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

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

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]

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, … , 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)

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)

Rare Concurrence of Apical Hypertrophic Cardiomyopathy and Effusive Constrictive Pericarditis

A 78-year-old man with a history of pulmonary tuberculosis was referred for preoperative evaluation of cardiac function. Echocardiography and cardiac cine magnetic resonance imaging (MRI) indicated apical hypertrophic cardiomyopathy (HCM), a thickened visceral pericardium, and a large pericardial effusion. Cardiac late gadolinium-enhanced MRI revealed pericardial inflammation or fibrosis. Apical HCM with concurrent effusive constrictive pericarditis was diagnosed. Further studies are required to elucidate the pathophysiology of this condition

Validation of Two MODIS Aerosols Algorithms with SKYNET and Prospects for Future Climate Satellites Such as the GCOM-C/SGLI

Potential improvements of aerosols algorithms for future climate-oriented satellites such as the coming Global Change Observation Mission Climate/Second generation Global Imager (GCOM-C/SGLI) are discussed based on a validation study of three years’ (2008–2010) daily aerosols properties, that is, the aerosol optical thickness (AOT) and the Ångström exponent (AE) retrieved from two MODIS algorithms. The ground-truth data used for this validation study are aerosols measurements from 3 SKYNET ground sites. The results obtained show a good agreement between the ground-truth data AOT and that of one of the satellites’ algorithms, then a systematic overestimation (around 0.2) by the other satellites’ algorithm. The examination of the AE shows a clear underestimation (by around 0.2–0.3) by both satellites’ algorithms. The uncertainties explaining these ground-satellites’ algorithms discrepancies are examined: the cloud contamination affects differently the aerosols properties (AOT and AE) of both satellites’ algorithms due to the retrieval scale differences between these algorithms. The deviation of the real part of the refractive index values assumed by the satellites’ algorithms from that of the ground tends to decrease the accuracy of the AOT of both satellites’ algorithms. The asymmetry factor (AF) of the ground tends to increase the AE ground-satellites discrepancies as well

片側性非腫瘤性腎疾患における腎RIアンギオグラフィ

金沢大学大学院医学系研究