The rr-coloring and maximum stable set problem in hypergraphs with bounded matching number and edge size

Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting $\nu(G)$ denote the maximum size of a matching in $H$, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed $r, k, s \in \mathbb{N}$: $r$-Coloring in hypergraphs $G$ with edge size at most $k$ and $\nu(G) \leq s$; $r$-Precoloring Extension in $k$-uniform hypergraphs $G$ with $\nu(G) \leq s$; $r$-Precoloring Extension in hypergraphs $G$ with edge size at most $k$ and $\nu(G) \leq s$; Maximum Stable Set in $k$-uniform hypergraphs $G$ with $\nu(G) \leq s$; Maximum Weight Stable Set in $k$-uniform hypergraphs with $\nu(G) \leq s$; as well as partial results for $r$-Coloring in $k$-uniform hypergraphs $\nu(G) \leq s$. We then turn our attention to $2$-Coloring in 3-uniform hypergraphs with forbidden induced subhypergraphs, and give a polynomial-time algorithm when restricting the input to hypergraphs excluding a fixed one-edge hypergraph. Finally, we consider linear 3-uniform hypergraphs (in which every two edges share at most one vertex), and show that excluding an induced matching in $G$ implies that $\nu(G)$ is bounded by a constant; and that $3$-coloring linear $3$-uniform hypergraphs $G$ with $\nu(G) \leq 532$ is NP-hard

Coloring Algorithms for Graphs and Hypergraphs with Forbidden Substructures

This thesis mainly focus on complexity results of the generalized version of the $r$-Coloring Problem, the $r$-Pre-Coloring Extension Problem and the List $r$-Coloring Problem restricted to hypergraphs and ordered graphs with forbidden substructures. In the context of forbidding non-induced substructure in hypergraphs, we obtain complete complexity dichotomies of the $r$-Coloring Problem and the $r$-Pre-Coloring Extension Problem in hypergraphs with bounded edge size and bounded matching number, as well as the $r$-Pre-Coloring Extension Problem in hypergraphs with uniform edge size and bounded matching number. We also get partial complexity result of the $r$-Coloring Problem in hypergraphs with uniform edge size and bounded matching number. Additionally, we study the Maximum Stable Set Problem and the Maximum Weight Stable Set Problem in hypergraphs. We obtain complexity dichotomies of these problems in hypergraphs with uniform edge size and bounded matching number. We then give a polynomial-time algorithm of the 2-Coloring Problem restricted to the class of 3-uniform hypergraphs excluding a fixed one-edge induced subhypergraph. We also consider linear hypergraphs and show that 3-Coloring in linear 3-uniform hypergraphs with either bounded matching size or bounded induced matching size is NP-hard if the bound is a large enough constant. This thesis also contains a near-dichotomy of complexity results for ordered graphs. We prove that the List-3-Coloring Problem in ordered graphs with a forbidden induced ordered subgraph is polynomial-time solvable if the ordered subgraph contains only one edge, or it is isomorphic to some fixed ordered 3-vertex path plus isolated vertices. On the other hand, it is NP-hard if the ordered subgraph contains at least three edges, or contains a vertex of degree two and does not satisfy the polynomial-time case mentioned before, or contains two non-adjacent edges with a specific ordering. The complexity result when forbidding a few ordered subgraphs with exactly two edges is still unknown

Fake News Detection with Heterogeneous Transformer

The dissemination of fake news on social networks has drawn public need for effective and efficient fake news detection methods. Generally, fake news on social networks is multi-modal and has various connections with other entities such as users and posts. The heterogeneity in both news content and the relationship with other entities in social networks brings challenges to designing a model that comprehensively captures the local multi-modal semantics of entities in social networks and the global structural representation of the propagation patterns, so as to classify fake news effectively and accurately. In this paper, we propose a novel Transformer-based model: HetTransformer to solve the fake news detection problem on social networks, which utilises the encoder-decoder structure of Transformer to capture the structural information of news propagation patterns. We first capture the local heterogeneous semantics of news, post, and user entities in social networks. Then, we apply Transformer to capture the global structural representation of the propagation patterns in social networks for fake news detection. Experiments on three real-world datasets demonstrate that our model is able to outperform the state-of-the-art baselines in fake news detection

