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

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

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

Near-Optimal Deterministic Algorithms for Volume Computation and Lattice Problems via M-Ellipsoids

We give a deterministic 2^{O(n)} algorithm for computing an M-ellipsoid of a convex body, matching a known lower bound. This has several interesting consequences including improved deterministic algorithms for volume estimation of convex bodies and the shortest and closest lattice vector problems under general norms

Faster Deterministic Volume Estimation in the Oracle Model via Thin Lattice Coverings

We give a 2O(n)(1+1/")n time and poly(n)-space deterministic algorithm for computing a (1+")n approximation to the volume of a general convex body K, which comes close to matching the (1+c/")n/2 lower bound for volume estimation in the oracle model by Bárány and Füredi (STOC 1986, Proc. Amer. Math. Soc. 1988). This improves on the previous results of Dadush and Vempala (Proc. Nat’l Acad. Sci. 2013), which gave the above result only for symmetric bodies and achieved a dependence of 2O(n)(1 + log5/2(1/")/"3)n. For our methods, we reduce the problem of volume estimation in K to counting lattice points in K Rn (via enumeration) for a specially constructed lattice L: a so-called thin covering of space with respect to K (more precisely, for which L + K = Rn and voln(K)/ det(L) = 2O(n)). The trade off between time and approximation ratio is achieved by scaling down the lattice. As our main technical contribution, we give the first deterministic 2O(n)-time and poly(n)- space construction of thin covering lattices for general convex bodies. This improves on a recent construction of Alon et al. (STOC 2013) which requires exponential space and only works for symmetric bodies. For our construction, we combine the use of the M-ellipsoid from convex geometry (Milman, C. R. Math. Acad. Sci. Paris 1986) together with lattice sparsification and densification techniques (Dadush and Kun, SODA 2013; Rogers, J. London Math. Soc. 1950)

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