We consider the Online Facility Location (OFL) problem in the framework of learning-augmented online algorithms. In Online Facility Location (OFL), demands arrive one-by-one in a metric space and must be (irrevocably) assigned to an open facility upon arrival, without any knowledge about future demands. We focus on uniform facility opening costs and present an online algorithm for OFL that exploits potentially imperfect predictions on the locations of the optimal facilities. We prove that the competitive ratio decreases from sublogarithmic in the number n of demands to constant as the so-called η1 error, i.e., the sum of distances of the predicted locations to the optimal facility locations, decreases towards zero. E.g., our analysis implies that if for some ε > 0, η1 = OPT / n^ε, where OPT is the cost of the optimal solution, the competitive ratio is O(1/ε). We complement our analysis with a matching lower bound establishing that the dependence of the algorithm's competitive ratio on the η1 error is optimal, up to constant factors.

We provide efficient algorithms for the problem of distribution learning from high-dimensional Gaussian data where in each sample, some of the variable values are missing. We suppose that the variables are {\em missing not at random (MNAR)}. The missingness model, denoted by $\mathbb{S}(\mathbf{y})$, is the function that maps any point $\mathbf{y}\in \mathbb{R}^d$ to the subsets of its coordinates that are seen. In this work, we assume that it is known. We study the following two settings: - [\textbf{Self-censoring}] An observation $\mathbf{x}$ is generated by first sampling the true value $\mathbf{y}$ from a $d$-dimensional Gaussian $\mathcal{N}(\mathbf{\mu}^*, \Sigma^*)$ with unknown $\mathbf{\mu}^*$ and $\Sigma^*$. For each coordinate $i$, there exists a set $S_i\subseteq \mathbb{R}^d$ such that $x_i=y_i$ if and only if $y_i\in S_i$. Otherwise, $x_i$ is missing and takes a generic value (e.g “?"). We design an algorithm that learns $\mathcal{N}(\mathbf{\mu}^*, \Sigma^*)$ up to TV distance $\varepsilon$, using $\textup{poly}(d, 1/\varepsilon)$ samples, assuming only that each pair of coordinates is observed with sufficiently high probability. - [\textbf{Linear thresholding}] An observation $\mathbf{x}$ is generated by first sampling $\mathbf{y}$ from a $d$-dimensional Gaussian $\mathcal{N}(\mathbf{\mu}^*, \Sigma)$ with unknown $\mathbf{\mu}^*$ and known $\Sigma$, and then applying the missingness model $\mathbb{S}$ where $\mathbb{S}(\mathbf{y}) = \{i \in [d]: \mathbf{v}_i^T \mathbf{y} \leq b_i\}$ for some $\mathbf{v}_1, …, \mathbf{v}_d \in \mathbb{R}^d$ and $b_1, …, b_d \in \mathbb{R}$. We design an efficient mean estimation algorithm, assuming that none of the possible missingness patterns is very rare conditioned on the values of the observed coordinates and that any small subset of coordinates is observed with sufficiently high probability.

We consider the task of Gaussian mean testing, that is, of testing whether a high-dimensional vector perturbed by white noise has large magnitude, or is the zero vector. This question, originating from the signal processing community, has recently seen a surge of interest from the machine learning and theoretical computer science community, and is by now fairly well understood. What is much less understood, and the focus of our work, is how to perform this task under \emph{truncation}: that is, when the observations (i.i.d. samples from the underlying high-dimensional Gaussian) are only observed when they fall in an given subset of the domain $\mathbb{R}^d$. This truncation model, previously studied in the context of \emph{learning} (instead of \emph{testing}) the mean vector, has a range of applications, in particular in Economics and Social Sciences. As our work shows, sample truncations affect the complexity of the testing task in a rather subtle and surprising way.

