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We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant $\Delta$, and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between $1$ and $1+\Delta$. Our goal is to maximize the number of near exact covered bins. If $\Delta=0$ or $\Delta>0$ is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that $P\neq NP$). However, for the case where $\Delta>0$ is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.

We present the first algorithm for regular expression matching that can take advantage of sparsity in the input instance. Our main result is a new algorithm that solves regular expression matching in $O\left(\Delta \log \log \frac{nm}{\Delta} + n + m\right)$ time, where $m$ is the number of positions in the regular expression, $n$ is the length of the string, and $\Delta$ is the \emph{density} of the instance, defined as the total number of states in a simulation of the position automaton. This measure is a lower bound on the total number of states in simulations of all classic polynomial sized finite automata. Our bound improves the best known bounds for regular expression matching by almost a linear factor in the density of the problem. The key component in the result is a novel linear space representation of the position automaton that supports state-set transition computation in near-linear time in the size of the input and output state sets.

This paper studies the problem of matching two complete graphs with edge weights correlated through latent geometries, extending a recent line of research on random graph matching with independent edge weights to geometric models. Specifically, given a random permutation $\pi^*$ on $[n]$ and $n$ iid pairs of correlated Gaussian vectors $\{X_{\pi^*(i)}, Y_i\}$ in $\mathbb{R}^d$ with noise parameter $\sigma$, the edge weights are given by $A_{ij}=\kappa(X_i,X_j)$ and $B_{ij}=\kappa(Y_i,Y_j)$ for some link function $\kappa$. The goal is to recover the hidden vertex correspondence $\pi^*$ based on the observation of $A$ and $B$. We focus on the dot-product model with $\kappa(x,y)=\langle x, y \rangle$ and Euclidean distance model with $\kappa(x,y)=\|x-y\|^2$, in the low-dimensional regime of $d=o(\log n)$ wherein the underlying geometric structures are most evident. We derive an approximate maximum likelihood estimator, which provably achieves, with high probability, perfect recovery of $\pi^*$ when $\sigma=o(n^{-2/d})$ and almost perfect recovery with a vanishing fraction of errors when $\sigma=o(n^{-1/d})$. Furthermore, these conditions are shown to be information-theoretically optimal even when the latent coordinates $\{X_i\}$ and $\{Y_i\}$ are observed, complementing the recent results of [DCK19] and [KNW22] in geometric models of the planted bipartite matching problem. As a side discovery, we show that the celebrated spectral algorithm of [Ume88] emerges as a further approximation to the maximum likelihood in the geometric model.

We introduce a notion of \emph{generic local algorithm} which strictly generalizes existing frameworks of local algorithms such as \emph{factors of i.i.d.} by capturing local \emph{quantum} algorithms such as the Quantum Approximate Optimization Algorithm (QAOA). Motivated by a question of Farhi et al. [arXiv:1910.08187, 2019] we then show limitations of generic local algorithms including QAOA on random instances of constraint satisfaction problems (CSPs). Specifically, we show that any generic local algorithm whose assignment to a vertex depends only on a local neighborhood with $o(n)$ other vertices (such as the QAOA at depth less than $\epsilon\log(n)$) cannot arbitrarily-well approximate boolean CSPs if the problem satisfies a geometric property from statistical physics called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019]. We show that the random MAX-k-XOR problem has this property when $k\geq4$ is even by extending the corresponding result for diluted $k$-spin glasses. Our concentration lemmas confirm a conjecture of Brandao et al. [arXiv:1812.04170, 2018] asserting that the landscape independence of QAOA extends to logarithmic depth -- in other words, for every fixed choice of QAOA angle parameters, the algorithm at logarithmic depth performs almost equally well on almost all instances. One of these concentration lemmas is a strengthening of McDiarmid's inequality, applicable when the random variables have a highly biased distribution, and may be of independent interest.

