The Longest Common Subsequence (LCS) of two strings is a fundamental string similarity measure with a classical dynamic programming solution taking quadratic time. Despite significant efforts, little progress was made in improving the runtime. Even in the realm of approximation, not much was known for linear time algorithms beyond the trivial $\sqrt{n}$-approximation. Recent breakthrough result provided a $n^{0.497}$-factor approximation algorithm [HSSS19], which was more recently improved to a $n^{0.4}$-factor one [BCD21]. The latter paper also showed a $n^{2-2.5\alpha}$ time algorithm which outputs a $n^{\alpha}$ approximation to the LCS, but so far no sub-polynomial approximation is known in truly subquadratic time. In this work, we show an algorithm which runs in $O(n)$ time, and outputs a $n^{o(1)}$-factor approximation to LCS$(x,y)$, with high probability, for any pair of length $n$ input strings. Our entire algorithm is merely an efficient black-box reduction to the Block-LIS problem, introduced very recently in [ANSS21], and solving the Block-LIS problem directly.
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In the Strip Packing problem (SP), we are given a vertical half-strip $[0,W]\times[0,\infty)$ and a set of $n$ axis-aligned rectangles of width at most $W$. The goal is to find a non-overlapping packing of all rectangles into the strip such that the height of the packing is minimized. A well-studied and frequently used practical constraint is to allow only those packings that are guillotine separable, i.e., every rectangle in the packing can be obtained by recursively applying a sequence of edge-to-edge axis-parallel cuts (guillotine cuts) that do not intersect any item of the solution. In this paper, we study approximation algorithms for the Guillotine Strip Packing problem (GSP), i.e., the Strip Packing problem where we require additionally that the packing needs to be guillotine separable. This problem generalizes the classical Bin Packing problem and also makespan minimization on identical machines, and thus it is already strongly NP-hard. Moreover, due to a reduction from the Partition problem, it is NP-hard to obtain a polynomial-time $(3/2-\varepsilon)$-approximation algorithm for GSP for any $\varepsilon>0$ (exactly as Strip Packing). We provide a matching polynomial time $(3/2+\varepsilon)$-approximation algorithm for GSP. Furthermore, we present a pseudo-polynomial time $(1+\varepsilon)$-approximation algorithm for GSP. This is surprising as it is NP-hard to obtain a $(5/4-\varepsilon)$-approximation algorithm for (general) Strip Packing in pseudo-polynomial time. Thus, our results essentially settle the approximability of GSP for both the polynomial and the pseudo-polynomial settings.
Consider a matrix $\mathbf{F} \in \mathbb{K}^{m \times n}$ of univariate polynomials over a field~$\mathbb{K}$. We study the problem of computing the column rank profile of $\mathbf{F}$. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of $\mathbf{F}$ with a rank-sensitive complexity of $O\tilde{~}(r^{\omega-2} n (m+D))$ operations in $\mathbb{K}$. Here, $D$ is the sum of row degrees of $\mathbf{F}$, $\omega$ is the exponent of matrix multiplication, and $O\tilde{~}(\cdot)$ hides logarithmic factors.
