Daniel Dadush

The split closure of a strictly convex body

The Chvatal-Gomory closure and the split closure of a rational polyhedron are rational polyhedra. It has been recently shown that the Chvatal-Gomory closure of a strictly convex body is also a rational polytope. In this note, we show that the split closure of a strictly convex body is defined by a finite number of split disjunctions, but is not necessarily polyhedral. We also give a closed form expression in the original variable space of a split cut for full-dimensional ellipsoids.

Solving the Shortest Vector Problem in 2 n Time Using Discrete Gaussian Sampling: Extended Abstract

We give a randomized 2n+o(n)-time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic ~O(4n)-time and ~O(2n)-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp. 2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample 2n/2 vectors from the discrete Gaussian distribution at any parameter in 2n+o(n) time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS. In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate 2n/2 discrete Gaussian samples in just 2n/2+o(n) time and space. Among other things, this implies a 2n/2+o(n)-time and space algorithm for 1.93-approximate decision SVP.

On the Existence of 0/1 Polytopes with High Semidefinite Extension Complexity

Rothvoss [1] showed that there exists a 0/1 polytope (a polytope whose vertices are in {0,1} n ) such that any higher-dimensional polytope projecting to it must have 2Ω(n) facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high PSD extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension 2Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations.

Solving the Closest Vector Problem in 2^n Time -- The Discrete Gaussian Strikes Again!

We give a 2n+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 Õ(4n)-time and Õ(2n)-space algorithm of Micciancio and Voulgaris [1]. We achieve our main result in three steps. First, we show how to modify the sampling algorithm from [2] 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 [2], the analysis uses substantial new ideas. This yields a 2n+o(n)-time algorithm for approximate CVP with the very good approximation factor γ = 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.

Unconditional differentially private mechanisms for linear queries

We investigate the problem of designing differentially private mechanisms for a set of d linear queries over a database, while adding as little error as possible. Hardt and Talwar [HT10] related this problem to geometric properties of a convex body defined by the set of queries and gave a O(log3 d)-approximation to the minimum l22 error, assuming a conjecture from convex geometry called the Slicing or Hyperplane conjecture. In this work we give a mechanism that works unconditionally, and also gives an improved O(log2 d) approximation to the expected l22 error. We remove the dependence on the Slicing conjecture by using a result of Klartag [Kla06] that shows that any convex body is close to one for which the conjecture holds; our main contribution is in making this result constructive by using recent techniques of Dadush, Peikert and Vempala [DPV10]. The improvement in approximation ratio relies on a stronger lower bound we derive on the optimum. This new lower bound goes beyond the packing argument that has traditionally been used in Differential Privacy and allows us to add the packing lower bounds obtained from orthogonal subspaces. We are able to achieve this via a symmetrization argument which argues that there always exists a near optimal differentially private mechanism which adds noise that is independent of the input database! We believe this result should be of independent interest, and also discuss some interesting consequences.

On the Chvátal-Gomory closure of a compact convex set

In this paper, we show that the Chvatal-Gomory closure of any compact convex set is a rational polytope. This resolves an open question of Schrijver [15] for irrational polytopes, and generalizes the same result for the case of rational polytopes [15], rational ellipsoids [7] and strictly convex bodies [6].

The Chvátal-Gomory Closure of a Strictly Convex Body

In this paper, we prove that the Chvatal-Gomory closure of a set obtained as an intersection of a strictly convex body and a rational polyhedron is a polyhedron. Thus, we generalize a result of Schrijver [Schrijver, A. 1980. On cutting planes. Ann. Discrete Math.9 291--296], which shows that the Chvatal-Gomory closure of a rational polyhedron is a polyhedron.

On the Closest Vector Problem with a Distance Guarantee

We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n/√log n), improving on the previous best of O(n1.5) due to Lag arias, Lenstra, and Schnorr [1]. We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(√n/log n) due to Aharonov and Regev [2]. To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio [3] gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+∈)γ approximate CVP to γ approximate CVP where the target is at distance at most 1+1/∈ times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun [4]. As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice [5]. We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide Ω(√n) factor larger.

An Algorithm for Komlós 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(t1/2 log n) bound. Our result also extends to the more general Komlós setting and gives an algorithmic O(log1/2 n) bound.

Near-optimal deterministic algorithms for volume computation via M-ellipsoids

We give a deterministic algorithm for computing an M-ellipsoid of a convex body, matching a known lower bound. This leads to a nearly optimal deterministic algorithm for estimating the volume of a convex body and improved deterministic algorithms for fundamental lattice problems under general norms.