Lattice points in lattice cubes
Faculty of Mathematics lattice points
Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well? If the polytope is convex and also centrally symmetric then what is the situation for 1. Update If you know the number of lattice points approximately then we can guess volume approximately. Below the line I address the question after "Update" about the relationship between volume and the number of lattice points. Here are three special cases in which a positive result is known:.
This problem is also studied in a setting analogous to the Gauss circle problem. Check this thesis by Guo for references.
In fact, SVP is widely believed to be hard but not NP-hard to approximate even to within any polynomial approximation factor, and a lot of cryptography is based on this assumption. Some cryptography is based on the presumed hardness for even superpolynomial approximation factors.
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This algorithm is in the Euclidean norm, so to extend even this to arbitrary norms, you must be able to calculate a not-too-terrible approximating ellipsoid. It would be a major breakthrough to improve this. For centrally symmetric bodies, the parity is always odd, so the problem is only interesting for shifted bodies or asymmetric bodies.
This follows from the simple reduction from MaxSAT in  with Bennett and Golovnev, which preserves the number of solutions, as we mention at the end of Section 6. There is probably an earlier reduction that also has this property. Finally, there are many of upper bounds on the number of lattice points in a convex body based on certain geometric parameters of the body. For example, Henk's bound .
The one that Sasho described is probably the easiest to work with. GitHub issue tracker. Personal blog. What can we improve? The page or its content looks wrong. I can't find what I'm looking for.
Tag: lattice points
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