104 research outputs found
On streaming approximation algorithms for constraint satisfaction problems
In this thesis, we explore streaming algorithms for approximating constraint
satisfaction problems (CSPs). The setup is roughly the following: A computer
has limited memory space, sees a long "stream" of local constraints on a set of
variables, and tries to estimate how many of the constraints may be
simultaneously satisfied. The past ten years have seen a number of works in
this area, and this thesis includes both expository material and novel
contributions. Throughout, we emphasize connections to the broader theories of
CSPs, approximability, and streaming models, and highlight interesting open
problems.
The first part of our thesis is expository: We present aspects of previous
works that completely characterize the approximability of specific CSPs like
Max-Cut and Max-Dicut with -space streaming algorithm (on
-variable instances), while characterizing the approximability of all CSPs
in space in the special case of "composable" (i.e., sketching)
algorithms, and of a particular subclass of CSPs with linear-space streaming
algorithms.
In the second part of the thesis, we present two of our own joint works. We
begin with a work with Madhu Sudan and Santhoshini Velusamy in which we prove
linear-space streaming approximation-resistance for all ordering CSPs (OCSPs),
which are "CSP-like" problems maximizing over sets of permutations. Next, we
present joint work with Joanna Boyland, Michael Hwang, Tarun Prasad, and
Santhoshini Velusamy in which we investigate the -space streaming
approximability of symmetric Boolean CSPs with negations. We give explicit
-space sketching approximability ratios for several families of CSPs,
including Max-AND; develop simpler optimal sketching approximation
algorithms for threshold predicates; and show that previous lower bounds fail
to characterize the -space streaming approximability of Max-AND.Comment: Harvard College senior thesis; 119 pages plus references; abstract
shortened for arXiv; formatted with Dissertate template (feel free to copy!);
exposits papers arXiv:2105.01782 (APPROX 2021) and arXiv:2112.06319 (APPROX
2022
Streaming Approximation Resistance of Every Ordering CSP
An ordering constraint satisfaction problem (OCSP) is given by a positive
integer and a constraint predicate mapping permutations on
to . Given an instance of OCSP on
variables and constraints, the goal is to find an ordering of the
variables that maximizes the number of constraints that are satisfied, where a
constraint specifies a sequence of distinct variables and the constraint is
satisfied by an ordering on the variables if the ordering induced on the
variables in the constraint satisfies . OCSPs capture natural problems
including "Maximum acyclic subgraph (MAS)" and "Betweenness".
In this work we consider the task of approximating the maximum number of
satisfiable constraints in the (single-pass) streaming setting, where an
instance is presented as a stream of constraints. We show that for every ,
OCSP is approximation-resistant to -space streaming algorithms.
This space bound is tight up to polylogarithmic factors. In the case of MAS our
result shows that for every , MAS is not
-approximable in space. The previous best
inapproximability result only ruled out a -approximation in
space.
Our results build on recent works of Chou, Golovnev, Sudan, Velingker, and
Velusamy who show tight, linear-space inapproximability results for a broad
class of (non-ordering) constraint satisfaction problems over arbitrary
(finite) alphabets. We design a family of appropriate CSPs (one for every )
from any given OCSP, and apply their work to this family of CSPs. We show that
the hard instances from this earlier work have a particular "small-set
expansion" property. By exploiting this combinatorial property, in combination
with the hardness results of the resulting families of CSPs, we give optimal
inapproximability results for all OCSPs.Comment: 23 pages, 1 figure. Replaces earlier version with lower
bound, using new bounds from arXiv:2106.13078. To appear in APPROX'2
Streaming beyond sketching for Maximum Directed Cut
We give an -space single-pass -approximation
streaming algorithm for estimating the maximum directed cut size
() in a directed graph on vertices. This improves over
an -space approximation algorithm due to Chou,
Golovnev, Velusamy (FOCS 2020), which was known to be optimal for
-space algorithms.
is a special case of a constraint satisfaction problem
(CSP). In this broader context, our work gives the first CSP for which
algorithms with space can provably outperform
-space algorithms on general instances. Previously, this was shown
in the restricted case of bounded-degree graphs in a previous work of the
authors (SODA 2023). Prior to that work, the only algorithms for any CSP were
based on generalizations of the -space algorithm for
, and were in particular so-called "sketching" algorithms.
In this work, we demonstrate that more sophisticated streaming algorithms can
outperform these algorithms even on general instances.
