The Complexity of the Empire Colouring Problem

We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in 1890) for maps containing empires formed by exactly $r > 1$ countries each. We prove that the problem can be solved in polynomial time using $s$ colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than $s/r$. However, if $s \geq 3$, the problem is NP-hard even if the graph is a forest of paths of arbitrary lengths (for any $r \geq 2$, provided $s < 2r - \sqrt(2r + 1/4+ 3/2)$. Furthermore we obtain a complete characterization of the problem's complexity for the case when the input graph is a tree, whereas our result for arbitrary planar graphs fall just short of a similar dichotomy. Specifically, we prove that the empire colouring problem is NP-hard for trees, for any $r \geq 2$, if $3 \leq s \leq 2r-1$ (and polynomial time solvable otherwise). For arbitrary planar graphs we prove NP-hardness if $s<7$ for $r=2$, and $s < 6r-3$, for $r \geq 3$. The result for planar graphs also proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than $6r-3$ colours graphs of thickness $r \geq 3$.Comment: 23 pages, 12 figure

Human leukocyte antigen alleles associate with COVID-19 vaccine immunogenicity and risk of breakthrough infection

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) vaccine immunogenicity varies between individuals, and immune responses correlate with vaccine efficacy. Using data from 1,076 participants enrolled in ChAdOx1 nCov-19 vaccine efficacy trials in the United Kingdom, we found that inter-individual variation in normalized antibody responses against SARS-CoV-2 spike and its receptor-binding domain (RBD) at 28 days after first vaccination shows genome-wide significant association with major histocompatibility complex (MHC) class II alleles. The most statistically significant association with higher levels of anti-RBD antibody was HLA-DQB1*06 (P = 3.2 × 10−9), which we replicated in 1,677 additional vaccinees. Individuals carrying HLA-DQB1*06 alleles were less likely to experience PCR-confirmed breakthrough infection during the ancestral SARS-CoV-2 virus and subsequent Alpha variant waves compared to non-carriers (hazard ratio = 0.63, 0.42–0.93, P = 0.02). We identified a distinct spike-derived peptide that is predicted to bind differentially to HLA-DQB1*06 compared to other similar alleles, and we found evidence of increased spike-specific memory B cell responses in HLA-DQB1*06 carriers at 84 days after first vaccination. Our results demonstrate association of HLA type with Coronavirus Disease 2019 (COVID-19) vaccine antibody response and risk of breakthrough infection, with implications for future vaccine design and implementation

Whole-genome sequencing reveals host factors underlying critical COVID-19

Critical COVID-19 is caused by immune-mediated inflammatory lung injury. Host genetic variation influences the development of illness requiring critical care1 or hospitalization2,3,4 after infection with SARS-CoV-2. The GenOMICC (Genetics of Mortality in Critical Care) study enables the comparison of genomes from individuals who are critically ill with those of population controls to find underlying disease mechanisms. Here we use whole-genome sequencing in 7,491 critically ill individuals compared with 48,400 controls to discover and replicate 23 independent variants that significantly predispose to critical COVID-19. We identify 16 new independent associations, including variants within genes that are involved in interferon signalling (IL10RB and PLSCR1), leucocyte differentiation (BCL11A) and blood-type antigen secretor status (FUT2). Using transcriptome-wide association and colocalization to infer the effect of gene expression on disease severity, we find evidence that implicates multiple genes—including reduced expression of a membrane flippase (ATP11A), and increased expression of a mucin (MUC1)—in critical disease. Mendelian randomization provides evidence in support of causal roles for myeloid cell adhesion molecules (SELE, ICAM5 and CD209) and the coagulation factor F8, all of which are potentially druggable targets. Our results are broadly consistent with a multi-component model of COVID-19 pathophysiology, in which at least two distinct mechanisms can predispose to life-threatening disease: failure to control viral replication; or an enhanced tendency towards pulmonary inflammation and intravascular coagulation. We show that comparison between cases of critical illness and population controls is highly efficient for the detection of therapeutically relevant mechanisms of disease

ON BALANCING SUPERCONDUCTING GRADIOMETRIC MAGNETOMETERS

Nous décrivons un système d'équilibrage de gradiomètres supraconducteurs et un équipement pour mesurer l'équilibrage en laboratoire avec une précision de l'ordre du ppm.A system for balancing superconducting gradiometric magnetometers and equipment for ppm-level testing of the balance in a laboratory are described

Low Discrepancy Sets Yield Approximate Min-Wise Independent Permutation Families

Motivated by a problem of filtering near-duplicate Web documents, Broder, Charikar, Frieze &amp; Mitzenmacher defined the following notion of ffl-approximate min-wise independent permutation families. A multiset F of permutations of f0; 1; : : : ; n \Gamma 1g is such a family if for all K ` f0; 1; : : : ; n \Gamma 1g and any x 2 K, a permutation chosen uniformly at random from F satisfies j Pr[minf(K)g = (x)] \Gamma 1 jKj j ffl jKj : We show connections of such families with low discrepancy sets for geometric rectangles, and give explicit constructions of such families F of size n O( p log n) for ffl = 1=n \Theta(1) , improving upon the previously best-known bound of Indyk. We also present polynomialsize constructions when the min-wise condition is required only for jKj 2 O(log 2=3 n) , with ffl 2 \GammaO(log 2=3 n) . Keywords: Combinatorial problems; min-wise independent permutations; information retrieval; document filtering; pseudorandom permutations; explicit constructions