We revisit the problem of distribution learning within the framework of learning-augmented algorithms. In this setting, we explore the scenario where a probability distribution is provided as potentially inaccurate advice on the true, unknown distribution. Our objective is to develop learning algorithms whose sample complexity decreases as the quality of the advice improves, thereby surpassing standard learning lower bounds when the advice is sufficiently accurate. Specifically, we demonstrate that this outcome is achievable for the problem of learning a multivariate Gaussian distribution $N(\mu, \Sigma)$ in the PAC learning setting. Classically, in the advice-free setting, $\widetilde{\Theta}(d^2/\varepsilon^2)$ samples are sufficient and worst case necessary to learn $d$-dimensional Gaussians up to TV distance $\varepsilon$ with constant probability. When we are additionally given a parameter $\widetilde{\Sigma}$ as advice, we show that $\widetilde{\mathcal{O}}(d^{2-\beta}/\varepsilon^2)$ samples suffices whenever $|| \widetilde{\Sigma}^{-1/2} \Sigma \widetilde{\Sigma}^{-1/2} - I_d ||_1 \leq \varepsilon d^{1-\beta}$ (where $||\cdot||_1$ denotes the entrywise $\ell_1$ norm) for any $\beta > 0$, yielding a polynomial improvement over the advice-free setting.

Given i.i.d.~samples from an unknown distribution $P$, the goal of distribution learning is to recover the parameters of a distribution that is close to $P$. When $P$ belongs to the class of product distributions on the Boolean hypercube $\{0,1\}^d$, it is known that $\Omega(d/\epsilon^2)$ samples are necessary to learn $P$ within total variation (TV) distance $\epsilon$. We revisit this problem when the learner is also given as advice the parameters of a product distribution $Q$. We show that there is an efficient algorithm to learn $P$ within TV distance $\epsilon$ that has sample complexity $\tilde{O}(d^{1-\eta}/\epsilon^2)$, if $\|\mathbf{p} - \mathbf{q}\|_1<\epsilon d^{0.5 - \Omega(\eta)}$. Here, $\mathbf{p}$ and $\mathbf{q}$ are the mean vectors of $P$ and $Q$ respectively, and no bound on $\|\mathbf{p} - \mathbf{q}\|_1$ is known to the algorithm a priori.

We study the problem of online unweighted bipartite matching with $n$ offline vertices and $n$ online vertices where one wishes to be competitive against the optimal offline algorithm. While the classic RANKING algorithm of (Karp et al., 1990) provably attains competitive ratio of $1-1/e > 1/2$, we show that no learning-augmented method can be both 1-consistent and strictly better than 1/2-robust under the adversarial arrival model. Meanwhile, under the random arrival model, we show how one can utilize methods from distribution testing to design an algorithm that takes in external advice about the online vertices and provably achieves competitive ratio interpolating between any ratio attainable by advice-free methods and the optimal ratio of 1, depending on the advice quality.

We study the online Traveling Salesman Problem (TSP) on the line augmented with machine-learned predictions. In the classical problem, there is a stream of requests released over time along the real line. The goal is to minimize the makespan of the algorithm. We distinguish between the open variant and the closed one, in which we additionally require the algorithm to return to the origin after serving all requests. The state of the art is a 1.64-competitive algorithm and a 2.04-competitive algorithm for the closed and open variants, respectively. In both cases, a tight lower bound is known. In both variants, our primary prediction model involves predicted positions of the requests. We introduce algorithms that (i) obtain a tight 1.5 competitive ratio for the closed variant and a 1.66 competitive ratio for the open variant in the case of perfect predictions, (ii) are robust against unbounded prediction error, and (iii) are smooth, i.e., their performance degrades gracefully as the prediction error increases. Moreover, we further investigate the learning-augmented setting in the open variant by additionally considering a prediction for the last request served by the optimal offline algorithm. Our algorithm for this enhanced setting obtains a 1.33 competitive ratio with perfect predictions while also being smooth and robust, beating the lower bound of 1.44 we show for our original prediction setting for the open variant. Also, we provide a lower bound of 1.25 for this enhanced setting.

We study the Online Traveling Salesperson Problem (OLTSP) with predictions. In OLTSP, a sequence of initially unknown requests arrive over time at points (locations) of a metric space. The goal is, starting from a particular point of the metric space (the origin), to serve all these requests while minimizing the total time spent. The server moves with unit speed or is "waiting" (zero speed) at some location. We consider two variants: in the open variant, the goal is achieved when the last request is served. In the closed one, the server additionally has to return to the origin. We adopt a prediction model, introduced for OLTSP on the line [Gouleakis et al., 2023], in which the predictions correspond to the locations of the requests and extend it to more general metric spaces. We first propose an oracle-based algorithmic framework, inspired by previous work [Bampis et al., 2023]. This framework allows us to design online algorithms for general metric spaces that provide competitive ratio guarantees which, given perfect predictions, beat the best possible classical guarantee (consistency). Moreover, they degrade gracefully along with the increase in error (smoothness), but always within a constant factor of the best known competitive ratio in the classical case (robustness). Having reduced the problem to designing suitable efficient oracles, we describe how to achieve this for general metric spaces as well as specific metric spaces (rings, trees and flowers), the resulting algorithms being tractable in the latter case. The consistency guarantees of our algorithms are tight in almost all cases, and their smoothness guarantees only suffer a linear dependency on the error, which we show is necessary. Finally, we provide robustness guarantees improving previous results.

