Terrain Visibility Graphs: Persistence Is Not Enough

In this paper, we consider the Visibility Graph Recognition and Reconstruction problems in the context of terrains. Here, we are given a graph $G$ with labeled vertices $v_0, v_1, \ldots, v_{n-1}$ such that the labeling corresponds with a Hamiltonian path $H$. $G$ also may contain other edges. We are interested in determining if there is a terrain $T$ with vertices $p_0, p_1, \ldots, p_{n-1}$ such that $G$ is the visibility graph of $T$ and the boundary of $T$ corresponds with $H$. $G$ is said to be persistent if and only if it satisfies the so-called X-property and Bar-property. It is known that every "pseudo-terrain" has a persistent visibility graph and that every persistent graph is the visibility graph for some pseudo-terrain. The connection is not as clear for (geometric) terrains. It is known that the visibility graph of any terrain $T$ is persistent, but it has been unclear whether every persistent graph $G$ has a terrain $T$ such that $G$ is the visibility graph of $T$. There actually have been several papers that claim this to be the case (although no formal proof has ever been published), and recent works made steps towards building a terrain reconstruction algorithm for any persistent graph. In this paper, we show that there exists a persistent graph $G$ that is not the visibility graph for any terrain $T$. This means persistence is not enough by itself to characterize the visibility graphs of terrains, and implies that pseudo-terrains are not stretchable.Comment: To appear in SoCG 202

The Complexity of Guarding Monotone Polygons

Abstract A polygon P is x-monotone if any line orthogonal to the x-axis has a simply connected intersection with P . A set G of points inside P or on the boundary of P is said to guard the polygon if every point inside P or on the boundary of P is seen by a point in G. An interior guard can lie anywhere inside or on the boundary of the polygon. Using a reduction from Monotone 3SAT, we prove that interior guarding a monotone polygon is NP-hard. Because interior guards can be placed anywhere inside the polygon, a clever gadget is introduced that forces interior guards to be placed at very specific locations

Half-Guarding Weakly-Visible Polygons and Terrains

We consider a variant of the art gallery problem where all guards are limited to seeing 180degree. Guards that can only see in one direction are called half-guards. We give a polynomial time approximation scheme for vertex guarding the vertices of a weakly-visible polygon with half-guards. We extend this to vertex guarding the boundary of a weakly-visible polygon with half-guards. We also show NP-hardness for vertex guarding a weakly-visible polygon with half-guards. Lastly, we show that the orientation of half-guards is critical in terrain guarding. Depending on the orientation of the half-guards, the problem is either very easy (polynomial time solvable) or very hard (NP-hard)

Sequencing of prostate cancers identifies new cancer genes, routes of progression and drug targets

Prostate cancer represents a substantial clinical challenge because it is difficult to predict outcome and advanced disease is often fatal. We sequenced the whole genomes of 112 primary and metastatic prostate cancer samples. From joint analysis of these cancers with those from previous studies (930 cancers in total), we found evidence for 22 previously unidentified putative driver genes harboring coding mutations, as well as evidence for NEAT1 and FOXA1 acting as drivers through noncoding mutations. Through the temporal dissection of aberrations, we identified driver mutations specifically associated with steps in the progression of prostate cancer, establishing, for example, loss of CHD1 and BRCA2 as early events in cancer development of ETS fusion-negative cancers. Computational chemogenomic (canSAR) analysis of prostate cancer mutations identified 11 targets of approved drugs, 7 targets of investigational drugs, and 62 targets of compounds that may be active and should be considered candidates for future clinical trials

Mendelian randomization integrating GWAS and eQTL data reveals genetic determinants of complex and clinical traits

Genome-wide association studies (GWAS) have identified thousands of variants associated with complex traits, but their biological interpretation often remains unclear. Most of these variants overlap with expression QTLs, indicating their potential involvement in regulation of gene expression. Here, we propose a transcriptome-wide summary statistics-based Mendelian Randomization approach (TWMR) that uses multiple SNPs as instruments and multiple gene expression traits as exposures, simultaneously. Applied to 43 human phenotypes, it uncovers 3,913 putatively causal gene-trait associations, 36% of which have no genome-wide significant SNP nearby in previous GWAS. Using independent association summary statistics, we find that the majority of these loci were missed by GWAS due to power issues. Noteworthy among these links is educational attainment-associated BSCL2, known to carry mutations leading to a Mendelian form of encephalopathy. We also find pleiotropic causal effects suggestive of mechanistic connections. TWMR better accounts for pleiotropy and has the potential to identify biological mechanisms underlying complex traits

Refining Attention-Deficit/Hyperactivity Disorder and Autism Spectrum Disorder Genetic Loci by Integrating Summary Data From Genome-wide Association, Gene Expression, and DNA Methylation Studies

Background: Recent genome-wide association studies (GWASs) identified the first genetic loci associated with attention-deficit/hyperactivity disorder (ADHD) and autism spectrum disorder (ASD). The next step is to use these results to increase our understanding of the biological mechanisms involved. Most of the identified variants likely influence gene regulation. The aim of the current study is to shed light on the mechanisms underlying the genetic signals and prioritize genes by integrating GWAS results with gene expression and DNA methylation (DNAm) levels. Methods: We applied summary-data–based Mendelian randomization to integrate ADHD and ASD GWAS data with fetal brain expression and methylation quantitative trait loci, given the early onset of these disorders. We also analyzed expression and methylation quantitative trait loci datasets of adult brain and blood, as these provide increased statistical power. We subsequently used summary-data–based Mendelian randomization to investigate if the same variant influences both DNAm and gene expression levels. Results: We identified multiple gene expression and DNAm levels in fetal brain at chromosomes 1 and 17 that were associated with ADHD and ASD, respectively, through pleiotropy at shared genetic variants. The analyses in brain and blood showed additional associated gene expression and DNAm levels at the same and additional loci, likely because of increased statistical power. Several of the associated genes have not been identified in ADHD and ASD GWASs before. Conclusions: Our findings identified the genetic variants associated with ADHD and ASD that likely act through gene regulation. This facilitates prioritization of candidate genes for functional follow-up studies