Elena Grigorescu

TCS Journal 2023 Journal Article

On computing discretized Ricci curvatures of graphs: Local algorithms and (localized) fine-grained reductions

• Bhaskar DasGupta
• Elena Grigorescu
• Tamalika Mukherjee

Characterizing shapes of high-dimensional objects via Ricci curvatures plays a critical role in many research areas in mathematics and physics. However, even though several discretizations of Ricci curvatures for discrete combinatorial objects such as networks have been proposed and studied by mathematicians, the computational complexity aspects of these discretizations have escaped the attention of theoretical computer scientists to a large extent. In this paper, we study one such discretization, namely the Ollivier-Ricci curvature, from the perspective of efficient computation by fine-grained reductions and local query-based algorithms. Our main contributions are the following. ▹ We relate our curvature computation problem to minimum weight perfect matching problem on complete bipartite graphs via fine-grained reduction. ▹ We formalize the computational aspects of the curvature computation problems in suitable frameworks so that they can be studied by researchers in local algorithms. ▹ We provide the first known lower and upper bounds on queries for query-based algorithms for the curvature computation problems in our local algorithms framework. En route, we also illustrate a localized version of our fine-grained reduction. We believe that our results bring forth an intriguing set of research questions, motivated both in theory and practice, regarding designing efficient algorithms for curvatures of geometrical objects.

NeurIPS Conference 2022 Conference Paper

Learning-Augmented Algorithms for Online Linear and Semidefinite Programming

• Elena Grigorescu
• Young-San Lin
• Sandeep Silwal
• Maoyuan Song
• Samson Zhou

Semidefinite programming (SDP) is a unifying framework that generalizes both linear programming and quadratically-constrained quadratic programming, while also yielding efficient solvers, both in theory and in practice. However, there exist known impossibility results for approximating the optimal solution when constraints for covering SDPs arrive in an online fashion. In this paper, we study online covering linear and semidefinite programs in which the algorithm is augmented with advice from a possibly erroneous predictor. We show that if the predictor is accurate, we can efficiently bypass these impossibility results and achieve a constant-factor approximation to the optimal solution, i. e. , consistency. On the other hand, if the predictor is inaccurate, under some technical conditions, we achieve results that match both the classical optimal upper bounds and the tight lower bounds up to constant factors, i. e. , robustness. More broadly, we introduce a framework that extends both (1) the online set cover problem augmented with machine-learning predictors, studied by Bamas, Maggiori, and Svensson (NeurIPS 2020), and (2) the online covering SDP problem, initiated by Elad, Kale, and Naor (ICALP 2016). Specifically, we obtain general online learning-augmented algorithms for covering linear programs with fractional advice and constraints, and initiate the study of learning-augmented algorithms for covering SDP problems. Our techniques are based on the primal-dual framework of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and can be further adjusted to handle constraints where the variables lie in a bounded region, i. e. , box constraints.

FOCS Conference 2021 Conference Paper

Exponential Lower Bounds for Locally Decodable and Correctable Codes for Insertions and Deletions

• Jeremiah Blocki
• Kuan Cheng
• Elena Grigorescu
• Xin Li 0006
• Yu Zheng 0014
• Minshen Zhu

Locally Decodable Codes (LDCs) are error-correcting codes for which individual message symbols can be quickly recovered despite errors in the codeword. LDCs for Hamming errors have been studied extensively in the past few decades, where a major goal is to understand the amount of redundancy that is necessary and sufficient to decode from large amounts of error, with small query complexity. Despite exciting progress, we still don't have satisfactory answers in several important parameter regimes. For example, in the case of 3-query LDCs, the gap between existing constructions and lower bounds is superpolynomial in the message length. In this work we study LDCs for insertion and deletion errors, called Insdel LDCs. Their study was initiated by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), who gave a reduction from Hamming LDCs to Insdel LDCs with a small blowup in the code parameters. On the other hand, the only known lower bounds for Insdel LDCs come from those for Hamming LDCs, thus there is no separation between them. Here we prove new, strong lower bounds for the existence of Insdel LDCs. In particular, we show that 2-query linear Insdel LDCs do not exist, and give an exponential lower bound for the length of all q-query Insdel LDCs with constant q. For $q$ ≥ 3 our bounds are exponential in the existing lower bounds for Hamming LDCs. Furthermore, our exponential lower bounds continue to hold for adaptive decoders, and even in private-key settings where the encoder and decoder share secret randomness. This exhibits a strict separation between Hamming LDCs and Insdel LDCs. Our strong lower bounds also hold for the related notion of Insdel LCCs (except in the private-key setting), due to an analogue to the Insdel notions of a reduction from Hamming LCCs to LDCs. Our techniques are based on a delicate design and analysis of hard distributions of insertion and deletion errors, which depart significantly from typical techniques used in analyzing Hamming LDCs.