We introduce the problem of active causal structure learning with advice. In the typical well-studied setting, the learning algorithm is given the essential graph for the observational distribution and is asked to recover the underlying causal directed acyclic graph (DAG) $G^*$ while minimizing the number of interventions made. In our setting, we are additionally given side information about $G^*$ as advice, e.g. a DAG $G$ purported to be $G^*$. We ask whether the learning algorithm can benefit from the advice when it is close to being correct, while still having worst-case guarantees even when the advice is arbitrarily bad. Our work is in the same space as the growing body of research on _algorithms with predictions_. When the advice is a DAG $G$, we design an adaptive search algorithm to recover $G^*$ whose intervention cost is at most $\mathcal{O}(\max\{1, \log \psi\})$ times the cost for verifying $G^*$; here, $\psi$ is a distance measure between $G$ and $G^*$ that is upper bounded by the number of variables $n$, and is exactly 0 when $G=G^*$. Our approximation factor matches the state-of-the-art for the advice-less setting.

Abstract In this paper, we refine the (almost) existentially optimal distributed Laplacian solver of Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS ‘21) into an (almost) universally optimal distributed Laplacian solver. Specifically, when the topology is known (i.e., the Supported-CONGEST model), we show that any Laplacian system on an n -node graph with shortcut quality $$\textrm{SQ}(G)$$ SQ ( G ) can be solved after $$n^{o(1)} \text {SQ}(G) \log (1/\epsilon )$$ n o ( 1 ) SQ ( G ) log ( 1 / ϵ ) rounds, where $$\epsilon >0$$ ϵ > 0 is the required accuracy. This almost matches our lower bound that guarantees that any correct algorithm on G requires $$\widetilde{\Omega }(\textrm{SQ}(G))$$ Ω ~ ( SQ ( G ) ) rounds, even for a crude solution with $$\epsilon \le 1/2$$ ϵ ≤ 1 / 2 . Several important implications hold in the unknown-topology (i.e., standard CONGEST) case: for excluded-minor graphs we get an almost universally optimal algorithm that terminates in $$D \cdot n^{o(1)} \log (1/\epsilon )$$ D · n o ( 1 ) log ( 1 / ϵ ) rounds, where D is the hop-diameter of the network; as well as $$n^{o(1)} \log (1/\epsilon )$$ n o ( 1 ) log ( 1 / ϵ ) -round algorithms for the case of $$\textrm{SQ}(G) \le n^{o(1)}$$ SQ ( G ) ≤ n o ( 1 ) , which holds for most networks of interest. Moreover, following a recent line of work in distributed algorithms, we consider a hybrid communication model which enhances CONGEST with limited global power in the form of the node-capacitated cli

This paper refines the distributed Laplacian solver recently developed by Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS '21) via the Ghaffari-Haeupler framework (SODA '16) of low-congestion shortcuts. Specifically, if ε > 0 is the error of the Laplacian solver, we obtain two main results.

This work provides several new insights on the robustness of Kearns’ statistical query framework against challenging label-noise models. First, we build on a recent result by \cite{DBLP:journals/corr/abs-2006-04787} that showed noise tolerance of distribution-independently evolvable concept classes under Massart noise. Specifically, we extend their characterization to more general noise models, including the Tsybakov model which considerably generalizes the Massart condition by allowing the flipping probability to be arbitrarily close to $\frac{1}{2}$ for a subset of the domain. As a corollary, we employ an evolutionary algorithm by \cite{DBLP:conf/colt/KanadeVV10} to obtain the first polynomial time algorithm with arbitrarily small excess error for learning linear threshold functions over any spherically symmetric distribution in the presence of spherically symmetric Tsybakov noise. Moreover, we posit access to a stronger oracle, in which for every labeled example we additionally obtain its flipping probability. In this model, we show that every SQ learnable class admits an efficient learning algorithm with $\opt + \epsilon$ misclassification error for a broad class of noise models. This setting substantially generalizes the widely-studied problem of classification under RCN with known noise rate, and corresponds to a non-convex optimization problem even when the noise function – i.e. the flipping probabilities of all points – is known in advance.

