Proof of the Goldberg-Seymour Conjecture on Edge-Colorings of Multigraphs

Given a multigraph $G=(V,E)$, the {\em edge-coloring problem} (ECP) is to color the edges of $G$ with the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer program, and its linear programming relaxation is called the {\em fractional edge-coloring problem} (FECP). In the literature, the optimal value of ECP (resp. FECP) is called the {\em chromatic index} (resp. {\em fractional chromatic index}) of $G$, denoted by $\chi'(G)$ (resp. $\chi^*(G)$). Let $\Delta(G)$ be the maximum degree of $G$ and let $$\Gamma(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm {\rm and \hskip 2mm odd} \Big\},$$ where $E(U)$ is the set of all edges of $G$ with both ends in $U$. Clearly, $\max\{\Delta(G), \, \lceil \Gamma(G) \rceil \}$ is a lower bound for $\chi'(G)$. As shown by Seymour, $\chi^*(G)=\max\{\Delta(G), \, \Gamma(G)\}$. In the 1970s Goldberg and Seymour independently conjectured that $\chi'(G) \le \max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$. Over the past four decades this conjecture, a cornerstone in modern edge-coloring, has been a subject of extensive research, and has stimulated a significant body of work. In this paper we present a proof of this conjecture. Our result implies that, first, there are only two possible values for $\chi'(G)$, so an analogue to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in graph theory, holds for multigraphs; second, although it is $NP$-hard in general to determine $\chi'(G)$, we can approximate it within one of its true value, and find it exactly in polynomial time when $\Gamma(G)>\Delta(G)$; third, every multigraph $G$ satisfies $\chi'(G)-\chi^*(G) \le 1$, so FECP has a fascinating integer rounding property

Automated Measurement of Midline Shift in Brain CT Images and its Application in Computer-Aided Medical Decision Making

The severity of traumatic brain injury (TBI) is known to be characterized by the shift of the middle line in brain as the ventricular system often changes in size and position, depending on the location of the original injury. In this thesis, the focus is given to processing of the CT (Computer Tomography) brain images to automatically calculate midline shift in pathological cases and use it to predict Intracranial Pressure (ICP). The midline shift measurement can be divided into three steps. First the ideal midline of the brain, i.e., the midline before injury, is found via a hierarchical search based on skull symmetry and tissue features. Second, the ventricular system is segmented from the brain CT slices. Third, the actual midline is estimated from the deformed ventricles by shape matching method. The horizontal shift in the ventricles is then calculated based on the ideal midline and the actual midline in TBI CT images. The proposed method presents accurate detection of the ideal midline using anatomical features in the skull, accurate segmentation of ventricles for actual midline estimation using the information of anatomical features with a spatial template derived from a magnetic resonance imaging (MRI) scan, and an accurate estimation of the actual midline based on the robust proposed multiple regions shape matching algorithm. After the midline shift is successively measured, features including midline shift, texture information of CT images, as well as other demographic information are used to predict ICP. Machine learning algorithms are used to model the relation between the ICP and the extracted features. By using systematic feature selection and parameter selection of the learning model, promising results on ICP prediction are achieved. The prediction results also indicate the reliability of the proposed midline shift estimation

Ranking tournaments with no errors I: Structural description

In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments

Ranking tournaments with no errors II: Minimax relation

A tournament T=(V,A) is called cycle Mengerian (CM) if it satisfies the minimax relation on packing and covering cycles, for every nonnegative integral weight function defined on A. The purpose of this series of two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In the first paper we have given a structural description of all Möbius-free tournaments, and have proved that every CM tournament is Möbius-free. In this second paper we establish the converse by using our structural theorems and linear programming approach

Metabolic signaling directs the reciprocal lineage decisions of αβ and γδ T cells

Wiring metabolic signaling circuits in thymocytes Cell differentiation is often accompanied by metabolic changes. Yang et al. report that generation of double-positive (DP) thymocytes from double-negative (DN) cells coincides with dynamic regulation of glycolytic and oxidative metabolism. Given the central role of mechanistic target of rapamycin complex 1 (mTORC1) signaling in regulating metabolic changes, they examined the role of mTORC1 pathway in thymocyte development by conditionally deleting RAPTOR, the key component of the mTORC1 complex, in thymocytes. Loss of RAPTOR impaired the DN-to-DP transition, but unexpectedly also perturbed the balance between αβ and γδ T cells and promoted the generation of γδ T cells. Their studies highlight an unappreciated role for mTORC1-dependent metabolic changes in controlling thymocyte fates. The interaction between extrinsic factors and intrinsic signal strength governs thymocyte development, but the mechanisms linking them remain elusive. We report that mechanistic target of rapamycin complex 1 (mTORC1) couples microenvironmental cues with metabolic programs to orchestrate the reciprocal development of two fundamentally distinct T cell lineages, the αβ and γδ T cells. Developing thymocytes dynamically engage metabolic programs including glycolysis and oxidative phosphorylation, as well as mTORC1 signaling. Loss of RAPTOR-mediated mTORC1 activity impairs the development of αβ T cells but promotes γδ T cell generation, associated with disrupted metabolic remodeling of oxidative and glycolytic metabolism. Mechanistically, we identify mTORC1-dependent control of reactive oxygen species production as a key metabolic signal in mediating αβ and γδ T cell development, and perturbation of redox homeostasis impinges upon thymocyte fate decisions and mTORC1-associated phenotypes. Furthermore, single-cell RNA sequencing and genetic dissection reveal that mTORC1 links developmental signals from T cell receptors and NOTCH to coordinate metabolic activity and signal strength. Our results establish mTORC1-driven metabolic signaling as a decisive factor for reciprocal αβ and γδ T cell development and provide insight into metabolic control of cell signaling and fate decisions. Development of αβ and γδ T cells requires coupling of environmental signals with metabolic and redox regulation by mTORC1. Development of αβ and γδ T cells requires coupling of environmental signals with metabolic and redox regulation by mTORC1