The scope of this paper is the analysis and approximation of an optimal control problem related to the Allen-Cahn equation. A tracking functional is minimized subject to the Allen-Cahn equation using distributed controls that satisfy point-wise control constraints. First and second order necessary and sufficient conditions are proved. The lowest order discontinuous Galerkin - in time - scheme is considered for the approximation of the control to state and adjoint state mappings. Under a suitable restriction on maximum size of the temporal and spatial discretization parameters $k$, $h$ respectively in terms of the parameter $\epsilon$ that describes the thickness of the interface layer, a-priori estimates are proved with constants depending polynomially upon $1/ \epsilon$. Unlike to previous works for the uncontrolled Allen-Cahn problem our approach does not rely on a construction of an approximation of the spectral estimate, and as a consequence our estimates are valid under low regularity assumptions imposed by the optimal control setting. These estimates are also valid in cases where the solution and its discrete approximation do not satisfy uniform space-time bounds independent of $\epsilon$. These estimates and a suitable localization technique, via the second order condition (see \cite{Arada-Casas-Troltzsch_2002,Casas-Mateos-Troltzsch_2005,Casas-Raymond_2006,Casas-Mateos-Raymond_2007}), allows to prove error estimates for the difference between local optimal controls and their discrete approximation as well as between the associated state and adjoint state variables and their discrete approximations
Linear Convergence of Entropy-Regularized Natural Policy Gradient with Linear Function Approximation
Natural policy gradient (NPG) methods with entropy regularization achieve impressive empirical success in reinforcement learning problems with large state-action spaces. However, their convergence properties and the impact of entropy regularization remain elusive in the function approximation regime. In this paper, we establish finite-time convergence analyses of entropy-regularized NPG with linear function approximation under softmax parameterization. In particular, we prove that entropy-regularized NPG with averaging satisfies the \emph{persistence of excitation} condition, and achieves a fast convergence rate of $\tilde{O}(1/T)$ up to a function approximation error in regularized Markov decision processes. This convergence result does not require any a priori assumptions on the policies. Furthermore, under mild regularity conditions on the concentrability coefficient and basis vectors, we prove that entropy-regularized NPG exhibits \emph{linear convergence} up to a function approximation error.
We study algorithms for approximating pairwise similarity matrices that arise in natural language processing. Generally, computing a similarity matrix for $n$ data points requires $\Omega(n^2)$ similarity computations. This quadratic scaling is a significant bottleneck, especially when similarities are computed via expensive functions, e.g., via transformer models. Approximation methods reduce this quadratic complexity, often by using a small subset of exactly computed similarities to approximate the remainder of the complete pairwise similarity matrix. Significant work focuses on the efficient approximation of positive semidefinite (PSD) similarity matrices, which arise e.g., in kernel methods. However, much less is understood about indefinite (non-PSD) similarity matrices, which often arise in NLP. Motivated by the observation that many of these matrices are still somewhat close to PSD, we introduce a generalization of the popular Nystr\"{o}m method to the indefinite setting. Our algorithm can be applied to any similarity matrix and runs in sublinear time in the size of the matrix, producing a rank-$s$ approximation with just $O(ns)$ similarity computations. We show that our method, along with a simple variant of CUR decomposition, performs very well in approximating a variety of similarity matrices arising in NLP tasks. We demonstrate high accuracy of the approximated similarity matrices in the downstream tasks of document classification, sentence similarity, and cross-document coreference.
We revisit the task of computing the edit distance in sublinear time. In the $(k,K)$-gap edit distance problem the task is to distinguish whether the edit distance of two strings is at most $k$ or at least $K$. It has been established by Goldenberg, Krauthgamer and Saha (FOCS '19), with improvements by Kociumaka and Saha (FOCS '20), that the $(k,k^2)$-gap problem can be solved in time $\widetilde O(n/k+\operatorname{poly}(k))$. One of the most natural questions in this line of research is whether the $(k,k^2)$-gap is best-possible for the running time $\widetilde O(n/k+\operatorname{poly}(k))$. In this work we answer this question by significantly improving the gap. Specifically, we show that in time $O(n/k+\operatorname{poly}(k))$ we can even solve the $(k,k^{1+o(1)})$-gap problem. This is the first algorithm that breaks the $(k,k^2)$-gap in this running time. Our algorithm is almost optimal in the following sense: In the low distance regime ($k\le n^{0.19}$) our running time becomes $O(n/k)$, which matches a known $n/k^{1+o(1)}$ lower bound for the $(k,k^{1+o(1)})$-gap problem up to lower order factors. Our result also reveals a surprising similarity of Hamming distance and edit distance in the low distance regime: For both, the $(k,k^{1+o(1)})$-gap problem has time complexity $n/k^{1\pm o(1)}$ for small $k$. In contrast to previous work, which employed a subsampled variant of the Landau-Vishkin algorithm, we instead build upon the algorithm of Andoni, Krauthgamer and Onak (FOCS '10). We first simplify their approach and then show how to to effectively prune their computation tree in order to obtain a sublinear-time algorithm in the given time bound. Towards that, we use a variety of structural insights on the (local and global) patterns that can emerge during this process and design appropriate property testers to effectively detect these patterns.
