The planted densest subgraph detection problem refers to the task of testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on $n$ nodes. Under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi graph with edge probability (or, density) $q$. Under the alternative, there is a subgraph on $k$ vertices with edge probability $p>q$. The statistical as well as the computational barriers of this problem are well-understood for a wide range of the edge parameters $p$ and $q$. In this paper, we consider a natural variant of the above problem, where one can only observe a small part of the graph using adaptive edge queries. For this model, we determine the number of queries necessary and sufficient for detecting the presence of the planted subgraph. Specifically, we show that any (possibly randomized) algorithm must make $\mathsf{Q} = \Omega(\frac{n^2}{k^2\chi^4(p||q)}\log^2n)$ adaptive queries (on expectation) to the adjacency matrix of the graph to detect the planted subgraph with probability more than $1/2$, where $\chi^2(p||q)$ is the Chi-Square distance. On the other hand, we devise a quasi-polynomial-time algorithm that finds the planted subgraph with high probability by making $\mathsf{Q} = O(\frac{n^2}{k^2\chi^4(p||q)}\log^2n)$ adaptive queries. We then propose a polynomial-time algorithm which is able to detect the planted subgraph using $\mathsf{Q} = O(\frac{n^4}{k^4\chi^2(p||q)}\log n)$ queries. We conjecture that in the leftover regime, where $\frac{n^2}{k^2}\ll\mathsf{Q}\ll \frac{n^4}{k^4}$, no polynomial-time algorithms exist; we give an evidence for this hypothesis using the planted clique conjecture. Our results resolve three questions posed in \cite{racz2020finding}, where the special case of adaptive detection and recovery of a planted clique was considered.
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We propose a new representation of $k$-partite, $k$-uniform hypergraphs (i.e. a hypergraph with a partition of vertices into $k$ parts such that each hyperedge contains exactly one vertex of each type; we call them $k$-hypergraphs for short) by a finite set $P$ of points in $\mathbb{R}^d$ and a parameter $\ell\leq d-1$. Each point in $P$ is covered by $k={d\choose\ell}$ many axis-aligned affine $\ell$-dimensional subspaces of $\mathbb{R}^d$, which we call $\ell$-subspaces for brevity. We interpret each point in $P$ as a hyperedge that contains each of the covering $\ell$-subspaces as a vertex. The class of $(d,\ell)$-hypergraphs is the class of $k$-hypergraphs that can be represented in this way, where $k={d\choose\ell}$. The resulting classes of hypergraphs are fairly rich: Every $k$-hypergraph is a $(k,k-1)$-hypergraph. On the other hand, $(d,\ell)$-hypergraphs form a proper subclass of the class of all $d\choose\ell$-hypergraphs for $\ell<d-1$. In this paper we give a natural structural characterization of $(d,\ell)$-hypergraphs based on vertex cuts. This characterization leads to a polynomial-time recognition algorithm that decides for a given $d\choose\ell$-hypergraph whether or not it is a $(d,\ell)$-hypergraph and that computes a representation if existing. We assume that the dimension $d$ is constant and that the partitioning of the vertex set is prescribed.
Given an undirected graph $G=(V,E)$, a vertex $v\in V$ is edge-vertex (ev) dominated by an edge $e\in E$ if $v$ is either incident to $e$ or incident to an adjacent edge of $e$. A set $S^{ev}\subseteq E$ is an edge-vertex dominating set (referred to as ev-dominating set) of $G$ if every vertex of $G$ is ev-dominated by at least one edge of $S^{ev}$. The minimum cardinality of an ev-dominating set is the ev-domination number. The edge-vertex dominating set problem is to find a minimum ev-domination number. In this paper we prove that the ev-dominating set problem is {\tt NP-hard} on unit disk graphs. We also prove that this problem admits a polynomial-time approximation scheme on unit disk graphs. Finally, we give a simple 5-factor linear-time approximation algorithm.
