266 research outputs found
An Algorithm for Koml\'os Conjecture Matching Banaszczyk's bound
We consider the problem of finding a low discrepancy coloring for sparse set
systems where each element lies in at most t sets. We give an efficient
algorithm that finds a coloring with discrepancy O((t log n)^{1/2}), matching
the best known non-constructive bound for the problem due to Banaszczyk. The
previous algorithms only achieved an O(t^{1/2} log n) bound. The result also
extends to the more general Koml\'{o}s setting and gives an algorithmic
O(log^{1/2} n) bound
THE STATE OF CHINA-EUROPEAN UNION ECONOMIC RELATIONS. Bruegel Working Paper Issue 09 20 November 2019
China and the European Union have an extensive and growing economic
relationship. The relationship is problematic because of the distortions
caused by Chinaâs state capitalist system and the diversity of interests within
the EUâs incomplete federation. More can be done to capture the untapped
trade and investment opportunities that exist between the parties. Chinaâs
size and dynamism, and its recent shift from an export-led to a domesticdemand-
led growth model, mean that these opportunities are likely to grow
with time. As the Chinese economy matures, provided appropriate policy
steps are taken, it is likely to become a less disruptive force in world markets
than during its extraordinary breakout period
The Law of Identity Harm
Identity harm refers to the anguish experienced by consumers who learn that their efforts to consume in line with their personal values have been undermined by a companyâs false or exaggerated promises about its wares. When broken, other-regarding âvirtuous promisesâ about products (e.g., eco-friendly, responsible, fair-trade, cruelty free, conflict free) give rise to identity harm by making consumers unwittingly complicit in hurting others. A leading example is the Volkswagen emissions scandal: when environmentally-conscious purchasers of Volkswagenâs âclean dieselâ cars learned that the vehicles were in fact hyper-polluting, they experienced identity harm because of their complicity in a scheme that hurt the planet and the health of their communities.
As more people become sensitized to environmental and social (labor and human rights) sustainability challenges, they are also becoming increasingly concerned about their role in aggravating these challenges through their individual consumption. Identity harm surfaces against the backdrop of an under-regulated market for virtuous goods that is expanding to meet the demands of conscious consumers. Troublingly, those who experience identity harm currently have little recourse in private law, which reveals a serious deficit in our legal regime. This Article, one in a series, recommends correcting this protective deficit by operationalizing identity harm under tort, contract, and state consumer law, with a particular focus on the latter
On the Lattice Distortion Problem
We introduce and study the \emph{Lattice Distortion Problem} (LDP). LDP asks
how "similar" two lattices are. I.e., what is the minimal distortion of a
linear bijection between the two lattices? LDP generalizes the Lattice
Isomorphism Problem (the lattice analogue of Graph Isomorphism), which simply
asks whether the minimal distortion is one.
As our first contribution, we show that the distortion between any two
lattices is approximated up to a factor by a simple function of
their successive minima. Our methods are constructive, allowing us to compute
low-distortion mappings that are within a factor
of optimal in polynomial time and within a factor of optimal in
singly exponential time. Our algorithms rely on a notion of basis reduction
introduced by Seysen (Combinatorica 1993), which we show is intimately related
to lattice distortion. Lastly, we show that LDP is NP-hard to approximate to
within any constant factor (under randomized reductions), by a reduction from
the Shortest Vector Problem.Comment: This is the full version of a paper that appeared in ESA 201
Solving the Closest Vector Problem in Time--- The Discrete Gaussian Strikes Again!
We give a -time and space randomized algorithm for solving the
exact Closest Vector Problem (CVP) on -dimensional Euclidean lattices. This
improves on the previous fastest algorithm, the deterministic
-time and -space algorithm of
Micciancio and Voulgaris.
We achieve our main result in three steps. First, we show how to modify the
sampling algorithm from [ADRS15] to solve the problem of discrete Gaussian
sampling over lattice shifts, , with very low parameters. While the
actual algorithm is a natural generalization of [ADRS15], the analysis uses
substantial new ideas. This yields a -time algorithm for
approximate CVP for any approximation factor .
Second, we show that the approximate closest vectors to a target vector can
be grouped into "lower-dimensional clusters," and we use this to obtain a
recursive reduction from exact CVP to a variant of approximate CVP that
"behaves well with these clusters." Third, we show that our discrete Gaussian
sampling algorithm can be used to solve this variant of approximate CVP.
The analysis depends crucially on some new properties of the discrete
Gaussian distribution and approximate closest vectors, which might be of
independent interest
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