Dynamic spin-lattice coupling and nematic fluctuations in NaFeAs

We use inelastic neutron scattering to study acoustic phonons and spin excitations in single crystals of NaFeAs, a parent compound of iron pnictide superconductors. NaFeAs exhibits a tetragonal-to-orthorhombic structural transition at $T_s\approx 58$ K and a collinear antiferromagnetic (AF) order at $T_N\approx 45$ K. While longitudinal and out-of-plane transverse acoustic phonons behave as expected, the in-plane transverse acoustic phonons reveal considerable softening on cooling to $T_s$, and then harden on approaching $T_N$ before saturating below $T_N$. In addition, we find that spin-spin correlation lengths of low-energy magnetic excitations within the FeAs layer and along the $c$-axis increase dramatically below $T_s$, and show weak anomaly across $T_N$. These results suggest that the electronic nematic phase present in the paramagnetic tetragonal phase is closely associated with dynamic spin-lattice coupling, possibly arising from the one-phonon-two-magnon mechanism

MethylPCA: a toolkit to control for confounders in methylome-wide association studies

Background In methylome-wide association studies (MWAS) there are many possible differences between cases and controls (e.g. related to life style, diet, and medication use) that may affect the methylome and produce false positive findings. An effective approach to control for these confounders is to first capture the major sources of variation in the methylation data and then regress out these components in the association analyses. This approach is, however, computationally very challenging due to the extremely large number of methylation sites in the human genome. Result We introduce MethylPCA that is specifically designed to control for potential confounders in studies where the number of methylation sites is extremely large. MethylPCA offers a complete and flexible data analysis including 1) an adaptive method that performs data reduction prior to PCA by empirically combining methylation data of neighboring sites, 2) an efficient algorithm that performs a principal component analysis (PCA) on the ultra high-dimensional data matrix, and 3) association tests. To accomplish this MethylPCA allows for parallel execution of tasks, uses C++ for CPU and I/O intensive calculations, and stores intermediate results to avoid computing the same statistics multiple times or keeping results in memory. Through simulations and an analysis of a real whole methylome MBD-seq study of 1,500 subjects we show that MethylPCA effectively controls for potential confounders. Conclusions MethylPCA provides users a convenient tool to perform MWAS. The software effectively handles the challenge in memory and speed to perform tasks that would be impossible to accomplish using existing software when millions of sites are interrogated with the sample sizes required for MWAS

Polygenic risk scores for major depressive disorder and neuroticism as predictors of antidepressant response:Meta-analysis of three treatment cohorts

There are currently no reliable approaches for correctly identifying which patients with major depressive disorder (MDD) will respond well to antidepressant therapy. However, recent genetic advances suggest that Polygenic Risk Scores (PRS) could allow MDD patients to be stratified for antidepressant response. We used PRS for MDD and PRS for neuroticism as putative predictors of antidepressant response within three treatment cohorts: The Genome-based Therapeutic Drugs for Depression (GENDEP) cohort, and 2 sub-cohorts from the Pharmacogenomics Research Network Antidepressant Medication Pharmacogenomics Study PRGN-AMPS (total patient number = 760). Results across cohorts were combined via meta-analysis within a random effects model. Overall, PRS for MDD and neuroticism did not significantly predict antidepressant response but there was a consistent direction of effect, whereby greater genetic loading for both MDD (best MDD result, p < 5*10–5 MDD-PRS at 4 weeks, β = -0.019, S.E = 0.008, p = 0.01) and neuroticism (best neuroticism result, p < 0.1 neuroticism-PRS at 8 weeks, β = -0.017, S.E = 0.008, p = 0.03) were associated with less favourable response. We conclude that the PRS approach may offer some promise for treatment stratification in MDD and should now be assessed within larger clinical cohorts