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Graph Convolutional Networks (GCNs) are one of the most popular architectures that are used to solve classification problems accompanied by graphical information. We present a rigorous theoretical understanding of the effects of graph convolutions in multi-layer networks. We study these effects through the node classification problem of a non-linearly separable Gaussian mixture model coupled with a stochastic block model. First, we show that a single graph convolution expands the regime of the distance between the means where multi-layer networks can classify the data by a factor of at least $1/\sqrt[4]{\mathbb{E}{\rm deg}}$, where $\mathbb{E}{\rm deg}$ denotes the expected degree of a node. Second, we show that with a slightly stronger graph density, two graph convolutions improve this factor to at least $1/\sqrt[4]{n}$, where $n$ is the number of nodes in the graph. Finally, we provide both theoretical and empirical insights into the performance of graph convolutions placed in different combinations among the layers of a network, concluding that the performance is mutually similar for all combinations of the placement. We present extensive experiments on both synthetic and real-world data that illustrate our results.

We consider the problem of enumerating optimal solutions for two hypergraph $k$-partitioning problems -- namely, Hypergraph-$k$-Cut and Minmax-Hypergraph-$k$-Partition. The input in hypergraph $k$-partitioning problems is a hypergraph $G=(V, E)$ with positive hyperedge costs along with a fixed positive integer $k$. The goal is to find a partition of $V$ into $k$ non-empty parts $(V_1, V_2, \ldots, V_k)$ -- known as a $k$-partition -- so as to minimize an objective of interest. 1. If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-$k$-Partition. A subset of hyperedges is a minmax-$k$-cut-set if it is the subset of hyperedges crossing an optimum $k$-partition for Minmax-Hypergraph-$k$-Partition. 2. If the objective of interest is the total cost of hyperedges crossing the $k$-partition, then the problem is known as Hypergraph-$k$-Cut. A subset of hyperedges is a min-$k$-cut-set if it is the subset of hyperedges crossing an optimum $k$-partition for Hypergraph-$k$-Cut. We give the first polynomial bound on the number of minmax-$k$-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed $k$. Our technique is strong enough to also enable an $n^{O(k)}p$-time deterministic algorithm to enumerate all min-$k$-cut-sets in hypergraphs, thus improving on the previously known $n^{O(k^2)}p$-time deterministic algorithm, where $n$ is the number of vertices and $p$ is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-$k$-Cut and Minmax-Hypergraph-$k$-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs).

We describe a polynomial-time algorithm which, given a graph $G$ with treewidth $t$, approximates the pathwidth of $G$ to within a ratio of $O(t\sqrt{\log t})$. This is the first algorithm to achieve an $f(t)$-approximation for some function $f$. Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least $th+2$ has treewidth at least $t$ or contains a subdivision of a complete binary tree of height $h+1$. The bound $th+2$ is best possible up to a multiplicative constant. This result was motivated by, and implies (with $c=2$), the following conjecture of Kawarabayashi and Rossman (SODA'18): there exists a universal constant $c$ such that every graph with pathwidth $\Omega(k^c)$ has treewidth at least $k$ or contains a subdivision of a complete binary tree of height $k$. Our main technical algorithm takes a graph $G$ and some (not necessarily optimal) tree decomposition of $G$ of width $t'$ in the input, and it computes in polynomial time an integer $h$, a certificate that $G$ has pathwidth at least $h$, and a path decomposition of $G$ of width at most $(t'+1)h+1$. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height $h$. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth.

We study the problem of testing whether a function $f: \mathbb{R}^n \to \mathbb{R}$ is a polynomial of degree at most $d$ in the \emph{distribution-free} testing model. Here, the distance between functions is measured with respect to an unknown distribution $\mathcal{D}$ over $\mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we do not assume that $\mathcal{D}$ has finite support. We design a tester that given query access to $f$, and sample access to $\mathcal{D}$, makes $(d/\varepsilon)^{O(1)}$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $\varepsilon$ with respect to $\mathcal{D}$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest.

Category theory can be used to state formulas in First-Order Logic without using set membership. Several notable results in logic such as proof of the continuum hypothesis can be elegantly rewritten in category theory. We propose in this paper a reformulation of the usual set-theoretical semantics of the description logic $\mathcal{ALC}$ by using categorical language. In this setting, ALC concepts are represented as objects, concept subsumptions as arrows, and memberships as logical quantifiers over objects and arrows of categories. Such a category-theoretical semantics provides a more modular representation of the semantics of $\mathcal{ALC}$ and a new way to design algorithms for reasoning.

For a connected graph $G=(V,E)$, a matching $M\subseteq E$ is a matching cut of $G$ if $G-M$ is disconnected. It is known that for an integer $d$, the corresponding decision problem Matching Cut is polynomial-time solvable for graphs of diameter at most $d$ if $d\leq 2$ and NP-complete if $d\geq 3$. We prove the same dichotomy for graphs of bounded radius. For a graph $H$, a graph is $H$-free if it does not contain $H$ as an induced subgraph. As a consequence of our result, we can solve Matching Cut in polynomial time for $P_6$-free graphs, extending a recent result of Feghali for $P_5$-free graphs. We then extend our result to hold even for $(sP_3+P_6)$-free graphs for every $s\geq 0$ and initiate a complexity classification of Matching Cut for $H$-free graphs.

Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time. This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a {\em dynamic setting}, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [HWC17]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of $(1+\epsilon)r^2$ and an update time of $O(\text{poly} (r, \log n))$, where $r$ denotes the maximum rank of the input across all the updates. We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of $(1+\epsilon)$ that is independent of $r$, and a similar update time of $O(\text{poly} (r, \log n))$. It is the first $(1+\epsilon)$-approximation algorithm even for the special case of weighted simple graphs. To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [HWC17] both in terms of accuracy and efficiency.

The presence of noise is an intrinsic problem in acquisition processes for digital images. One way to enhance images is to combine the forward and backward diffusion equations. However, the latter problem is well known to be exponentially unstable with respect to any small perturbations on the final data. In this scenario, the final data can be regarded as a blurred image obtained from the forward process, and that image can be pixelated as a network. Therefore, we study in this work a regularization framework for the backward diffusion equation on graphs. Our aim is to construct a spectral graph-based solution based upon a cut-off projection. Stability and convergence results are provided together with some numerical experiments.

In this short note, we show that for any $\epsilon >0$ and $k<n^{0.5-\epsilon}$ the choice number of the Kneser graph $KG_{n,k}$ is $\Theta (n\log n)$.

Automatic text summarization has experienced substantial progress in recent years. With this progress, the question has arisen whether the types of summaries that are typically generated by automatic summarization models align with users' needs. Ter Hoeve et al (2020) answer this question negatively. Amongst others, they recommend focusing on generating summaries with more graphical elements. This is in line with what we know from the psycholinguistics literature about how humans process text. Motivated from these two angles, we propose a new task: summarization with graphical elements, and we verify that these summaries are helpful for a critical mass of people. We collect a high quality human labeled dataset to support research into the task. We present a number of baseline methods that show that the task is interesting and challenging. Hence, with this work we hope to inspire a new line of research within the automatic summarization community.