亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

We establish a bijection between marginal independence models on $n$ random variables and split closed order ideals in the poset of partial set partitions. We also establish that every discrete marginal independence model is toric in cdf coordinates. This generalizes results of Boege, Petrovic, and Sturmfels and Drton and Richardson, and provides a unified framework for discussing marginal independence models.

Given a graph $G$ with a vertex threshold function $\tau$, consider a dynamic process in which any inactive vertex $v$ becomes activated whenever at least $\tau(v)$ of its neighbors are activated. A vertex set $S$ is called a target set if all vertices of $G$ would be activated when initially activating vertices of $S$. In the Minmax Target Set Reconfiguration problem, for a graph $G$ and its two target sets $X$ and $Y$, we wish to transform $X$ into $Y$ by repeatedly adding or removing a single vertex, using only target sets of $G$, so as to minimize the maximum size of any intermediate target set. We prove that it is NP-hard to approximate Minmax Target Set Reconfiguration within a factor of $2-o\left(\frac{1}{\operatorname{polylog} n}\right)$, where $n$ is the number of vertices. Our result establishes a tight lower bound on approximability of Minmax Target Set Reconfiguration, which admits a $2$-factor approximation algorithm. The proof is based on a gap-preserving reduction from Target Set Selection to Minmax Target Set Reconfiguration, where NP-hardness of approximation for the former problem is proven by Chen (SIAM J. Discrete Math., 2009) and Charikar, Naamad, and Wirth (APPROX/RANDOM 2016).

We present a parallel algorithm for the $(1-\epsilon)$-approximate maximum flow problem in capacitated, undirected graphs with $n$ vertices and $m$ edges, achieving $O(\epsilon^{-3}\text{polylog} n)$ depth and $O(m \epsilon^{-3} \text{polylog} n)$ work in the PRAM model. Although near-linear time sequential algorithms for this problem have been known for almost a decade, no parallel algorithms that simultaneously achieved polylogarithmic depth and near-linear work were known. At the heart of our result is a polylogarithmic depth, near-linear work recursive algorithm for computing congestion approximators. Our algorithm involves a recursive step to obtain a low-quality congestion approximator followed by a "boosting" step to improve its quality which prevents a multiplicative blow-up in error. Similar to Peng [SODA'16], our boosting step builds upon the hierarchical decomposition scheme of R\"acke, Shah, and T\"aubig [SODA'14]. A direct implementation of this approach, however, leads only to an algorithm with $n^{o(1)}$ depth and $m^{1+o(1)}$ work. To get around this, we introduce a new hierarchical decomposition scheme, in which we only need to solve maximum flows on subgraphs obtained by contracting vertices, as opposed to vertex-induced subgraphs used in R\"acke, Shah, and T\"aubig [SODA'14]. In particular, we are able to directly extract congestion approximators for the subgraphs from a congestion approximator for the entire graph, thereby avoiding additional recursion on those subgraphs. Along the way, we also develop a parallel flow-decomposition algorithm that is crucial to achieving polylogarithmic depth and may be of independent interest.

An unbiased $m$-sparsification of a vector $p\in \mathbb{R}^n$ is a random vector $Q\in \mathbb{R}^n$ with mean $p$ that has at most $m<n$ nonzero coordinates. Unbiased sparsification compresses the original vector without introducing bias; it arises in various contexts, such as in federated learning and sampling sparse probability distributions. Ideally, unbiased sparsification should also minimize the expected value of a divergence function $\mathsf{Div}(Q,p)$ that measures how far away $Q$ is from the original $p$. If $Q$ is optimal in this sense, then we call it efficient. Our main results describe efficient unbiased sparsifications for divergences that are either permutation-invariant or additively separable. Surprisingly, the characterization for permutation-invariant divergences is robust to the choice of divergence function, in the sense that our class of optimal $Q$ for squared Euclidean distance coincides with our class of optimal $Q$ for Kullback-Leibler divergence, or indeed any of a wide variety of divergences.

We present Clifford-Steerable Convolutional Neural Networks (CS-CNNs), a novel class of $\mathrm{E}(p, q)$-equivariant CNNs. CS-CNNs process multivector fields on pseudo-Euclidean spaces $\mathbb{R}^{p,q}$. They cover, for instance, $\mathrm{E}(3)$-equivariance on $\mathbb{R}^3$ and Poincar\'e-equivariance on Minkowski spacetime $\mathbb{R}^{1,3}$. Our approach is based on an implicit parametrization of $\mathrm{O}(p,q)$-steerable kernels via Clifford group equivariant neural networks. We significantly and consistently outperform baseline methods on fluid dynamics as well as relativistic electrodynamics forecasting tasks.

Sparse linear regression (SLR) is a well-studied problem in statistics where one is given a design matrix $X\in\mathbb{R}^{m\times n}$ and a response vector $y=X\theta^*+w$ for a $k$-sparse vector $\theta^*$ (that is, $\|\theta^*\|_0\leq k$) and small, arbitrary noise $w$, and the goal is to find a $k$-sparse $\widehat{\theta} \in \mathbb{R}^n$ that minimizes the mean squared prediction error $\frac{1}{m}\|X\widehat{\theta}-X\theta^*\|^2_2$. While $\ell_1$-relaxation methods such as basis pursuit, Lasso, and the Dantzig selector solve SLR when the design matrix is well-conditioned, no general algorithm is known, nor is there any formal evidence of hardness in an average-case setting with respect to all efficient algorithms. We give evidence of average-case hardness of SLR w.r.t. all efficient algorithms assuming the worst-case hardness of lattice problems. Specifically, we give an instance-by-instance reduction from a variant of the bounded distance decoding (BDD) problem on lattices to SLR, where the condition number of the lattice basis that defines the BDD instance is directly related to the restricted eigenvalue condition of the design matrix, which characterizes some of the classical statistical-computational gaps for sparse linear regression. Also, by appealing to worst-case to average-case reductions from the world of lattices, this shows hardness for a distribution of SLR instances; while the design matrices are ill-conditioned, the resulting SLR instances are in the identifiable regime. Furthermore, for well-conditioned (essentially) isotropic Gaussian design matrices, where Lasso is known to behave well in the identifiable regime, we show hardness of outputting any good solution in the unidentifiable regime where there are many solutions, assuming the worst-case hardness of standard and well-studied lattice problems.

