442 research outputs found
Limits on Counting Triangles using Bipartite Independent Set Queries
Beame et al. [ITCS 2018 & TALG 2021] introduced and used the Bipartite
Independent Set (BIS) and Independent Set (IS) oracle access to an unknown,
simple, unweighted and undirected graph and solved the edge estimation problem.
The introduction of this oracle set forth a series of works in a short span of
time that either solved open questions mentioned by Beame et al. or were
generalizations of their work as in Dell and Lapinskas [STOC 2018], Dell,
Lapinskas and Meeks [SODA 2020], Bhattacharya et al. [ISAAC 2019 & Theory
Comput. Syst. 2021], and Chen et al. [SODA 2020]. Edge estimation using BIS can
be done using polylogarithmic queries, while IS queries need sub-linear but
more than polylogarithmic queries. Chen et al. improved Beame et al.'s upper
bound result for edge estimation using IS and also showed an almost matching
lower bound. Beame et al. in their introductory work asked a few open questions
out of which one was on estimating structures of higher order than edges, like
triangles and cliques, using BIS queries. Motivated by this question, we
completely resolve the query complexity of estimating triangles using BIS
oracle. While doing so, we prove a lower bound for an even stronger query
oracle called Edge Emptiness (EE) oracle, recently introduced by Assadi,
Chakrabarty and Khanna [ESA 2021] to test graph connectivity.Comment: 30 page
The Geometry of Differential Privacy: the Sparse and Approximate Cases
In this work, we study trade-offs between accuracy and privacy in the context
of linear queries over histograms. This is a rich class of queries that
includes contingency tables and range queries, and has been a focus of a long
line of work. For a set of linear queries over a database , we
seek to find the differentially private mechanism that has the minimum mean
squared error. For pure differential privacy, an approximation to
the optimal mechanism is known. Our first contribution is to give an approximation guarantee for the case of (\eps,\delta)-differential
privacy. Our mechanism is simple, efficient and adds correlated Gaussian noise
to the answers. We prove its approximation guarantee relative to the hereditary
discrepancy lower bound of Muthukrishnan and Nikolov, using tools from convex
geometry.
We next consider this question in the case when the number of queries exceeds
the number of individuals in the database, i.e. when . It is known that better mechanisms exist in this setting. Our second
main contribution is to give an (\eps,\delta)-differentially private
mechanism which is optimal up to a \polylog(d,N) factor for any given query
set and any given upper bound on . This approximation is
achieved by coupling the Gaussian noise addition approach with a linear
regression step. We give an analogous result for the \eps-differential
privacy setting. We also improve on the mean squared error upper bound for
answering counting queries on a database of size by Blum, Ligett, and Roth,
and match the lower bound implied by the work of Dinur and Nissim up to
logarithmic factors.
The connection between hereditary discrepancy and the privacy mechanism
enables us to derive the first polylogarithmic approximation to the hereditary
discrepancy of a matrix
Triangle Estimation Using Tripartite Independent Set Queries
Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an approximate triangle counting algorithm using only polylogarithmic queries when the number of triangles on any edge in the graph is polylogarithmically bounded. Our query oracle Tripartite Independent Set (TIS) takes three disjoint sets of vertices A, B and C as input, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class of group queries (Ron and Tsur, ACM ToCT, 2016; Dell and Lapinskas, STOC 2018) and in particular is inspired by the Bipartite Independent Set (BIS) query oracle of Beame et al. (ITCS 2018). We extend the algorithmic framework of Beame et al., with TIS replacing BIS, for triangle counting using ideas from color coding due to Alon et al. (J. ACM, 1995) and a concentration inequality for sums of random variables with bounded dependency (Janson, Rand. Struct. Alg., 2004)
Quantum Recommendation Systems
A recommendation system uses the past purchases or ratings of products by
a group of users, in order to provide personalized recommendations to
individual users. The information is modeled as an preference
matrix which is assumed to have a good rank- approximation, for a small
constant .
In this work, we present a quantum algorithm for recommendation systems that
has running time . All known classical
algorithms for recommendation systems that work through reconstructing an
approximation of the preference matrix run in time polynomial in the matrix
dimension. Our algorithm provides good recommendations by sampling efficiently
from an approximation of the preference matrix, without reconstructing the
entire matrix. For this, we design an efficient quantum procedure to project a
given vector onto the row space of a given matrix. This is the first algorithm
for recommendation systems that runs in time polylogarithmic in the dimensions
of the matrix and provides an example of a quantum machine learning algorithm
for a real world application.Comment: 22 page
Fine-Grained Reductions from Approximate Counting to Decision
In this paper, we introduce a general framework for fine-grained reductions
of approximate counting problems to their decision versions. (Thus we use an
oracle that decides whether any witness exists to multiplicatively approximate
the number of witnesses with minimal overhead.) This mirrors a foundational
result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the
polynomial-time setting, and a similar result of M\"uller (IWPEC 2006) in the
FPT setting. Using our framework, we obtain such reductions for some of the
most important problems in fine-grained complexity: the Orthogonal Vectors
problem, 3SUM, and the Negative-Weight Triangle problem (which is closely
related to All-Pairs Shortest Path).
We also provide a fine-grained reduction from approximate #SAT to SAT.
Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for
some and all there is an -time algorithm for k-SAT. Then we
prove that for all , there is an -time algorithm for
approximate #-SAT. In particular, our result implies that the Exponential
Time Hypothesis (ETH) is equivalent to the seemingly-weaker statement that
there is no algorithm to approximate #3-SAT to within a factor of
in time (taking as part of the input).Comment: An extended abstract was presented at STOC 201
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