Complexity Dichotomy for List-5-Coloring with a Forbidden Induced Subgraph

First Published in the Journal of Discrete Mathematics in Volume 36, Issue 3, 2022, published by the Society for Industrial and Applied Mathematics (SIAM). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.For a positive integer r and graphs G and H, we denote by G+H the disjoint union of G and H and by rH the union of r mutually disjoint copies of H. Also, we say G is H-free if H is not isomorphic to an induced subgraph of G. We use Pt to denote the path on t vertices. For a fixed positive integer k, the List-k-Coloring Problem is to decide, given a graph G and a list L(v)⊆{1,…,k} of colors assigned to each vertex v of G, whether G admits a proper coloring ϕ with ϕ(v)∈L(v) for every vertex v of G, and the k-Coloring Problem is the List-k-Coloring Problem restricted to instances with L(v)={1,…,k} for every vertex v of G. We prove that, for every positive integer r, the List-5-Coloring Problem restricted to rP3-free graphs can be solved in polynomial time. Together with known results, this gives a complete dichotomy for the complexity of the List-5-Coloring Problem restricted to H-free graphs: For every graph H, assuming P≠NP, the List-5-Coloring Problem restricted to H-free graphs can be solved in polynomial time if and only if, H is an induced subgraph of either rP3 or P5+rP1 for some positive integer r. As a hardness counterpart, we also show that the k-Coloring Problem restricted to rP4-free graphs is NP-complete for all k≥5 and r≥2.This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2020-03912

Research on in-plane lateral performance of a new-type composite timber wall panel with cold-formed steel frames

A series of researches on the behaviour of a new structural system, composite timber wall panel with cold-formed steel frames, are investigated under monotonic and reversed cyclic loading. In order to improve the in-plane lateral performance of the composite timber panels, sixteen different optimized composite timber panels were proposed and tested, including increasing the thickness of the sheathings, improvement with steel X-bracings, filling with straw and advance of connection between sheathing and wood framing. The main objective of the investigation is to explore the pervasive mode of failure, determine the quantification of the improvement in lateral performance of these optimized composite timber wall panels and evaluate the benefits of each optimization during the process of failure

Identifying the Alteration Patterns of Brain Functional Connectivity in Progressive Mild Cognitive Impairment Patients: A Longitudinal Whole-Brain Voxel-Wise Degree Analysis

Patients with mild cognitive impairment (MCI) are at high risk for developing Alzheimer’s disease (AD), while some of them may remain stable over decades. The underlying mechanism is still not fully understood. In this study, we aimed to explore the connectivity differences between progressive MCI (PMCI) and stable MCI (SMCI) individuals on a whole-brain scale and on a voxel-wise basis, and we also aimed to reveal the differential dynamic alternation patterns between these two disease subtypes. The resting-state functional magnetic resonance images of PMCI and SMCI patients at baseline and year-one were obtained from the Alzheimer’s Disease Neuroimaging Initiative dataset, and the progression was determined based on a three-year follow-up. A whole-brain voxel-wise degree map that was calculated based on graph-theory was constructed for each subject, and then the cross-sectional and longitudinal analyses on the degree maps were performed between PMCI and SMCI patients. In longitudinal analyses, compared with SMCI group, PMCI group showed decreased long-range degree in the left middle occipital/supramarginal gyrus, while the short-range degree was increased in the left supplementary motor area and middle frontal gyrus and decreased in the right middle temporal pole. A significant longitudinal alteration of decreased short-range degree in the right middle occipital was found in PMCI group. Taken together with previous evidence, our current findings may suggest that PMCI, compared with SMCI, might be a severe presentation of disease along the AD continuum, and the rapidly reduced degree in the right middle occipital gyrus may have indicative value for the disease progression. Moreover, the cross-sectional comparison results and corresponding receiver-operator characteristic-curves analyses may indicate that the baseline degree difference is not a good predictor of disease progression in MCI patients. Overall, these findings may provide objective evidence and an indicator to characterize the progression-related brain connectivity changes in MCI patients