We study the problem of testing discrete distributions with a focus on the high probability regime. Specifically, given samples from one or more discrete distributions, a property P, and parameters 0< є, δ <1, we want to distinguish with probability at least 1−δ whether these distributions satisfy P or are є-far from P in total variation distance. Most prior work in distribution testing studied the constant confidence case (corresponding to δ = Ω(1)), and provided sample-optimal testers for a range of properties. While one can always boost the confidence probability of any such tester by black-box amplification, this generic boosting method typically leads to sub-optimal sample bounds.

The $\hybrid$ model was recently introduced by Augustine et al. \cite{DBLP:conf/soda/AugustineHKSS20} in order to characterize from an algorithmic standpoint the capabilities of networks which combine multiple communication modes. Concretely, it is assumed that the standard $\local$ model of distributed computing is enhanced with the feature of all-to-all communication, but with very limited bandwidth, captured by the node-capacitated clique ($\ncc$). In this work we provide several new insights on the power of hybrid networks for fundamental problems in distributed algorithms. First, we present a deterministic algorithm which solves any problem on a sparse $n$-node graph in $\widetilde{\mathcal{O}}(\sqrt{n})$ rounds of $\hybrid$. We combine this primitive with several sparsification techniques to obtain efficient distributed algorithms for general graphs. Most notably, for the all-pairs shortest paths problem we give deterministic $(1 + ε)$- and $\log n/\log \log n$-approximate algorithms for unweighted and weighted graphs respectively with round complexity $\widetilde{\mathcal{O}}(\sqrt{n})$ in $\hybrid$, closely matching the performance of the state of the art randomized algorithm of Kuhn and Schneider \cite{10.1145/3382734.3405719}. Moreover, we make a connection with the Ghaffari-Haeupler framework of low-congestion shortcuts \cite{DBLP:conf/soda/GhaffariH16}, leading -- among others -- to a $(1 + ε)$-approximate algorithm for Min-Cut after $\log^{\mathcal{O}(1)}n$ rounds, with high probability, even if we restrict local edges to transfer $\mathcal{O}(\log n)$-bits per round. Finally, we prove via a reduction from the set disjointness problem that $\widetildeΩ(n^{1/3})$ rounds are required to determine the radius of an unweighted graph, as well as a $(3/2 - ε)$-approximation for weighted graphs.

The classical analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. Often this is not an issue with machine learning approaches, which shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take them into account. In particular, we study the following online selection problems: (i) the classical secretary problem, (ii) online bipartite matching and (iii) the graphic matroid secretary problem. Our algorithms still come with a worst-case performance guarantee in the case that predictions are subpar while obtaining an improved competitive ratio (over the best-known classical online algorithm for each problem) when the predictions are sufficiently accurate. For each algorithm, we establish a trade-off between the competitive ratios obtained in the two respective cases.

We consider the task of designing Local Computation Algorithms (LCA) for applications of the Lovasz Local Lemma (LLL). LCA is a class of sublinear algorithms proposed by Rubinfeld et al. that have received a lot of attention in recent years. The LLL is an existential, sufficient condition for a collection of sets to have non-empty intersection (in applications, often, each set comprises all objects having a certain property). The ground-breaking algorithm of Moser and Tardos made the LLL fully constructive, following earlier results by Beck and Alon giving algorithms under significantly stronger LLL-like conditions. LCAs under those stronger conditions were given in Rubinfeld et al., where it was asked if the Moser-Tardos algorithm can be used to design LCAs under the standard LLL condition. The main contribution of this paper is to answer this question affirmatively. In fact, our techniques yield LCAs for settings beyond the standard LLL condition.