The compression of highly repetitive strings (i.e., strings with many repetitions) has been a central research topic in string processing, and quite a few compression methods for these strings have been proposed thus far. Among them, an efficient compression format gathering increasing attention is the run-length Burrows--Wheeler transform (RLBWT), which is a run-length encoded BWT as a reversible permutation of an input string on the lexicographical order of suffixes. State-of-the-art construction algorithms of RLBWT have a serious issue with respect to (i) non-optimal computation time or (ii) a working space that is linearly proportional to the length of an input string. In this paper, we present \emph{r-comp}, the first optimal-time construction algorithm of RLBWT in BWT-runs bounded space. That is, the computational complexity of r-comp is $O(n + r \log{r})$ time and $O(r\log{n})$ bits of working space for the length $n$ of an input string and the number $r$ of equal-letter runs in BWT. The computation time is optimal (i.e., $O(n)$) for strings with the property $r=O(n/\log{n})$, which holds for most highly repetitive strings. Experiments using a real-world dataset of highly repetitive strings show the effectiveness of r-comp with respect to computation time and space.
$\newcommand{\NP}{\mathsf{NP}}\newcommand{\GapSVP}{\textrm{GapSVP}}$We give a simple proof that the (approximate, decisional) Shortest Vector Problem is $\NP$-hard under a randomized reduction. Specifically, we show that for any $p \geq 1$ and any constant $\gamma < 2^{1/p}$, the $\gamma$-approximate problem in the $\ell_p$ norm ($\gamma$-$\GapSVP_p$) is not in $\mathsf{RP}$ unless $\NP \subseteq \mathsf{RP}$. Our proof follows an approach pioneered by Ajtai (STOC 1998), and strengthened by Micciancio (FOCS 1998 and SICOMP 2000), for showing hardness of $\gamma$-$\GapSVP_p$ using locally dense lattices. We construct such lattices simply by applying "Construction A" to Reed-Solomon codes with suitable parameters, and prove their local density via an elementary argument originally used in the context of Craig lattices. As in all known $\NP$-hardness results for $\GapSVP_p$ with $p < \infty$, our reduction uses randomness. Indeed, it is a notorious open problem to prove $\NP$-hardness via a deterministic reduction. To this end, we additionally discuss potential directions and associated challenges for derandomizing our reduction. In particular, we show that a close deterministic analogue of our local density construction would improve on the state-of-the-art explicit Reed-Solomon list-decoding lower bounds of Guruswami and Rudra (STOC 2005 and IEEE Trans. Inf. Theory 2006). As a related contribution of independent interest, we also give a polynomial-time algorithm for decoding $n$-dimensional "Construction A Reed-Solomon lattices" (with different parameters than those used in our hardness proof) to a distance within an $O(\sqrt{\log n})$ factor of Minkowski's bound. This asymptotically matches the best known distance for decoding near Minkowski's bound, due to Mook and Peikert (IEEE Trans. Inf. Theory 2022), whose work we build on with a somewhat simpler construction and analysis.
Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}(\log n/\epsilon^{1.75})$ iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with $\tilde{O}((\log n)^{4}/\epsilon^{2})$ or $\tilde{O}((\log n)^{6}/\epsilon^{1.75})$ iterations, our algorithm is polynomially better in terms of $\log n$ and matches their complexities in terms of $1/\epsilon$. For the stochastic setting, our algorithm outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}((\log n)^{2}/\epsilon^{4})$ iterations. Technically, our main contribution is an idea of implementing a robust Hessian power method using only gradients, which can find negative curvature near saddle points and achieve the polynomial speedup in $\log n$ compared to the perturbed gradient descent methods. Finally, we also perform numerical experiments that support our results.
In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate $\mathcal{M}$ a $d$-dimensional $C^{m+1}$ smooth submanifold of $\mathbb{R}^n$ ($d \ll n$) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located "near" the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating $d$-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., $O(h^{m+1})$, where $h$ is the fill distance and $m$ is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension $n$. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.
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