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We consider the all pairs all shortest paths (APASP) problem, which maintains all of the multiple shortest paths for every vertex pair in a directed graph $G=(V,E)$ with a positive real weight on each edge. We present two fully dynamic algorithms for this problem in which an update supports either weight increases or weight decreases on a subset of edges incident to a vertex. Our first algorithm runs in amortized $O({\nu^*}^2 \cdot \log^3 n)$ time per update, where $n = |V|$, and $\nu^*$ bounds the number of edges that lie on shortest paths through any single vertex. Our APASP algorithm leads to the same amortized bound for the fully dynamic computation of betweenness centrality (BC), which is a parameter widely used in the analysis of large complex networks. Our method is a generalization and a variant of the fully dynamic algorithm of Demetrescu and Italiano [DI04] for unique shortest path, and it builds on our recent decremental APASP [NPR14]. Our second (faster) algorithm reduces the amortized cost per operation by a logarithmic factor, and uses new data structures and techniques that are extensions of methods in a fully dynamic algorithm by Thorup.

A partial orientation $\vec{H}$ of a graph $G$ is a weak $r$-guidance system if for any two vertices at distance at most $r$ in $G$, there exists a shortest path $P$ between them such that $\vec{H}$ directs all but one edge in $P$ towards this edge. In case $\vec{H}$ has bounded maximum outdegree, this gives an efficient representation of shortest paths of length at most $r$ in $G$. We show that graphs from many natural graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion.

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.

While algorithms for planar graphs have received a lot of attention, few papers have focused on the additional power that one gets from assuming an embedding of the graph is available. While in the classic sequential setting, this assumption gives no additional power (as a planar graph can be embedded in linear time), we show that this is far from being the case in other settings. We assume that the embedding is straight-line, but our methods also generalize to non-straight-line embeddings. Specifically, we focus on sublinear-time computation and massively parallel computation (MPC). Our main technical contribution is a sublinear-time algorithm for computing a relaxed version of an $r$-division. We then show how this can be used to estimate Lipschitz additive graph parameters. This includes, for example, the maximum matching, maximum independent set, or the minimum dominating set. We also show how this can be used to solve some property testing problems with respect to the vertex edit distance. In the second part of our paper, we show an MPC algorithm that computes an $r$-division of the input graph. We show how this can be used to solve various classical graph problems with space per machine of $O(n^{2/3+\epsilon})$ for some $\epsilon>0$, and while performing $O(1)$ rounds. This includes for example approximate shortest paths or the minimum spanning tree. Our results also imply an improved MPC algorithm for Euclidean minimum spanning tree.

Given a set $P$ of $n$ points in the plane, we consider the problem of computing the number of points of $P$ in a query unit disk (i.e., all query disks have the same radius). We show that the main techniques for simplex range searching in the plane can be adapted to this problem. For example, by adapting Matou\v{s}ek's results, we can build a data structure of $O(n)$ space so that each query can be answered in $O(\sqrt{n})$ time. Our techniques lead to improvements for several other classical problems, such as batched range searching, counting/reporting intersecting pairs of unit circles, distance selection, discrete 2-center, etc. For example, given a set of $n$ unit disks and a set of $n$ points in the plane, the batched range searching problem is to compute for each disk the number of points in it. Previous work [Katz and Sharir, 1997] solved the problem in $O(n^{4/3}\log n)$ time while our new algorithm runs in $O(n^{4/3})$ time.

In this paper, we propose a depth-first search (DFS) algorithm for searching maximum matchings in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating paths in the super-vertices shrunk from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending time on shrinking and expanding blossoms. This DFS algorithm can determine a maximum matching of a general graph with $m$ edges and $n$ vertices in $O(mn)$ time with space complexity $O(n)$.

SVD (singular value decomposition) is one of the basic tools of machine learning, allowing to optimize basis for a given matrix. However, sometimes we have a set of matrices $\{A_k\}_k$ instead, and would like to optimize a single common basis for them: find orthogonal matrices $U$, $V$, such that $\{U^T A_k V\}$ set of matrices is somehow simpler. For example DCT-II is orthonormal basis of functions commonly used in image/video compression - as discussed here, this kind of basis can be quickly automatically optimized for a given dataset. While also discussed gradient descent optimization might be computationally costly, there is proposed CSVD (common SVD): fast general approach based on SVD. Specifically, we choose $U$ as built of eigenvectors of $\sum_i (w_k)^q (A_k A_k^T)^p$ and $V$ of $\sum_k (w_k)^q (A_k^T A_k)^p$, where $w_k$ are their weights, $p,q>0$ are some chosen powers e.g. 1/2, optionally with normalization e.g. $A \to A - rc^T$ where $r_i=\sum_j A_{ij}, c_j =\sum_i A_{ij}$.

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 stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to $\sqrt{n}$, the gradient complexity to achieve an $\epsilon$-precision is $\tilde{O}((n+dn^{1/2}\epsilon^{-1})\gamma^2 L^2\alpha^{-2})$, which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.

Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.