In this work, we consider the sample complexity required for testing the monotonicity of distributions over partial orders. A distribution $p$ over a poset is {\em monotone} if, for any pair of domain elements $x$ and $y$ such that $x \preceq y$, $p(x) \leq p(y)$. To understand the sample complexity of this problem, we introduce a new property called \emph{bigness} over a finite domain, where the distribution is $T$-big if the minimum probability for any domain element is at least $T$. We establish a lower bound of $\Omega(n/\log n)$ for testing bigness of distributions on domains of size $n$. We then build on these lower bounds to give $\Omega(n/\log{n})$ lower bounds for testing monotonicity over a matching poset of size $n$ and significantly improved lower bounds over the hypercube poset. We give sublinear sample complexity bounds for testing bigness and for testing monotonicity over the matching poset. We then give a number of tools for analyzing upper bounds on the sample complexity of the monotonicity testing problem.

We provide a computationally and statistically efficient estimator for the classical problem of truncated linear regression, where the dependent variable $y = \vec{w}^{\rm T} \vec{x}+{\varepsilon}$ and its corresponding vector of covariates $\vec{x} \in \mathbb{R}^k$ are only revealed if the dependent variable falls in some subset $S \subseteq \mathbb{R}$; otherwise the existence of the pair $(\vec{x},y)$ is hidden. This problem has remained a challenge since the early works of Tobin 1958, Amemiya et al. 1973, HAusman et al 1977, Breen et al. 1996, its applications are abundant, and its history dates back even further to the work of Galton 1897, Pearson 1908, Lee 1915, and Fisher 1931. While consistent estimators of the regression coefficients have been identified, the error rates are not well-understood, especially in high-dimensional settings. Under a “thickness assumption” about the covariance matrix of the covariates in the revealed sample, we provide a computationally efficient estimator for the coefficient vector $\vec{w}$ from $n$ revealed samples that attains $\ell_2$ error $\tilde{O}(\sqrt{k / n})$, almost recovering the guarantees of least squares in the standard (untruncated) linear regression setting. Our estimator uses Projected Stochastic Gradient Descent (PSGD) on the negative log-likelihood of the truncated sample, and only needs oracle access to the set $S$, which may otherwise be arbitrary, and in particular may be non-convex. PSGD must be restricted to an appropriately defined convex cone to guarantee that the negative log-likelihood is strongly convex, which in turn is established using concentration of matrices on variables with sub-exponential tails. We perform experiments on simulated data to illustrate the accuracy of our estimator. As a corollary of our work, we show that SGD provably learns the parameters of single-layer neural networks with noisy Relu activation functions (see Nair and Hinton 2010, Bengio et al. 2013, Gulcehre et al. 2016), given linearly many, in the number of network parameters, input-output pairs in the realizable setting.

We study distribution testing with communication and memory constraints in the following computational models: (1) The {\em one-pass streaming model} where the goal is to minimize the sample complexity of the protocol subject to a memory constraint, and (2) A {\em distributed model} where the data samples reside at multiple machines and the goal is to minimize the communication cost of the protocol. In both these models, we provide efficient algorithms for uniformity/identity testing (goodness of fit) and closeness testing (two sample testing). Moreover, we show nearly-tight lower bounds on (1) the sample complexity of any one-pass streaming tester for uniformity, subject to the memory constraint, and (2) the communication cost of any uniformity testing protocol, in a restricted “one-pass” model of communication.

We study the problem of {\em distribution-independent} PAC learning of halfspaces in the presence of Massart noise. Specifically, we are given a set of labeled examples $(\bx, y)$ drawn from a distribution $\D$ on $\R^{d+1}$ such that the marginal distribution on the unlabeled points $\bx$ is arbitrary and the labels $y$ are generated by an unknown halfspace corrupted with Massart noise at noise rate $\eta<1/2$. The goal is to find a hypothesis $h$ that minimizes the misclassification error $\pr_{(\bx, y) \sim \D} \left[ h(\bx) \neq y \right]$. We give a $\poly\left(d, 1/\eps\right)$ time algorithm for this problem with misclassification error $\eta+\eps$. We also provide evidence that improving on the error guarantee of our algorithm might be computationally hard. Prior to our work, no efficient weak (distribution-independent) learner was known in this model, even for the class of disjunctions. The existence of such an algorithm for halfspaces (or even disjunctions) has been posed as an open question in various works, starting with Sloan (1988), Cohen (1997), and was most recently highlighted in Avrim Blum's FOCS 2003 tutorial.