NeurIPS Conference 2017 Conference Paper

Communication-Efficient Distributed Learning of Discrete Distributions

• Ilias Diakonikolas
• Elena Grigorescu
• Jerry Li
• Abhiram Natarajan
• Krzysztof Onak
• Ludwig Schmidt

We initiate a systematic investigation of distribution learning (density estimation) when the data is distributed across multiple servers. The servers must communicate with a referee and the goal is to estimate the underlying distribution with as few bits of communication as possible. We focus on non-parametric density estimation of discrete distributions with respect to the l1 and l2 norms. We provide the first non-trivial upper and lower bounds on the communication complexity of this basic estimation task in various settings of interest. Specifically, our results include the following: 1. When the unknown discrete distribution is unstructured and each server has only one sample, we show that any blackboard protocol (i. e. , any protocol in which servers interact arbitrarily using public messages) that learns the distribution must essentially communicate the entire sample. 2. For the case of structured distributions, such as k-histograms and monotone distributions, we design distributed learning algorithms that achieve significantly better communication guarantees than the naive ones, and obtain tight upper and lower bounds in several regimes. Our distributed learning algorithms run in near-linear time and are robust to model misspecification. Our results provide insights on the interplay between structure and communication efficiency for a range of fundamental distribution estimation tasks.

FOCS Conference 2016 Conference Paper

NP-Hardness of Reed-Solomon Decoding and the Prouhet-Tarry-Escott Problem

• Venkata Gandikota
• Badih Ghazi
• Elena Grigorescu

Establishing the complexity of Bounded Distance Decoding for Reed-Solomon codes is a fundamental open problem in coding theory, explicitly asked by Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005). The problem is motivated by the large current gap between the regime when it is NP-hard, and the regime when it is efficiently solvable (i. e. , the Johnson radius). We show the first NP-hardness results for asymptotically smaller decoding radii than the maximum likelihood decoding radius of Guruswami and Vardy. Specifically, for Reed-Solomon codes of length N and dimension K = O(N), we show that it is NP-hard to decode more than N-K-O/log N log log N) errors. Moreover, we show that the problem is NP-hard under quasipolynomial-time reductions for an error amount > N-K-c log N (with c > 0 an absolute constant). An alternative natural reformulation of the Bounded Distance Decoding problem for Reed-Solomon codes is as a Polynomial Reconstruction problem. In this view, our results show that it is NP-hard to decide whether there exists a degree K polynomial passing through K + O(log N / log log N) points from a given set of points (a1, b1), (a2, b2). .. , (aN, bN). Furthermore, it is NP-hard under quasipolynomial-time reductions to decide whether there is a degree K polynomial passing through K + c log N many points (with c > 0 an absolute constant). These results follow from the NP-hardness of a generalization of the classical Subset Sum problem to higher moments, called Moments Subset Sum, which has been a known open problem, and which may be of independent interest. We further reveal a strong connection with the well-studied Prouhet-Tarry-Escott problem in Number Theory, which turns out to capture a main barrier in extending our techniques. We believe the Prouhet-Tarry-Escott problem deserves further study in the theoretical computer science community.

STOC Conference 2013 Conference Paper

Statistical algorithms and a lower bound for detecting planted cliques

• Vitaly Feldman
• Elena Grigorescu
• Lev Reyzin
• Santosh S. Vempala
• Ying Xiao 0003

We introduce a framework for proving lower bounds on computational problems over distributions, based on a class of algorithms called statistical algorithms . For such algorithms, access to the input distribution is limited to obtaining an estimate of the expectation of any given function on a sample drawn randomly from the input distribution, rather than directly accessing samples. Most natural algorithms of interest in theory and in practice, e.g., moments-based methods, local search, standard iterative methods for convex optimization, MCMC and simulated annealing, are statistical algorithms or have statistical counterparts. Our framework is inspired by and generalize the statistical query model in learning theory [34]. Our main application is a nearly optimal lower bound on the complexity of any statistical algorithm for detecting planted bipartite clique distributions (or planted dense subgraph distributions) when the planted clique has size O(n 1/2-δ ) for any constant δ > 0. Variants of these problems have been assumed to be hard to prove hardness for other problems and for cryptographic applications. Our lower bounds provide concrete evidence of hardness, thus supporting these assumptions.

