205 research outputs found
Probabilistic existence of regular combinatorial structures
We show the existence of regular combinatorial objects which previously were
not known to exist. Specifically, for a wide range of the underlying
parameters, we show the existence of non-trivial orthogonal arrays, t-designs,
and t-wise permutations. In all cases, the sizes of the objects are optimal up
to polynomial overhead. The proof of existence is probabilistic. We show that a
randomly chosen structure has the required properties with positive yet tiny
probability. Our method allows also to give rather precise estimates on the
number of objects of a given size and this is applied to count the number of
orthogonal arrays, t-designs and regular hypergraphs. The main technical
ingredient is a special local central limit theorem for suitable lattice random
walks with finitely many steps.Comment: An extended abstract of this work [arXiv:1111.0492] appeared in STOC
2012. This version expands the literature discussio
Steiner configurations ideals: Containment and colouring
Given a homogeneous ideal I ⊆ k[x0, …, xn ], the Containment problem studies the relation between symbolic and regular powers of I, that is, it asks for which pairs m, r ∈ N, I(m) ⊆ Ir holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne–Huneke Conjecture, and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in Pnk. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demailly’s Conjectures. Moreover, given a hypergraph H, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to H. We apply these results in the case that H is a Steiner System
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Coding Theory
Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances
Quantum error mitigation and error correction for practical quantum computation
We are rapidly entering the era of potentially useful quantum computation. To keep on designing larger and more capable quantum computers, some form of algorithmic noise management will be necessary. In this thesis, I propose multiple practical advances in quantum error mitigation and error correction. First, I present a novel and intuitive way to mitigate errors using a strategy that assumes no or very minimal knowledge about the nature of errors. This strategy can deal with most complex noise profiles, including those that describe severe correlated errors. Second, I present proof that quantum computation is scalable on a defective planar array of qubits. This result is based on a two-dimensional surface code architecture for which I showed that a finite rate of fabrication defects is not a fundamental obstacle to maintaining a non-zero error-rate threshold. The same conclusions are supported by extensive numerical studies. Finally, I give a new perspective on how to view and construct quantum error-correcting codes tailored for modular architectures. Following a given recipe, one can design codes that are compatible with the qubit connectivity demanded by the architecture. In addition, I present several product code constructions, some of which correspond to the latest developments in quantum LDPC code design. These and other practical advancements in quantum error mitigation and error correction will be crucial in guiding the design of emerging quantum computers
A Tutorial on Clique Problems in Communications and Signal Processing
Since its first use by Euler on the problem of the seven bridges of
K\"onigsberg, graph theory has shown excellent abilities in solving and
unveiling the properties of multiple discrete optimization problems. The study
of the structure of some integer programs reveals equivalence with graph theory
problems making a large body of the literature readily available for solving
and characterizing the complexity of these problems. This tutorial presents a
framework for utilizing a particular graph theory problem, known as the clique
problem, for solving communications and signal processing problems. In
particular, the paper aims to illustrate the structural properties of integer
programs that can be formulated as clique problems through multiple examples in
communications and signal processing. To that end, the first part of the
tutorial provides various optimal and heuristic solutions for the maximum
clique, maximum weight clique, and -clique problems. The tutorial, further,
illustrates the use of the clique formulation through numerous contemporary
examples in communications and signal processing, mainly in maximum access for
non-orthogonal multiple access networks, throughput maximization using index
and instantly decodable network coding, collision-free radio frequency
identification networks, and resource allocation in cloud-radio access
networks. Finally, the tutorial sheds light on the recent advances of such
applications, and provides technical insights on ways of dealing with mixed
discrete-continuous optimization problems
Graph Neural Networks for Graphs with Heterophily: A Survey
Recent years have witnessed fast developments of graph neural networks (GNNs)
that have benefited myriads of graph analytic tasks and applications. In
general, most GNNs depend on the homophily assumption that nodes belonging to
the same class are more likely to be connected. However, as a ubiquitous graph
property in numerous real-world scenarios, heterophily, i.e., nodes with
different labels tend to be linked, significantly limits the performance of
tailor-made homophilic GNNs. Hence, GNNs for heterophilic graphs are gaining
increasing research attention to enhance graph learning with heterophily. In
this paper, we provide a comprehensive review of GNNs for heterophilic graphs.
Specifically, we propose a systematic taxonomy that essentially governs
existing heterophilic GNN models, along with a general summary and detailed
analysis. Furthermore, we discuss the correlation between graph heterophily and
various graph research domains, aiming to facilitate the development of more
effective GNNs across a spectrum of practical applications and learning tasks
in the graph research community. In the end, we point out the potential
directions to advance and stimulate more future research and applications on
heterophilic graph learning with GNNs.Comment: 22 page
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