A wide range of learning tasks require human input in labeling massive data. The collected data though are usually low quality and contain inaccuracies and errors. As a result, modern science and business face the problem of learning from unreliable data sets. In this work, we provide a generic approach that is based on \textit{verification} of only few records of the data set to guarantee high quality learning outcomes for various optimization objectives. Our method, identifies small sets of critical records and verifies their validity. We show that many problems only need $\text{poly}(1/\varepsilon)$ verifications, to ensure that the output of the computation is at most a factor of $(1 \pm \varepsilon)$ away from the truth. For any given instance, we provide an \textit{instance optimal} solution that verifies the minimum possible number of records to approximately certify correctness. Then using this instance optimal formulation of the problem we prove our main result: “every function that satisfies some Lipschitz continuity condition can be certified with a small number of verifications”. We show that the required Lipschitz continuity condition is satisfied even by some NP-complete problems, which illustrates the generality and importance of this theorem. In case this certification step fails, an invalid record will be identified. Removing these records and repeating until success, guarantees that the result will be accurate and will depend only on the verified records. Surprisingly, as we show, for several computation tasks more efficient methods are possible. These methods always guarantee that the produced result is not affected by the invalid records, since any invalid record that affects the output will be detected and verified.

We provide an efficient algorithm for the classical problem, going back to Galton, Pearson, and Fisher, of estimating, with arbitrary accuracy the parameters of a multivariate normal distribution from truncated samples. Truncated samples from a d-variate normal N(mu, Sigma) means a samples is only revealed if it falls in some subset S of the d-dimensional Euclidean space; otherwise the samples are hidden and their count in proportion to the revealed samples is also hidden. We show that the mean mu and covariance matrix Sigma can be estimated with arbitrary accuracy in polynomial-time, as long as we have oracle access to S, and S has non-trivial measure under the unknown d-variate normal distribution. Additionally we show that without oracle access to S, any non-trivial estimation is impossible.

We study the problem of testing identity against a given distribution with a focus on the high confidence regime. More precisely, given samples from an unknown distribution p over n elements, an explicitly given distribution q, and parameters 0< epsilon, delta < 1, we wish to distinguish, with probability at least 1-delta, whether the distributions are identical versus epsilon-far in total variation distance. Most prior work focused on the case that delta = Omega(1), for which the sample complexity of identity testing is known to be Theta(sqrt{n}/epsilon^2). Given such an algorithm, one can achieve arbitrarily small values of delta via black-box amplification, which multiplies the required number of samples by Theta(log(1/delta)). We show that black-box amplification is suboptimal for any delta = o(1), and give a new identity tester that achieves the optimal sample complexity. Our new upper and lower bounds show that the optimal sample complexity of identity testing is Theta((1/epsilon^2) (sqrt{n log(1/delta)} + log(1/delta))) for any n, epsilon, and delta. For the special case of uniformity testing, where the given distribution is the uniform distribution U_n over the domain, our new tester is surprisingly simple: to test whether p = U_n versus d_{TV} (p, U_n) >= epsilon, we simply threshold d_{TV}({p^}, U_n), where {p^} is the empirical probability distribution. The fact that this simple "plug-in" estimator is sample-optimal is surprising, even in the constant delta case. Indeed, it was believed that such a tester would not attain sublinear sample complexity even for constant values of epsilon and delta. An important contribution of this work lies in the analysis techniques that we introduce in this context. First, we exploit an underlying strong convexity property to bound from below the expectation gap in the completeness and soundness cases. Second, we give a new, fast method for obtaining provably correct empirical estimates of the true worst-case failure probability for a broad class of uniformity testing statistics over all possible input distributions - including all previously studied statistics for this problem. We believe that our novel analysis techniques will be useful for other distribution testing problems as well.