SODA Conference 2012 Conference Paper

Testing odd-cycle-freeness in Boolean functions

• Arnab Bhattacharyya 0001
• Elena Grigorescu
• Prasad Raghavendra
• Asaf Shapira

FOCS Conference 2010 Conference Paper

A Unified Framework for Testing Linear-Invariant Properties

• Arnab Bhattacharyya 0001
• Elena Grigorescu
• Asaf Shapira

There has been a sequence of recent papers devoted to understanding the relation between the testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Invariance with respect to F 2 -linear transformations is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties. This is explicitly posed as an open problem in a recent survey of Sudan. We obtain the following results: 1. We show that every linear-invariant property that can be characterized by forbidding induced solutions to a (possibly infinite) set of linear equations can be tested with one-sided error. 2. We show that every linear-invariant property that can be tested with one-sided error can be characterized by forbidding induced solutions to a (possibly infinite) set of systems of linear equations. We conjecture that our result from item (1) can be extended to cover systems of linear equations. We further show that the validity of this conjecture would have the following implications: 1. It would imply that every linear-invariant property that is closed under restrictions to linear subspaces is testable with one-sided error. Such a result would unify several previous results on testing Boolean functions, such as the testability of low-degree polynomials and of Fourier dimensionality. 2. It would imply that a linear-invariant property P is testable with one-sided error if and only if P is closed under restrictions to linear subspaces, thus resolving Sudan's problem.

SODA Conference 2009 Conference Paper

Transitive-closure spanners

• Arnab Bhattacharyya 0001
• Elena Grigorescu
• Kyomin Jung
• Sofya Raskhodnikova
• David P. Woodruff

We define the notion of a transitive-closure spanner of a directed graph. Given a directed graph G = ( V, E ) and an integer k ≥ 1, a k-transitive-closure-spanner ( k-TC-spanner ) of G is a directed graph H = ( V, E H ) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were studied implicitly in access control, property testing, and data structures, and properties of these spanners have been rediscovered over the span of 20 years. We bring these areas under the unifying framework of TC-spanners. We abstract the common task implicitly tackled in these diverse applications as the problem of constructing sparse TC-spanners. We study the approximability of the size of the sparsest k -TC-spanner for a given digraph. Our technical contributions fall into three categories: algorithms for general digraphs, inapproximability results, and structural bounds for a specific graph family which imply an efficient algorithm with a good approximation ratio for that family. Algorithms. We present two efficient deterministic algorithms that find k -TC-spanners of near optimal size. The first algorithm gives an -approximation for k > 2. Our method, based on a combination of convex programming and sampling, yields the first sublinear approximation ratios for (1) D irected k -S panner, a well-studied generalization of k -TC-S panner, and (2) its variants C lient /S erver D irected k -S panner, and the k -D iameter S panning S ubgraph. This resolves the main open question of Elkin and Peleg (IPCO, 2001). The second algorithm, specific to the k -TC-spanner problem, gives an -approximation. It shows that for, our problem has a provably better approximation ratio than D irected k -S panner and its variants. This algorithm also resolves an open question of Hesse (SODA, 2003).

STOC Conference 2008 Conference Paper

Decodability of group homomorphisms beyond the johnson bound

• Irit Dinur
• Elena Grigorescu
• Swastik Kopparty
• Madhu Sudan 0001

Given a pair of finite groups G and H, the set of homomorphisms from G to H form an error-correcting code where codewords differ in at least 1/2 the coordinates. We show that for every pair of abelian groups G and H, the resulting code is (locally) list-decodable from a fraction of errors arbitrarily close to its distance. At the heart of this result is the following combinatorial result: There is a fixed polynomial p(•) such that for every pair of abelian groups G and H, if the maximum fraction of agreement between two distinct homomorphisms from G to H is Λ, then for every ε> 0 and every function f:G -> H, the number of homomorphisms that have agreement Λ + ε with f is at most p(1/ε). We thus give a broad class of codes whose list-decoding radius exceeds the "Johnson bound". Examples of such codes are rare in the literature, and for the ones that do exist, "combinatorial" techniques to analyze their list-decodability are limited. Our work is an attempt to add to the body of such techniques. We use the fact that abelian groups decompose into simpler ones and thus codes derived from homomorphisms over abelian groups may be viewed as certain "compositions" of simpler codes. We give techniques to lift list-decoding bounds for the component codes to bounds for the composed code. We believe these techniques may be of general interest.