It is well known [Lov\'{a}sz, 1967] that up to isomorphism a graph $G$ is determined by the homomorphism counts $\hom(F, G)$, i.e., the number of homomorphisms from $F$ to $G$, where $F$ ranges over all graphs. Moreover, it suffices that $F$ ranges over the graphs with at most as many vertices as $G$. Thus in principle we can answer any query concerning $G$ with only accessing the $\hom(\cdot,G)$'s instead of $G$ itself. In this paper, we zoom in on those queries that can be answered using a constant number of $\hom(\cdot,G)$ for every graph $G$. We observe that if a query $\varphi$ is expressible as a Boolean combination of universal sentences in first-order logic, then whether a graph $G$ satisfies $\varphi$ can be determined by the vector \[\overrightarrow{\mathrm{hom}}_{F_1, \ldots, F_k}(G):= \big(\mathrm{hom}(F_1, G), \ldots, \mathrm{hom}(F_k, G)\big),\] where the graphs $F_1,\ldots,F_k$ only depend on $\varphi$. This leads to a query algorithm for $\varphi$ that is non-adaptive in the sense that those $F_i$ are independent of the input $G$. On the other hand, we prove that the existence of an isolated vertex, which is not definable by such a $\varphi$ but in first-order logic, cannot be determined by any $\overrightarrow{\mathrm{hom}}_{F_1, \ldots, F_k}(\cdot)$. These results provide a clear delineation of the power of non-adaptive query algorithms with access to a constant number of $\hom(\cdot, G)$'s. For adaptive query algorithms, i.e., algorithms that might access some $\hom(F_{i+1}, G)$ with $F_{i+1}$ depending on $\hom(F_1, G), \ldots, \hom(F_i, G)$, we show that three homomorphism counts $\hom(\cdot,G)$ are both sufficient and in general necessary to determine the graph $G$. In particular, by three adaptive queries we can answer any question on $G$. Moreover, adaptively accessing two $\hom(\cdot, G)$'s is already enough to detect an isolated vertex.
In this work we consider the well-known Secretary Problem -- a number $n$ of elements, each having an adversarial value, are arriving one-by-one according to some random order, and the goal is to choose the highest value element. The decisions are made online and are irrevocable -- if the algorithm decides to choose or not to choose the currently seen element, based on the previously observed values, it cannot change its decision later regarding this element. The measure of success is the probability of selecting the highest value element, minimized over all adversarial assignments of values. We show existential and constructive upper bounds on approximation of the success probability in this problem, depending on the entropy of the randomly chosen arrival order, including the lowest possible entropy $O(\log\log (n))$ for which the probability of success could be constant. We show that below entropy level $\mathcal{H}<0.5\log\log n$, all algorithms succeed with probability $0$ if random order is selected uniformly at random from some subset of permutations, while we are able to construct in polynomial time a non-uniform distribution with entropy $\mathcal{H}$ resulting in success probability of at least $\Omega\left(\frac{1}{(\log\log n +3\log\log\log n -\mathcal{H})^{2+\epsilon}}\right)$, for any constant $\epsilon>0$. We also prove that no algorithm using entropy $\mathcal{H}=O((\log\log n)^a)$ can improve our result by more than polynomially, for any constant $0<a<1$. For entropy $\log\log (n)$ and larger, our analysis precisely quantifies both multiplicative and additive approximation of the success probability. In particular, we improve more than doubly exponentially on the best previously known additive approximation guarantee for the secretary problem.
With the rapid expansion of graphs and networks and the growing magnitude of data from all areas of science, effective treatment and compression schemes of context-dependent data is extremely desirable. A particularly interesting direction is to compress the data while keeping the "structural information" only and ignoring the concrete labelings. Under this direction, Choi and Szpankowski introduced the structures (unlabeled graphs) which allowed them to compute the structural entropy of the Erd\H{o}s--R\'enyi random graph model. Moreover, they also provided an asymptotically optimal compression algorithm that (asymptotically) achieves this entropy limit and runs in expectation in linear time. In this paper, we consider the Stochastic Block Models with an arbitrary number of parts. Indeed, we define a partitioned structural entropy for Stochastic Block Models, which generalizes the structural entropy for unlabeled graphs and encodes the partition information as well. We then compute the partitioned structural entropy of the Stochastic Block Models, and provide a compression scheme that asymptotically achieves this entropy limit.
