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Enumerative Lattice Algorithms in any Norm Via M-ellipsoid Coverings
A novel algorithm for enumerating lattice points in any convex body known as the M-ellipsoid is given, and an expected O(f*(n))^n-time algorithm for Integer Programming, where f*( n) denotes the optimal bound in the so-calledflatnesstheorem, which is conjectured to be f* (n) = O(n). Expand
On the Closest Vector Problem with a Distance Guarantee
A substantially more efficient variant of the LLM algorithm is presented, and via an improved analysis, it is shown that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice. Expand
Integer programming, lattice algorithms, and deterministic volume estimation
The main subject of this thesis is the development of new geometric tools and techniques for solving classic problems within the geometry of numbers and convex geometry. At a high level, the problemsExpand
The split closure of a strictly convex body
It is shown that the split closure of a strictly convex body is defined by a finite number of split disjunctions, but is not necessarily polyhedral. Expand
The Gram-Schmidt walk: a cure for the Banaszczyk blues
This paper gives an efficient randomized algorithm to find a ± 1 combination of the vectors which lies in cK for c>0 an absolute constant, which leads to new efficient algorithms for several problems in discrepancy theory. Expand
An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound
An efficient algorithm is given that finds a coloring with discrepancy O((t log n)1/2), matching the best known nonconstructive bound for the problem due to Banaszczyk, and gives an algorithmic O(log 1/2n) bound. Expand
On the complexity of branching proofs
It is demonstrated that any branching proof can be recompiled so that the normals of the disjunctions have coefficients of size at most (nR)O(n2), where R ∈ N is the radius of an l1 ball containing K, while increasing the number of nodes in the branching tree by at most a factor O(n). Expand
Solving the Shortest Vector Problem in 2n Time Using Discrete Gaussian Sampling: Extended Abstract
The SVP result follows from a natural reduction from SVP to DGS, and 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 1.93-approximate decision SVP. Expand
Unconditional differentially private mechanisms for linear queries
This work gives a mechanism that works unconditionally, and also gives an improved O(log2 d) approximation to the expected l22 error, 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. Expand
On the Chvátal–Gomory closure of a compact convex set
In this paper, we show that the Chvátal–Gomory closure of any compact convex set is a rational polytope. This resolves an open question of Schrijver (Ann Discret Math 9:291–296, 1980) for irrationalExpand