that find an LO colouring with (log2 ()) colours, which is an exponential improvement