Excluding a large theta graph

A theta graph, denoted θa,b,c, is a graph of order a+b+c−1 consisting of a pair of vertices and three internally-disjoint paths between them of lengths a, b, and c. In this paper we study graphs that do not contain a large θa,b,c minor. More specifically, we describe the structure of θ1,2,t-, θ2,2,t-, θ1,t,t-, θ2,t,t-, and θt,t,t-free graphs where t is large. The main result is a characterization of θt,t,t-free graphs for large t. The 3-connected θt,t,t-free graphs are formed by 3-summing graphs without a long path to certain planar graphs. The 2-connected θt,t,t-free graphs are then built up in a similar fashion by 2- and 3-sums. This result implies a well-known theorem of Robertson and Chakravarti on graphs that do not have a bond containing three specified edges

Induced Subgraphs and Tree Decompositions III. Three-Path-Configurations and Logarithmic Treewidth.

A theta is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family H of graphs, we say a graph G is H-free if no induced subgraph of G is isomorphic to a member of H. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant c for which every (theta, triangle)-free graph G has treewidth at most c log(jV(G)j). A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth. Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in jV(G)j for every graph G excluding the so-called three-path-configurations as well as a fixed complete graph. It follows that several NP-hard problems such as STABLE SET, VERTEX COVER, DOMINATING SET and COLORING admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.Supported by NSF Grant DMS-1763817 and NSF-EPSRC Grant DMS-2120644. The authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number RGPIN-2020-03912]

Vertex elimination orderings for hereditary graph classes

We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vskovi\'c. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-hole-free graphs, square-theta-free Berge graphs, universally signable graphs and wheel-free graphs. Some other classes are new. It turns out that all the classes that we study in this paper can be defined by excluding some of the so-called Truemper configurations. For several classes of graphs, we obtain directly bounds on the chromatic number, or fast algorithms for the maximum clique problem or the coloring problem

Efficient Multi-way Theta-Join Processing Using MapReduce

Multi-way Theta-join queries are powerful in describing complex relations and therefore widely employed in real practices. However, existing solutions from traditional distributed and parallel databases for multi-way Theta-join queries cannot be easily extended to fit a shared-nothing distributed computing paradigm, which is proven to be able to support OLAP applications over immense data volumes. In this work, we study the problem of efficient processing of multi-way Theta-join queries using MapReduce from a cost-effective perspective. Although there have been some works using the (key,value) pair-based programming model to support join operations, efficient processing of multi-way Theta-join queries has never been fully explored. The substantial challenge lies in, given a number of processing units (that can run Map or Reduce tasks), mapping a multi-way Theta-join query to a number of MapReduce jobs and having them executed in a well scheduled sequence, such that the total processing time span is minimized. Our solution mainly includes two parts: 1) cost metrics for both single MapReduce job and a number of MapReduce jobs executed in a certain order; 2) the efficient execution of a chain-typed Theta-join with only one MapReduce job. Comparing with the query evaluation strategy proposed in [23] and the widely adopted Pig Latin and Hive SQL solutions, our method achieves significant improvement of the join processing efficiency.Comment: VLDB201

Pareto Smoothed Importance Sampling

Importance weighting is a general way to adjust Monte Carlo integration to account for draws from the wrong distribution, but the resulting estimate can be noisy when the importance ratios have a heavy right tail. This routinely occurs when there are aspects of the target distribution that are not well captured by the approximating distribution, in which case more stable estimates can be obtained by modifying extreme importance ratios. We present a new method for stabilizing importance weights using a generalized Pareto distribution fit to the upper tail of the distribution of the simulated importance ratios. The method, which empirically performs better than existing methods for stabilizing importance sampling estimates, includes stabilized effective sample size estimates, Monte Carlo error estimates and convergence diagnostics.Comment: Major revision: 1) proofs for consistency, finite variance, and asymptotic normality, 2) justification of k<0.7 with theoretical computational complexity analysis, 3) major rewrit

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

EEG-Based Emotion Recognition Using Regularized Graph Neural Networks

Electroencephalography (EEG) measures the neuronal activities in different brain regions via electrodes. Many existing studies on EEG-based emotion recognition do not fully exploit the topology of EEG channels. In this paper, we propose a regularized graph neural network (RGNN) for EEG-based emotion recognition. RGNN considers the biological topology among different brain regions to capture both local and global relations among different EEG channels. Specifically, we model the inter-channel relations in EEG signals via an adjacency matrix in a graph neural network where the connection and sparseness of the adjacency matrix are inspired by neuroscience theories of human brain organization. In addition, we propose two regularizers, namely node-wise domain adversarial training (NodeDAT) and emotion-aware distribution learning (EmotionDL), to better handle cross-subject EEG variations and noisy labels, respectively. Extensive experiments on two public datasets, SEED and SEED-IV, demonstrate the superior performance of our model than state-of-the-art models in most experimental settings. Moreover, ablation studies show that the proposed adjacency matrix and two regularizers contribute consistent and significant gain to the performance of our RGNN model. Finally, investigations on the neuronal activities reveal important brain regions and inter-channel relations for EEG-based emotion recognition