615 research outputs found
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Optimal and Near Optimal Configurations on Lattices and Manifolds
Optimal configurations of points arise in many contexts, for example classical ground states for interacting particle systems, Euclidean packings of convex bodies, as well as minimal discrete and continuous energy problems for general kernels. Relevant questions in this area include the understanding of asymptotic optimal configurations, of lattice and periodic configurations, the development of algorithmic constructions of near optimal configurations, and the application of methods in convex optimization such as linear and semidefinite programming
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
Online Discrepancy Minimization for Stochastic Arrivals
In the stochastic online vector balancing problem, vectors
chosen independently from an arbitrary distribution in
arrive one-by-one and must be immediately given a sign.
The goal is to keep the norm of the discrepancy vector, i.e., the signed
prefix-sum, as small as possible for a given target norm.
We consider some of the most well-known problems in discrepancy theory in the
above online stochastic setting, and give algorithms that match the known
offline bounds up to factors. This substantially
generalizes and improves upon the previous results of Bansal, Jiang, Singla,
and Sinha (STOC' 20). In particular, for the Koml\'{o}s problem where
for each , our algorithm achieves
discrepancy with high probability, improving upon the previous
bound. For Tusn\'{a}dy's problem of minimizing the
discrepancy of axis-aligned boxes, we obtain an bound for
arbitrary distribution over points. Previous techniques only worked for product
distributions and gave a weaker bound. We also consider the
Banaszczyk setting, where given a symmetric convex body with Gaussian
measure at least , our algorithm achieves discrepancy with
respect to the norm given by for input distributions with sub-exponential
tails.
Our key idea is to introduce a potential that also enforces constraints on
how the discrepancy vector evolves, allowing us to maintain certain
anti-concentration properties. For the Banaszczyk setting, we further enhance
this potential by combining it with ideas from generic chaining. Finally, we
also extend these results to the setting of online multi-color discrepancy
Binary perceptrons capacity via fully lifted random duality theory
We study the statistical capacity of the classical binary perceptrons with
general thresholds . After recognizing the connection between the
capacity and the bilinearly indexed (bli) random processes, we utilize a recent
progress in studying such processes to characterize the capacity. In
particular, we rely on \emph{fully lifted} random duality theory (fl RDT)
established in \cite{Stojnicflrdt23} to create a general framework for studying
the perceptrons' capacities. Successful underlying numerical evaluations are
required for the framework (and ultimately the entire fl RDT machinery) to
become fully practically operational. We present results obtained in that
directions and uncover that the capacity characterizations are achieved on the
second (first non-trivial) level of \emph{stationarized} full lifting. The
obtained results \emph{exactly} match the replica symmetry breaking predictions
obtained through statistical physics replica methods in \cite{KraMez89}. Most
notably, for the famous zero-threshold scenario, , we uncover the
well known scaled capacity
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
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Combinatorial Optimization
Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry
Online discrepancy minimization for stochastic arrivals
In the stochastic online vector balancing problem, vectors v1, v2,..., vT chosen independently from an arbitrary distribution in Rn arrive one-by-one and must be immediately given a ± sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm. We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to polylog(nT) factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC' 20). In particular, for the Komlós problem where kvtk2 ≤ 1 for each t, our algorithm achieves Oe(1) discrepancy with high probability, improving upon the previous Oe(n3/2) bound. For Tusnády's problem of minimizing the discrepancy of axis-aligned boxes, we obtain an O(logd+4 T) bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker O(log2d+1 T) bound. We also consider the Banaszczyk setting, where given a symmetric convex body K with Gaussian measure at least 1/2, our algorithm achieves Oe(1) discrepancy with respect to the norm given by K for input distributions with sub-exponential tails. Our results are based on a new potential function approach. Previous techniques consider a potential that penalizes large discrepancy, and greedily chooses the next color to minimize the increase in potential. Our key idea is to introduce a potential that also enforces constraints on how the discrepancy vector evolves, allowing us to maintain certain anti-concentration properties. We believe that our techniques to control the evolution of states could find other applications in stochastic processes and online algorithms. For the Banaszczyk setting, we further enhance this potential by combining it with ideas from generic chaining. Finally, we also extend these results to the setting of online multicolor discrepancy
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