In this paper, we consider two fundamental symmetric kernels in linear algebra: the Cholesky factorization and the symmetric rank-$k$ update (SYRK), with the classical three nested loops algorithms for these kernels. In addition, we consider a machine model with a fast memory of size $S$ and an unbounded slow memory. In this model, all computations must be performed on operands in fast memory, and the goal is to minimize the amount of communication between slow and fast memories. As the set of computations is fixed by the choice of the algorithm, only the ordering of the computations (the schedule) directly influences the volume of communications.We prove lower bounds of $\frac{1}{3\sqrt{2}}\frac{N^3}{\sqrt{S}}$ for the communication volume of the Cholesky factorization of an $N\times N$ symmetric positive definite matrix, and of $\frac{1}{\sqrt{2}}\frac{N^2M}{\sqrt{S}}$ for the SYRK computation of $\mat{A}\cdot\transpose{\mat{A}}$, where $\mathbf{A}$ is an $N\times M$ matrix. Both bounds improve the best known lower bounds from the literature by a factor $\sqrt{2}$.In addition, we present two out-of-core, sequential algorithms with matching communication volume: \TBS for SYRK, with a volume of $\frac{1}{\sqrt{2}}\frac{N^2M}{\sqrt{S}} + \bigo{NM\log N}$, and \LBC for Cholesky, with a volume of $\frac{1}{3\sqrt{2}}\frac{N^3}{\sqrt{S}} + \bigo{N^{5/2}}$. Both algorithms improve over the best known algorithms from the literature by a factor $\sqrt{2}$, and prove that the leading terms in our lower bounds cannot be improved further. This work shows that the operational intensity of symmetric kernels like SYRK or Cholesky is intrinsically higher (by a factor $\sqrt{2}$) than that of corresponding non-symmetric kernels (GEMM and LU factorization).

Given a point set $P\subset \mathbb{R}^d$, the kernel density estimate of $P$ is defined as \[ \overline{\mathcal{G}}_P(x) = \frac{1}{\left|P\right|}\sum_{p\in P}e^{-\left\lVert x-p \right\rVert^2} \] for any $x\in\mathbb{R}^d$. We study how to construct a small subset $Q$ of $P$ such that the kernel density estimate of $P$ is approximated by the kernel density estimate of $Q$. This subset $Q$ is called a coreset. The main technique in this work is constructing a $\pm 1$ coloring on the point set $P$ by discrepancy theory and we leverage Banaszczyk's Theorem. When $d>1$ is a constant, our construction gives a coreset of size $O\left(\frac{1}{\varepsilon}\right)$ as opposed to the best-known result of $O\left(\frac{1}{\varepsilon}\sqrt{\log\frac{1}{\varepsilon}}\right)$. It is the first result to give a breakthrough on the barrier of $\sqrt{\log}$ factor even when $d=2$.

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.

We consider the following oblivious sketching problem: given $\epsilon \in (0,1/3)$ and $n \geq d/\epsilon^2$, design a distribution $\mathcal{D}$ over $\mathbb{R}^{k \times nd}$ and a function $f: \mathbb{R}^k \times \mathbb{R}^{nd} \rightarrow \mathbb{R}$, so that for any $n \times d$ matrix $A$, $$\Pr_{S \sim \mathcal{D}} [(1-\epsilon) \|A\|_{op} \leq f(S(A),S) \leq (1+\epsilon)\|A\|_{op}] \geq 2/3,$$ where $\|A\|_{op}$ is the operator norm of $A$ and $S(A)$ denotes $S \cdot A$, interpreting $A$ as a vector in $\mathbb{R}^{nd}$. We show a tight lower bound of $k = \Omega(d^2/\epsilon^2)$ for this problem. Our result considerably strengthens the result of Nelson and Nguyen (ICALP, 2014), as it (1) applies only to estimating the operator norm, which can be estimated given any OSE, and (2) applies to distributions over general linear operators $S$ which treat $A$ as a vector and compute $S(A)$, rather than the restricted class of linear operators corresponding to matrix multiplication. Our technique also implies the first tight bounds for approximating the Schatten $p$-norm for even integers $p$ via general linear sketches, improving the previous lower bound from $k = \Omega(n^{2-6/p})$ [Regev, 2014] to $k = \Omega(n^{2-4/p})$. Importantly, for sketching the operator norm up to a factor of $\alpha$, where $\alpha - 1 = \Omega(1)$, we obtain a tight $k = \Omega(n^2/\alpha^4)$ bound, matching the upper bound of Andoni and Nguyen (SODA, 2013), and improving the previous $k = \Omega(n^2/\alpha^6)$ lower bound. Finally, we also obtain the first lower bounds for approximating Ky Fan norms.

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the problem of constructing a perfect matching in as few rounds as possible. In particular, we present an adaptive strategy for the player which achieves a perfect matching in $\beta n$ rounds, where the value of $\beta < 1.206$ is derived from a solution to some system of differential equations. This improves upon the previously best known upper bound of $(1+2/e+o(1)) \, n < 1.736 \, n$ rounds. We also improve the previously best lower bound of $(\ln 2 + o(1)) \, n > 0.693 \, n$ and show that the player cannot achieve the desired property in less than $\alpha n$ rounds, where the value of $\alpha > 0.932$ is derived from a solution to another system of differential equations. As a result, the gap between the upper and lower bounds is decreased roughly four times.

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.