In the $k$-Edit Circular Pattern Matching ($k$-Edit CPM) problem, we are given a length-$n$ text $T$, a length-$m$ pattern $P$, and a positive integer threshold $k$, and we are to report all starting positions of the substrings of $T$ that are at edit distance at most $k$ from some cyclic rotation of $P$. In the decision version of the problem, we are to check if any such substring exists. Very recently, Charalampopoulos et al. [ESA 2022] presented $O(nk^2)$-time and $O(nk \log^3 k)$-time solutions for the reporting and decision versions of $k$-Edit CPM, respectively. Here, we show that the reporting and decision versions of $k$-Edit CPM can be solved in $O(n+(n/m) k^6)$ time and $O(n+(n/m) k^5 \log^3 k)$ time, respectively, thus obtaining the first algorithms with a complexity of the type $O(n+(n/m) \mathrm{poly}(k))$ for this problem. Notably, our algorithms run in $O(n)$ time when $m=\Omega(k^6)$ and are superior to the previous respective solutions when $m=\omega(k^4)$. We provide a meta-algorithm that yields efficient algorithms in several other interesting settings, such as when the strings are given in a compressed form (as straight-line programs), when the strings are dynamic, or when we have a quantum computer. We obtain our solutions by exploiting the structure of approximate circular occurrences of $P$ in $T$, when $T$ is relatively short w.r.t. $P$. Roughly speaking, either the starting positions of approximate occurrences of rotations of $P$ form $O(k^4)$ intervals that can be computed efficiently, or some rotation of $P$ is almost periodic (is at a small edit distance from a string with small period). Dealing with the almost periodic case is the most technically demanding part of this work; we tackle it using properties of locked fragments (originating from [Cole and Hariharan, SICOMP 2002]).

We investigate pseudo-polynomial time algorithms for Subset Sum. Given a multi-set $X$ of $n$ positive integers and a target $t$, Subset Sum asks whether some subset of $X$ sums to $t$. Bringmann proposes an $\tilde{O}(n + t)$-time algorithm [Bringmann SODA'17], and an open question has naturally arisen: can Subset Sum be solved in $O(n + w)$ time? Here $w$ is the maximum integer in $X$. We make a progress towards resolving the open question by proposing an $\tilde{O}(n + \sqrt{wt})$-time algorithm.

We consider the problem of finding ``dissimilar'' $k$ shortest paths from $s$ to $t$ in an edge-weighted directed graph $D$, where the dissimilarity is measured by the minimum pairwise Hamming distances between these paths. More formally, given an edge-weighted directed graph $D = (V, A)$, two specified vertices $s, t \in V$, and integers $d, k$, the goal of Dissimilar Shortest Paths is to decide whether $D$ has $k$ shortest paths $P_1, \dots, P_k$ from $s$ to $t$ such that $|A(P_i) \mathbin{\triangle} A(P_j)| \ge d$ for distinct $P_i$ and $P_j$. We design a deterministic algorithm to solve Dissimilar Shortest Paths with running time $2^{O(3^kdk^2)}n^{O(1)}$, that is, Dissimilar Shortest Paths is fixed-parameter tractable parameterized by $k + d$. To complement this positive result, we show that Dissimilar Shortest Paths is W[1]-hard when parameterized by only $k$ and paraNP-hard parameterized by $d$.

In the Maximum Independent Set of Hyperrectangles problem, we are given a set of $n$ (possibly overlapping) $d$-dimensional axis-aligned hyperrectangles, and the goal is to find a subset of non-overlapping hyperrectangles of maximum cardinality. For $d=1$, this corresponds to the classical Interval Scheduling problem, where a simple greedy algorithm returns an optimal solution. In the offline setting, for $d$-dimensional hyperrectangles, polynomial time $(\log n)^{O(d)}$-approximation algorithms are known. However, the problem becomes notably challenging in the online setting, where the input objects (hyperrectangles) appear one by one in an adversarial order, and on the arrival of an object, the algorithm needs to make an immediate and irrevocable decision whether or not to select the object while maintaining the feasibility. Even for interval scheduling, an $\Omega(n)$ lower bound is known on the competitive ratio. To circumvent these negative results, in this work, we study the online maximum independent set of axis-aligned hyperrectangles in the random-order arrival model, where the adversary specifies the set of input objects which then arrive in a uniformly random order. Starting from the prototypical secretary problem, the random-order model has received significant attention to study algorithms beyond the worst-case competitive analysis. Surprisingly, we show that the problem in the random-order model almost matches the best-known offline approximation guarantees, up to polylogarithmic factors. In particular, we give a simple $(\log n)^{O(d)}$-competitive algorithm for $d$-dimensional hyperrectangles in this model, which runs in $\tilde{O_d}(n)$ time. Our approach also yields $(\log n)^{O(d)}$-competitive algorithms in the random-order model for more general objects such as $d$-dimensional fat objects and ellipsoids.