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 China’s state capitalist system and the diversity of interests within the EU’s incomplete federation. More can be done to capture the untapped trade and investment opportunities that exist between the parties. China’s 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 company’s false or exaggerated promises about its wares. When broken, other-regarding “virtuous promises” 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 Volkswagen’s “clean diesel” 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 $n^{O(\log n)}$ factor by a simple function of their successive minima. Our methods are constructive, allowing us to compute low-distortion mappings that are within a $2^{O(n \log \log n/\log n)}$ factor of optimal in polynomial time and within a $n^{O(\log n)}$ 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 2n2^n Time--- The Discrete Gaussian Strikes Again!

We give a $2^{n+o(n)}$-time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on $n$-dimensional Euclidean lattices. This improves on the previous fastest algorithm, the deterministic $\widetilde{O}(4^{n})$-time and $\widetilde{O}(2^{n})$-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, $L- t$, with very low parameters. While the actual algorithm is a natural generalization of [ADRS15], the analysis uses substantial new ideas. This yields a $2^{n+o(n)}$-time algorithm for approximate CVP for any approximation factor $\gamma = 1+2^{-o(n/\log n)}$. Second, we show that the approximate closest vectors to a target vector $t$ 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