191 research outputs found
Algorithms and complexity for approximately counting hypergraph colourings and related problems
The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Î, this problem falls into the regime of LovĂĄsz local lemma when Î âČ qá”. In prior, however, fast approximate counting algorithms exist when Î âČ qá”/Âł, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows.
âą When q, k â„ 4 are evens and Î â„ 5·qá”/ÂČ, approximating the number of hypergraph colourings is NP-hard.
âą When the input hypergraph is linear and Î âČ qá”/ÂČ, a fast approximate counting algorithm does exist
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Semidefinite Programming Relaxations of the Simplified Wasserstein Barycenter Problem: An ADMM Approach
The Simplified Wasserstein Barycenter problem, the problem of picking k points each chosen from a distinct set of n points as to minimize the sum of distances to their barycenter, finds applications in various areas of data science. Despite the simple formulation, it is a hard computational problem. The difficulty comes in the lack of efficient algorithms for approximating the solution. In this thesis, I propose a doubly non-negative relaxation to this problem and apply the alternating direction method of multipliers (ADMM) with intermediate update of multipliers, to efficiently compute tight lower and upper bounds on its optimal value for certain input data distributions. Our empirics show that generically the gap between upper and lower bounds is zero, though problems with symmetries exhibit positive gaps
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Optimal Dorfman Group Testing For Symmetric Distributions
We study Dorfman's classical group testing protocol in a novel setting where
individual specimen statuses are modeled as exchangeable random variables. We
are motivated by infectious disease screening. In that case, specimens which
arrive together for testing often originate from the same community and so
their statuses may exhibit positive correlation. Dorfman's protocol screens a
population of n specimens for a binary trait by partitioning it into
nonoverlapping groups, testing these, and only individually retesting the
specimens of each positive group. The partition is chosen to minimize the
expected number of tests under a probabilistic model of specimen statuses. We
relax the typical assumption that these are independent and indentically
distributed and instead model them as exchangeable random variables. In this
case, their joint distribution is symmetric in the sense that it is invariant
under permutations. We give a characterization of such distributions in terms
of a function q where q(h) is the marginal probability that any group of size h
tests negative. We use this interpretable representation to show that the set
partitioning problem arising in Dorfman's protocol can be reduced to an integer
partitioning problem and efficiently solved. We apply these tools to an
empirical dataset from the COVID-19 pandemic. The methodology helps explain the
unexpectedly high empirical efficiency reported by the original investigators.Comment: 20 pages w/o references, 2 figure
One-Message Secure Reductions: On the Cost of Converting Correlations
Correlated secret randomness is a useful resource for secure computation protocols, often enabling dramatic speedups compared to protocols in the plain model. This has motivated a line of work on identifying and securely generating useful correlations.
Different kinds of correlations can vary greatly in terms of usefulness and ease of generation. While there has been major progress on efficiently generating oblivious transfer (OT) correlations, other useful kinds of correlations are much more costly to generate. Thus, it is highly desirable to develop efficient techniques for securely converting copies of a given source correlation into copies of a given target correlation, especially when the former are cheaper to generate than the latter.
In this work, we initiate a systematic study of such conversions that only involve a single uni-directional message. We refer to such a conversion as a one-message secure reduction (OMSR).
Recent works (Agarwal et al, Eurocrypt 2022; Khorasgani et al, Eurocrypt 2022) studied a similar problem when no communication is allowed; this setting is quite restrictive, however, with few non-trivial conversions being feasible. The OMSR setting substantially expands the scope of feasible results, allowing for direct applications to existing MPC protocols.
We obtain the following positive and negative results.
- OMSR constructions. We present a general rejection-sampling based technique for OMSR with OT source correlations. We apply it to substantially improve in the communication complexity of optimized protocols for distributed symmetric cryptography (Dinur et al., Crypto 2021).
- OMSR lower bounds. We develop general techniques for proving lower bounds on the communication complexity of OMSR, matching our positive results up to small constant factors
Bayesian clustering of multiple zero-inflated outcomes
Several applications involving counts present a large proportion of zeros (excess-of-zeros data). A popular model for such data is the hurdle model, which explicitly models the probability of a zero count, while assuming a sampling distribution on the positive integers. We consider data from multiple count processes. In this context, it is of interest to study the patterns of counts and cluster the subjects accordingly. We introduce a novel Bayesian approach to cluster multiple, possibly related, zero-inflated processes. We propose a joint model for zero-inflated counts, specifying a hurdle model for each process with a shifted Negative Binomial sampling distribution. Conditionally on the model parameters, the different processes are assumed independent, leading to a substantial reduction in the number of parameters as compared with traditional multivariate approaches. The subject-specific probabilities of zero-inflation and the parameters of the sampling distribution are flexibly modelled via an enriched finite mixture with random number of components. This induces a two-level clustering of the subjects based on the zero/non-zero patterns (outer clustering) and on the sampling distribution (inner clustering). Posterior inference is performed through tailored Markov chain Monte Carlo schemes. We demonstrate the proposed approach on an application involving the use of the messaging service WhatsApp. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'
Approximate sampling and counting for spin models in graphs
En aquest treball abordem els problemes de mostreig i comptatge aproximat en models d'espins en grafs, recopilant els resultats mĂ©s significatius de l'Ă rea i introduĂŻnt els coneixements previs necessaris del camp de la fĂsica estadĂstica. En particular, prestem especial atenciĂł als mĂštodes generals de disseny d'algorismes desenvolupats per Weitz i Barvinok, aixĂ com els avenços recents en matĂšria de comptatge i mostreig de conjunts independents de mida donada. AixĂ mateix, discutim com es podrien adaptar aquests arguments als problemes de comptatge i mostreig de coloracions amb les mides de cada color fixades, explicant amb detall la lĂnia de recerca actual que estem duent a terme.En este trabajo abordamos los problemas de muestreo y conteo aproximado en modelos de espines en grafos, recopilando los resultados mĂĄs significativos del campo e introduciendo el conocimiento previo necesario del ĂĄrea de la fĂsica estadĂstica. En particular, prestamos especial atenciĂłn a los mĂ©todos generales de diseño de algorismos desarrollados por Weitz y Barvinok, asĂ como a los avances recientes en cuanto al conteo y muestreo de conjuntos independientes de un tamaño dado. AsĂ mismo, discutimos cĂłmo se podrĂan adaptar estos argumentos al problema de contar y muestrear coloraciones con el tamaño de cada color fijo, explicando en detalle la lĂnea de investigaciĂłn que estamos llevando a cabo actualmente.We approach the problems of approximate sampling and counting in spin models on graphs, surveying the most significant results in the area and introducing the necessary background from statistical physics. We pay particular attention to the general algorithm design frameworks developed by Weitz and Barvinok, as well as to the newer results on counting and sampling independent sets of given size. In addition, we discuss the adaptation of the arguments behind these results to count and sample colorings with fixed color sizes, explaining in detail the current research line we are undertaking.Outgoin
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
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