Sufficient conditions are provided under which the log-likelihood ratio test statistic fails to have a limiting chi-squared distribution under the null hypothesis when testing between one and two components under a general two-component mixture model, but rather tends to infinity in probability. These conditions are verified when the component densities describe continuous-time, discrete-statespace Markov chains and the results are illustrated via a parametric bootstrap simulation on an analysis of the migrations over time of a set of corporate bonds ratings. The precise limiting distribution is derived in a simple case with two states, one of which is absorbing which leads to a right-censored exponential scale mixture model. In that case, when centred by a function growing logarithmically in the sample size, the statistic has a limiting distribution of Gumbel extreme-value type rather than chi-squared.
3D object detection often involves complicated training and testing pipelines, which require substantial domain knowledge about individual datasets. Inspired by recent non-maximum suppression-free 2D object detection models, we propose a 3D object detection architecture on point clouds. Our method models 3D object detection as message passing on a dynamic graph, generalizing the DGCNN framework to predict a set of objects. In our construction, we remove the necessity of post-processing via object confidence aggregation or non-maximum suppression. To facilitate object detection from sparse point clouds, we also propose a set-to-set distillation approach customized to 3D detection. This approach aligns the outputs of the teacher model and the student model in a permutation-invariant fashion, significantly simplifying knowledge distillation for the 3D detection task. Our method achieves state-of-the-art performance on autonomous driving benchmarks. We also provide abundant analysis of the detection model and distillation framework.
Keypoint-based methods are a relatively new paradigm in object detection, eliminating the need for anchor boxes and offering a simplified detection framework. Keypoint-based CornerNet achieves state of the art accuracy among single-stage detectors. However, this accuracy comes at high processing cost. In this work, we tackle the problem of efficient keypoint-based object detection and introduce CornerNet-Lite. CornerNet-Lite is a combination of two efficient variants of CornerNet: CornerNet-Saccade, which uses an attention mechanism to eliminate the need for exhaustively processing all pixels of the image, and CornerNet-Squeeze, which introduces a new compact backbone architecture. Together these two variants address the two critical use cases in efficient object detection: improving efficiency without sacrificing accuracy, and improving accuracy at real-time efficiency. CornerNet-Saccade is suitable for offline processing, improving the efficiency of CornerNet by 6.0x and the AP by 1.0% on COCO. CornerNet-Squeeze is suitable for real-time detection, improving both the efficiency and accuracy of the popular real-time detector YOLOv3 (34.4% AP at 34ms for CornerNet-Squeeze compared to 33.0% AP at 39ms for YOLOv3 on COCO). Together these contributions for the first time reveal the potential of keypoint-based detection to be useful for applications requiring processing efficiency.
Structured queries expressed in languages (such as SQL, SPARQL, or XQuery) offer a convenient and explicit way for users to express their information needs for a number of tasks. In this work, we present an approach to answer these directly over text data without storing results in a database. We specifically look at the case of knowledge bases where queries are over entities and the relations between them. Our approach combines distributed query answering (e.g. Triple Pattern Fragments) with models built for extractive question answering. Importantly, by applying distributed querying answering we are able to simplify the model learning problem. We train models for a large portion (572) of the relations within Wikidata and achieve an average 0.70 F1 measure across all models. We also present a systematic method to construct the necessary training data for this task from knowledge graphs and describe a prototype implementation.
With the emergence of edge computing, there is an increasing need for running convolutional neural network based object detection on small form factor edge computing devices with limited compute and thermal budget for applications such as video surveillance. To address this problem, efficient object detection frameworks such as YOLO and SSD were proposed. However, SSD based object detection that uses VGG16 as backend network is insufficient to achieve real time speed on edge devices. To further improve the detection speed, the backend network is replaced by more efficient networks such as SqueezeNet and MobileNet. Although the speed is greatly improved, it comes with a price of lower accuracy. In this paper, we propose an efficient SSD named Fire SSD. Fire SSD achieves 70.7mAP on Pascal VOC 2007 test set. Fire SSD achieves the speed of 30.6FPS on low power mainstream CPU and is about 6 times faster than SSD300 and has about 4 times smaller model size. Fire SSD also achieves 22.2FPS on integrated GPU.
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