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

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

Are private capital flows to developing countries sustainable?

The remarkable surge in private capital flow to developing countries since 1990 has greatly facilitated their rapid growth, at a time when OECD countries have been in, or passed through, recession. The importance of these flows to the current account of severallarge developing countries has caused concern about their sustainability, especially if international interest rates continue rising. The form of these flows, and their source - investors rather than commercial banks - causes concern about their short-term volatility. To address the issue of sustainability, the authors draw on analyses of international financial flows and economic prospects carried out by the Bank's International Economics Department. They conclude that private capital flows to developing countries are likely to be sustained at, or near, current total levels for the following reasons: (a) Much of the private flow comes from direct investment. Foreign direct investment has increased as international businesses pursue globalization strategies. Firms are taking advantage of liberalization drives and rising incomes in developing countries, as well as dramatic changes in transport and telecommunications - factors that are structural rather than cyclical, and that are likely to be reinforced by implementation of Uruguay Round agreements. (b) Sources of finance are more diversified. There is greater risk-sharing between creditor and debtor. Funds are predominantly going to the private sector (not sovereign governments). Also, developing countries still account for less than 1 percent of the investment portfolios of OECD investors. In the 1970s, commercial loans accounted for proportionately more flows. Now, increasingly large roles are played by bondholders, equity investors, and money market funds. (c) A prolonged major increase in international interest rates would jeopardize continuation of the flows at current levels, but the likelihood of such an increase in the next three to five years is slim. Any rise in interest rates in industrial countries will largely reflect rising demand for credit because of increased economic activity, which will benefit developing country exports. Commodity prices have surged in the past six months, but measures of core inflation, including unit labor cost, are at a historic low. This scenario is very different from the combination of high interest rates and economic recession the developing would faced in the early 1980s, as high and rising inflation induced sudden tightening of monetary policies. Still, significant areas of risk deserve attention from developing country governments, international financial institutions, and industrial country investors. Some major recipients of private capital flow are vulnerable to sudden changes in both domestic or external environments. And portfolio equity flows are likely to be more volatile than other forms of private capital flows. The policy response to large capital inflows should depend on whether the current account deficit is sustainable and the degree to which it is over - or underfinanced. While the external environment is favorable, vulnerable countries have a window of opportunity to undertake adjustment.Economic Theory&Research,Banks&Banking Reform,Environmental Economics&Policies,Financial Intermediation,Macroeconomic Management

On the Closest Vector Problem with a Distance Guarantee

We present a substantially more efficient variant, both in terms of running time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky, and Micciancio for solving CVPP (the preprocessing version of the Closest Vector Problem, CVP) with a distance guarantee. For instance, for any $\alpha < 1/2$, our algorithm finds the (unique) closest lattice point for any target point whose distance from the lattice is at most $\alpha$ times the length of the shortest nonzero lattice vector, requires as preprocessing advice only $N \approx \widetilde{O}(n \exp(\alpha^2 n /(1-2\alpha)^2))$ vectors, and runs in time $\widetilde{O}(nN)$. As our second main contribution, we present reductions showing that it suffices to solve CVP, both in its plain and preprocessing versions, when the input target point is within some bounded distance of the lattice. The reductions are based on ideas due to Kannan and a recent sparsification technique due to Dadush and Kun. Combining our reductions with the LLM algorithm gives an approximation factor of $O(n/\sqrt{\log n})$ for search CVPP, improving on the previous best of $O(n^{1.5})$ due to Lagarias, Lenstra, and Schnorr. When combined with our improved algorithm we obtain, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve CVPP with (the only slightly worse) approximation factor of O(n).Comment: An early version of the paper was titled "On Bounded Distance Decoding and the Closest Vector Problem with Preprocessing". Conference on Computational Complexity (2014

Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem

An important theorem of Banaszczyk (Random Structures & Algorithms `98) states that for any sequence of vectors of $\ell_2$ norm at most $1/5$ and any convex body $K$ of Gaussian measure $1/2$ in $\mathbb{R}^n$, there exists a signed combination of these vectors which lands inside $K$. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination. We make progress towards this goal along several fronts. As our first contribution, we show an equivalence between Banaszczyk's theorem and the existence of $O(1)$-subgaussian distributions over signed combinations. For the case of symmetric convex bodies, our equivalence implies the existence of a universal signing algorithm (i.e. independent of the body), which simply samples from the subgaussian sign distribution and checks to see if the associated combination lands inside the body. For asymmetric convex bodies, we provide a novel recentering procedure, which allows us to reduce to the case where the body is symmetric. As our second main contribution, we show that the above framework can be efficiently implemented when the vectors have length $O(1/\sqrt{\log n})$, recovering Banaszczyk's results under this stronger assumption. More precisely, we use random walk techniques to produce the required $O(1)$-subgaussian signing distributions when the vectors have length $O(1/\sqrt{\log n})$, and use a stochastic gradient ascent method to implement the recentering procedure for asymmetric bodies

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