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Covering Convex Bodies and the Closest Vector Problem.

Márton Naszódi1, Moritz Venzin2

  • 1MTA-ELTE Lendület Combinatorial Geometry Research Group; Department of Geometry, Loránd Eötvös University, Budapest, Hungary.

Discrete & Computational Geometry
|May 16, 2022
PubMed
Summary

We developed faster algorithms for the approximate closest vector problem (CVP) using geometric covering and lattice sparsification. These improvements significantly reduce computation time for specific norms, advancing computational geometry research.

Keywords:
ApproximationClosest vector problemConvex body in d-dimensional spaceLattice sparsificationModulus of smoothness

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Area of Science:

  • Computational Geometry
  • Algorithm Analysis
  • Number Theory

Background:

  • The closest vector problem (CVP) is a fundamental problem in computational geometry with applications in cryptography and coding theory.
  • Existing algorithms for approximate CVP have running times that can be prohibitive for high-dimensional spaces.
  • Improvements in CVP algorithms are crucial for advancing various fields reliant on efficient geometric computations.

Purpose of the Study:

  • To develop novel algorithms for the ε-approximate closest vector problem (CVP) with improved running times.
  • To specifically enhance CVP algorithms for L_p-norms and polyhedral/zonotopal norms.
  • To explore the connection between geometric covering properties and lattice sparsification for CVP.

Main Methods:

  • Developed algorithms based on a geometric covering problem involving covering a norm ball with scaled homothets of itself.
  • Utilized the modulus of smoothness of L_p-balls to establish upper bounds for covering numbers.
  • Applied lattice sparsification techniques and existing CVP algorithms (e.g., Dadush's) to achieve improved approximation ratios and run times.

Main Results:

  • Achieved improved running times for ε-approximate CVP for L_p-norms, specifically O(n^(2p/(p-1))) and O(n^(p/(p-1))) for fixed p.
  • Presented a deterministic O(n) time algorithm for polyhedral and zonotopal norms in R^n.
  • Established a connection between the modulus of smoothness and lattice sparsification, leading to a simplified boosting procedure for L_p norms.

Conclusions:

  • The presented algorithms offer substantial improvements in running time for approximate CVP, particularly for specific classes of norms.
  • The novel approach using geometric covering numbers and modulus of smoothness provides a powerful tool for CVP approximation.
  • The established link between modulus of smoothness and lattice sparsification offers new avenues for algorithm design in computational geometry.