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Transference Theorems in the Geometry of Numbers

Transference Theorems in the Geometry of Numbers. Daniel Dadush New York University EPIT 2013. Convex Bodies. Convex body . (convex, full dimensional and bounded). Convexity: Line between and in . Equivalently . Non convex set. Integer Programming Problem (IP). Input:

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Transference Theorems in the Geometry of Numbers

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  1. Transference Theorems in the Geometry of Numbers Daniel Dadush New York University EPIT 2013

  2. Convex Bodies Convex body . (convex, full dimensional and bounded). Convexity: Line between and in . Equivalently Non convex set.

  3. Integer Programming Problem (IP) Input: Classic NP-Hard problem (integrality makes it hard) IP Problem:Decide whether above system has a solution. Focus for this talk:Geometry of Integer Programs convex set

  4. Integer Programming Problem (IP)Linear Programming (LP) Input: (integrality makes it hard) LP Problem:Decide whether above system has a solution. Polynomial Time Solvable:Khachiyan `79 (Ellipsoid Algorithm) convex set

  5. Integer Programming Problem (IP) Input: : Invertible Transformation Remark:can be restricted to any lattice .

  6. Integer Programming Problem (IP) Input: Remark:can be restricted to any lattice .

  7. Central Geometric Questions • When can we guarantee that a convex set contains a lattice point? (guarantee IP feasibility) • What do lattice free convex sets look like? (sets not containing integer points)

  8. Examples If a convex set very ``fat’’, then it will always contain a lattice point. “Hidden cube”

  9. Examples If a convex set very ``fat’’, then it will always contain a lattice point.

  10. Examples Volume does NOTguarantee lattice points (in contrast with Minkowski’s theorem). Infinite band

  11. Examples However, lattice point free sets must be ``flat’’ in some direction.

  12. Lattice Width For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes. …

  13. Lattice Width For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes.

  14. Lattice Width For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes. Note: axis parallel hyperperplanes do NOT suffice.

  15. Lattice Width Why is this useful? IP feasible regions:hyperplane decomposition enables reduction into dimensional sub-IPs. # intersections # subproblems subproblems subproblems

  16. Lattice Width Why is this useful? # intersections # subproblems If # intersections is small, can solve IP via recursion. subproblems subproblems

  17. Lattice Width Integer Hyperplane: Hyperplane where Fact: is an integer hyperplane, ( called primitive if )

  18. Lattice Width Hyperplane Decomposition of : For (parallel hyperplanes)

  19. Lattice Width Hyperplane Decomposition of : For (parallel hyperplanes) Note: If is not primitive, decomposition is finer than necessary.

  20. Lattice Width How many intersections with ? (parallel hyperplanes) # INTs }| + 1 (tight within +2)

  21. Lattice Width Width Norm of : for any Lattice Width: width

  22. Kinchine’s Flatness Theorem Theorem: For a convex body , , . Bounds improvements: Khinchine`48: Babai `86: Lenstra-Lagarias-Schnorr`87: Kannan-Lovasz`88: Banaszczyk et al `99: Rudelson `00: [Khinchine`48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr`87, Kannan-Lovasz`88, Banaszczyk`93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00] Bound conjectured to be (best possible).

  23. Properties of Width Norm of : for any Convex & Centrally Symmetric

  24. Properties of Width Norm of : for any Bounds: Convex & Centrally Symmetric

  25. Properties of Width Norm of : for any Bounds: Convex & Centrally Symmetric

  26. Properties of Width Norm of : for any Symmetry: By symmetry of Convex & Centrally Symmetric

  27. Properties of Width Norm of : for any Symmetry: Therefore Convex & Centrally Symmetric

  28. Properties of Width Norm of : for any Homogeneity: For (Trivial) Convex & Centrally Symmetric

  29. Properties of Width Norm of : for any Triangle Inequality: Convex & Centrally Symmetric

  30. Properties of Width Norm of : for any Triangle Inequality: Convex & Centrally Symmetric

  31. Properties of Width Norm of : for any Triangle Inequality: Convex & Centrally Symmetric

  32. Properties of Width Norm of : for any Triangle Inequality: Convex & Centrally Symmetric

  33. Properties of is invariant under translations of .

  34. Properties of is invariant under translations of . +

  35. Properties of is invariant under translations of . Also follows since . (width only looks at differences between vectors of . +

  36. Kinchine’s Flatness Theorem Theorem: For a convex body , such that , . Remark: Finding flatness direction is a general norm SVP! [Khinchine`48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr`87, Kannan-Lovasz`88, Banaszczyk`93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00] Bound conjectured to be (best possible).

  37. Kinchine’s Flatness Theorem Theorem: For a convex body , such that , . Easy generalize to arbitrary lattices. (note ) [Khinchine`48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr`87, Kannan-Lovasz`88, Banaszczyk`93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00] Bound conjectured to be (best possible).

  38. Kinchine’s Flatness Theorem Theorem: For a convex body , such that , . Easy generalize to arbitrary lattices. (note ) [Khinchine`48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr`87, Kannan-Lovasz`88, Banaszczyk`93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00] Bound conjectured to be (best possible).

  39. Kinchine’s Flatness Theorem Theorem: For a convex body and lattice , such that , . Easy generalize to arbitrary lattices. where is dual lattice. [Khinchine`48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr`87, Kannan-Lovasz`88, Banaszczyk`93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00] Bound conjectured to be (best possible).

  40. Kinchine’s Flatness Theorem Theorem: For a convex body and lattice , such that , . Homegeneity of Lattice Width: [Khinchine`48, Babai `86, Hastad `86, Lenstra-Lagarias-Schnorr`87, Kannan-Lovasz`88, Banaszczyk`93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00] Bound conjectured to be (best possible).

  41. Lower Bound: Simplex Bound cannot be improved to . No interior lattice points. (interior of S) Pf: If and , then a contradiction.

  42. Lower Bound: Simplex Bound cannot be improved to . Pf: For , then

  43. Flatness Theorem Theorem*: For a convex body and lattice , if such that , then . By shift invariance of .

  44. Flatness Theorem Theorem**: For a convex body and lattice , either1) , or2) .

  45. Covering Radius Definition: Covering radius of with respect to .

  46. Covering Radius Definition: Covering radius of with respect to .

  47. Covering Radius Definition: Covering radius of with respect to . Condition from Flatness Theorem

  48. Covering Radius Definition: Covering radius of with respect to . Condition from Flatness Theorem

  49. Covering Radius Definition: Covering radius of with respect to . Must scale by factor about to hit . Therefore .

  50. Covering Radius Definition: Covering radius of with respect to . contains a fundamental domain

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