Our algorithm constructs a "snapshot" of the graph and then applies a result
of Feige and Jozeph (Algorithmica, 2015) to approximately estimate the
value from this snapshot. Constructing this snapshot is
easy for bounded-degree graphs and the main contribution of our work is to
construct this snapshot in the general setting. This involves some delicate
sampling methods as well as a host of "continuity" results on the
behaviour in graphs.Comment: 57 pages, 2 figure
Streaming complexity of CSPs with randomly ordered constraints
We initiate a study of the streaming complexity of constraint satisfaction
problems (CSPs) when the constraints arrive in a random order. We show that
there exists a CSP, namely , for which random ordering
makes a provable difference. Whereas a approximation of
requires space with adversarial ordering,
we show that with random ordering of constraints there exists a
-approximation algorithm that only needs space. We also give
new algorithms for in variants of the adversarial ordering
setting. Specifically, we give a two-pass space
-approximation algorithm for general graphs and a single-pass
space -approximation algorithm for bounded degree
graphs.
On the negative side, we prove that CSPs where the satisfying assignments of
the constraints support a one-wise independent distribution require
-space for any non-trivial approximation, even when the
constraints are randomly ordered. This was previously known only for
adversarially ordered constraints. Extending the results to randomly ordered
constraints requires switching the hard instances from a union of random
matchings to simple Erd\"os-Renyi random (hyper)graphs and extending tools that
can perform Fourier analysis on such instances.
The only CSP to have been considered previously with random ordering is
where the ordering is not known to change the
approximability. Specifically it is known to be as hard to approximate with
random ordering as with adversarial ordering, for space
algorithms. Our results show a richer variety of possibilities and motivate
further study of CSPs with randomly ordered constraints
On sketching approximations for symmetric Boolean CSPs
A Boolean maximum constraint satisfaction problem, Max-CSP(), is specified
by a predicate . An -variable instance of
Max-CSP() consists of a list of constraints, each of which applies to
distinct literals drawn from the variables. For , Chou, Golovnev,
and Velusamy [CGV20, FOCS 2020] obtained explicit ratios characterizing the
-space streaming approximability of every predicate. For ,
Chou, Golovnev, Sudan, and Velusamy [CGSV21, arXiv:2102.12351] proved a general
dichotomy theorem for -space sketching algorithms: For every ,
there exists such that for every ,
Max-CSP() is -approximable by an -space
linear sketching algorithm, but -approximation sketching
algorithms require space.
In this work, we give closed-form expressions for the sketching approximation
ratios of multiple families of symmetric Boolean functions. Letting , we show that for odd ,
AND, and for even , AND. We also resolve the ratio for the "at-least--'s"
function for all even ; the "exactly--'s" function for odd
; and fifteen other functions. We stress here that for
general , according to [CGSV21], closed-form expressions for
need not have existed a priori.
Separately, for all threshold functions, we give optimal "bias-based"
approximation algorithms generalizing [CGV20] while simplifying [CGSV21].
Finally, we investigate the -space streaming lower bounds in [CGSV21],
and show that they are incomplete for AND.Comment: 27 pages; same results but significant changes in presentatio
Over-use of thyroid testing in Canadian and UK primary care in frequent attenders : a cross-sectional study
Dr Greiver is supported through the Gordon F. Cheesbrough Research Chair in Family and Community Medicine from North York General Hospital.Background Thyroid stimulating hormone (TSH) is a common test used to detect and monitor clinically significant hypo- and hyperthyroidism. Population based screening of asymptomatic adults for thyroid disorders is not recommended. Objective The research objectives were to determine patterns of TSH testing in Canadian and English primary care practices, as well as patient and physician practice characteristics associated with testing TSH for primary care patients with no identifiable indication. Methods In this two-year cross-sectional observational study, Canadian and English electronic medical record databases were used to identify patients and physician practices. Cohorts of patients aged 18 years or older, without identifiable indications for TSH testing, were generated from these databases. Analyses were performed using a random-effects logistic regression to determine patient and physician practice characteristics associated with increased testing. We determined the proportion of TSH tests done concurrently with at least one common screening blood test (lipid profile or hemoglobin A1c). Standardized proportions of TSH test per family practice were used to examine the heterogeneity in the populations. Results At least one TSH test was done in 35.97 % (N=489,663) of Canadian patients and 29.36% (N=1,030,489) of English patients. Almost all TSH tests in Canada and England (95.69% and 99.23% respectively) were within the normal range (0.40-5.00 mU/L). A greater number of patient-physician encounters was the strongest predictor of TSH testing. 51.40% of TSH tests in Canada and 76.55% in England were done on the same day as at least one other screening blood test. There was no association between practice size and proportion of asymptomatic patients tested. Conclusions This comparative binational study found TSH patterns suggestive of over-testing and potentially thyroid disorder screening in both countries. There may be significant opportunities to improve appropriateness of TSH ordering in Canada and England and therefore improve allocation of limited system resources.PostprintPeer reviewe
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