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Enumerative Lattice Algorithms in any Norm Via M-ellipsoid Coverings
- D. Dadush, Chris Peikert, S. Vempala
- Mathematics, Computer ScienceIEEE Annual Symposium on Foundations of Computer…
- 25 November 2010
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).
Integer programming, lattice algorithms, and deterministic volume estimation
A new 2O(n) nn time algorithm is given, which yields the fastest currently known algorithm for IP and improves on the classic works of Lenstra and Kannan, to give a new and tighter proof of the atness theorem.
The Gram-Schmidt walk: a cure for the Banaszczyk blues
- N. Bansal, D. Dadush, S. Garg, Shachar Lovett
- MathematicsSymposium on the Theory of Computing
- 3 August 2017
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.
Solving the Shortest Vector Problem in 2n Time Using Discrete Gaussian Sampling: Extended Abstract
- Divesh Aggarwal, D. Dadush, O. Regev, Noah Stephens-Davidowitz
- Computer Science, MathematicsSymposium on the Theory of Computing
- 26 December 2014
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.
On the Closest Vector Problem with a Distance Guarantee
- D. Dadush, O. Regev, Noah Stephens-Davidowitz
- Computer ScienceIEEE 29th Conference on Computational Complexity…
- 11 June 2014
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.
An Algorithm for Komlós Conjecture Matching Banaszczyk's Bound
- N. Bansal, D. Dadush, S. Garg
- Computer Science, MathematicsIEEE Annual Symposium on Foundations of Computer…
- 10 May 2016
An efficient algorithm is given 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, and gives an algorithmic O(log 1/2 n) bound.
The split closure of a strictly convex body
Rescaling Algorithms for Linear Conic Feasibility
The degenerate case $\rho_A=0$ is addressed by extending the algorithms to find maximum support nonnegative vectors in the kernel of A and in the image of A^\top by extending them to the oracle setting.
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 irrational…
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…