We present O(loglog n) -round algorithms in the Massively Parallel Computation (MPC) model, with Õ (n) memory per machine, that compute a maximal independent set, a 1+ε approximation of maximum matching, and a 2+εapproximation of minimum vertex cover, for any n-vertex graph and any constant \eps>0. These improve the state of the art as follows:

We study the question of testing structured properties (classes) of discrete distributions. Specifically, given sample access to an arbitrary distribution D over [n] and a property $\mathcal {P}$ , the goal is to distinguish between D ∈ $\mathcal {P}$ and ℓ1(D, $\mathcal {P}$ ) > ε. We develop a general algorithm for this question, which applies to a large range of “shape-constrained” properties, including monotone, log-concave, t-modal, piecewise-polynomial, and Poisson Binomial distributions. Moreover, for all cases considered, our algorithm has near-optimal sample complexity with regard to the domain size and is computationally efficient. For most of these classes, we provide the first non-trivial tester in the literature. In addition, we also describe a generic method to prove lower bounds for this problem, and use it to show our upper bounds are nearly tight. Finally, we extend some of our techniques to tolerant testing, deriving nearly–tight upper and lower bounds for the corresponding questions.

In many situations, sample data is obtained from a noisy or imperfect source. In order to address such corruptions, this paper introduces the concept of a sampling corrector. Such algorithms use structure that the distribution is purported to have, in order to allow one to make “on-the-fly” corrections to samples drawn from probability distributions. These algorithms then act as filters between the noisy data and the end user. We show connections between sampling correctors, distribution learning algorithms, and distribution property testing algorithms. We show that these connections can be utilized to expand the applicability of known distribution learning and property testing algorithms as well as to achieve improved algorithms for those tasks. As a first step, we show how to design sampling correctors using proper learning algorithms. We then focus on the question of whether algorithms for sampling correctors can be more efficient in terms of sample complexity than learning algorithms for the analogous families of distributions. When correcting monotonicity, we show that this is indeed the case when also granted query access to the cumulative distribution function. We also obtain sampling correctors for monotonicity even without this stronger type of access, provided that the distribution be originally very close to monotone (namely, at a distance $O(1/\log^2 n)$). In addition to that, we consider a restricted error model that aims at capturing “missing data” corruptions. In this model, we show that distributions that are close to monotone have sampling correctors that are significantly more efficient than achievable by the learning approach. We consider the question of whether an additional source of independent random bits is required by sampling correctors to implement the correction process. We show that for correcting close-to-uniform distributions and close-to-monotone distributions, no additional source of random bits is required, as the samples from the input source itself can be used to produce this randomness.

A conditional sampling oracle for a probability distribution D returns samples from the conditional distribution of D restricted to a specified subset of the domain. A recent line of work [7, 6] has shown that having access to such a conditional sampling oracle requires only polylogarithmic or even constant number of samples to solve distribution testing problems like identity and uniformity. This significantly improves over the standard sampling model where polynomially many samples are necessary. Inspired by these results, we introduce a computational model based on conditional sampling to develop sublinear algorithms with exponentially faster runtimes compared to standard sublinear algorithms. We focus on geometric optimization problems over points in high dimensional Euclidean space. Access to these points is provided via a conditional sampling oracle that takes as input a succinct representation of a subset of the domain and outputs a uniformly random point in that subset. We study two well studied problems: k-means clustering and estimating the weight of the minimum spanning tree. In contrast to prior algorithms for the classic model, our algorithms have time, space and sample complexity that is polynomial in the dimension and polylogarithmic in the number of points. Finally, we comment on the applicability of the model and compare with existing ones like streaming, parallel and distributed computational models.

In many situations, sample data is obtained from a noisy or imperfect source. In order to address such corruptions, this paper introduces the concept of a sampling corrector. Such algorithms use structure that the distribution is purported to have, in order to allow one to make "on-the-fly" corrections to samples drawn from probability distributions. These algorithms then act as filters between the noisy data and the end user.

The Lovasz Local Lemma (LLL) is a powerful tool that can be used to prove that an object having none of a set of bad properties exists, using the probabilistic method. In many applications of the LLL it is also desirable to explicitly construct the combinatorial object. Recently it was shown that this is possible using a randomized algorithm in the full asymmetric LLL setting [R. Moser and G. Tardos, 2010]. A strengthening of the LLL for the case of dense local neighborhoods proved in [R. Bissacot et al., 2010] was recently also made constructive in [W. Pegden, 2011]. In another recent work [B. Haupler, B. Saha, A. Srinivasan, 2010], it was proved that the algorithm of Moser and Tardos is still efficient even when the number of events is exponential. Here we prove that these last two contributions can be combined to yield a new version of the LLL.