Ethylene Oxide and Cancer: Digging for the Truth

Multiple studies have shown a relationship between EO exposure and an increased risk of cancer in humans, but the results have been inconsistent. Nonetheless, the association between EO and human cancer risk, especially in terms of dose-response, is poorly understood. Examining whether or not EO exposure is linked to increased cancer risk in the basic adult population in the U.S. was the primary focus of this study. The study included data from both the 2013–14 and 2015–16 waves of the National Health and Nutrition Examination Survey (NHANES), for a total of 3,448 people. Data including demographic characteristics, medical history, and serum EO biomarkers were retrieved from Serum EO biomarker (hemoglobin adduct of EO (HbEO)) concentrations evaluated. Odds ratios (ORs) and 95% confidence intervals (CIs) were determined by multiple logistic regression. The result shows that EO with the highest concentration between 1340 and 1780(OR = 19.12, 95% CI: 1.73-211.47) is statistically significant

The r-coloring and maximum stable set problem in hypergraphs with bounded matching number and edge size

The final publication is available at Elsevier via https://doi.org/10.1016/j.disc.2023.113342. © 2023. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting v(g) denote the maximum size of a matching in H, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed r, k, s, E N: • r-COLORING in hypergraphs G with edge size at most k and v(G) < s; • r-PRECOLORING EXTENSION in k-uniform hypergraphs G with v(G) < s; • r-PRECOLORING EXTENSION in hypergraphs G with edge size at most k and v(G) < s; • MAXIMUM STABLE SET in k-uniform hypergraphs G with v(G) < s; • MAXIMUM WEIGHT STABLE SET in k-uniform hypergraphs with v(G) < s; as well as partial results for r-COLORING in k-uniform hypergraphs v(G) < s. We then turn our attention to 2-COLORING in 3-uniform hypergraphs with forbidden induced subhypergraphs, and give a polynomial-time algorithm when restricting the input to hypergraphs excluding a fixed one-edge hypergraph. Finally, we consider linear 3-uniform hypergraphs (in which every two edges share at most one vertex), and show that excluding an induced matching in G implies that v(G) is bounded by a constant; and that 3-coloring linear 3-uniform hypergrpahs G with v(G) < 532 is NP-hard.Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2020-0391

List-3-Coloring ordered graphs with a forbidden induced subgraph

The List-3-Coloring Problem is to decide, given a graph $G$ and a list $L(v)\subseteq \{1,2,3\}$ of colors assigned to each vertex $v$ of $G$, whether $G$ admits a proper coloring $\phi$ with $\phi(v)\in L(v)$ for every vertex $v$ of $G$, and the $3$-Coloring Problem is the List-$3$-Coloring Problem on instances with $L(v)=\{1,2,3\}$ for every vertex $v$ of $G$. The List-$3$-Coloring Problem is a classical NP-complete problem, and it is well-known that while restricted to $H$-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph $H$), it remains NP-complete unless $H$ is isomorphic to an induced subgraph of a path. However, the current state of art is far from proving this to be sufficient for a polynomial time algorithm; in fact, the complexity of the $3$-Coloring Problem on $P_8$-free graphs (where $P_8$ denotes the eight-vertex path) is unknown. Here we consider a variant of the List-$3$-Coloring Problem called the Ordered Graph List-$3$-Coloring Problem, where the input is an ordered graph, that is, a graph along with a linear order on its vertex set. For ordered graphs $G$ and $H$, we say $G$ is $H$-free if $H$ is not isomorphic to an induced subgraph of $G$ with the isomorphism preserving the linear order. We prove, assuming $H$ to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-$3$-Coloring Problem restricted to $H$-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if $H$ has at most one edge, and remains NP-complete if $H$ has at least three edges. Moreover, in the case where $H$ has exactly two edges, we give a complete dichotomy when the two edges of $H$ share an end, and prove several NP-completeness results when the two edges of $H$ do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs