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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 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: 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
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
Integer Programming Problem (IP) Input: : Invertible Transformation Remark:can be restricted to any lattice .
Integer Programming Problem (IP) Input: Remark:can be restricted to any lattice .
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)
Examples If a convex set very ``fat’’, then it will always contain a lattice point. “Hidden cube”
Examples If a convex set very ``fat’’, then it will always contain a lattice point.
Examples Volume does NOTguarantee lattice points (in contrast with Minkowski’s theorem). Infinite band
Examples However, lattice point free sets must be ``flat’’ in some direction.
Lattice Width For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes. …
Lattice Width For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes.
Lattice Width For each example, there is hyperplane decomposition of with a small number of intersecting hyperplanes. Note: axis parallel hyperperplanes do NOT suffice.
Lattice Width Why is this useful? IP feasible regions:hyperplane decomposition enables reduction into dimensional sub-IPs. # intersections # subproblems subproblems subproblems
Lattice Width Why is this useful? # intersections # subproblems If # intersections is small, can solve IP via recursion. subproblems subproblems
Lattice Width Integer Hyperplane: Hyperplane where Fact: is an integer hyperplane, ( called primitive if )
Lattice Width Hyperplane Decomposition of : For (parallel hyperplanes)
Lattice Width Hyperplane Decomposition of : For (parallel hyperplanes) Note: If is not primitive, decomposition is finer than necessary.
Lattice Width How many intersections with ? (parallel hyperplanes) # INTs }| + 1 (tight within +2)
Lattice Width Width Norm of : for any Lattice Width: width
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).
Properties of Width Norm of : for any Convex & Centrally Symmetric
Properties of Width Norm of : for any Bounds: Convex & Centrally Symmetric
Properties of Width Norm of : for any Bounds: Convex & Centrally Symmetric
Properties of Width Norm of : for any Symmetry: By symmetry of Convex & Centrally Symmetric
Properties of Width Norm of : for any Symmetry: Therefore Convex & Centrally Symmetric
Properties of Width Norm of : for any Homogeneity: For (Trivial) Convex & Centrally Symmetric
Properties of Width Norm of : for any Triangle Inequality: Convex & Centrally Symmetric
Properties of Width Norm of : for any Triangle Inequality: Convex & Centrally Symmetric
Properties of Width Norm of : for any Triangle Inequality: Convex & Centrally Symmetric
Properties of Width Norm of : for any Triangle Inequality: Convex & Centrally Symmetric
Properties of is invariant under translations of .
Properties of is invariant under translations of . +
Properties of is invariant under translations of . Also follows since . (width only looks at differences between vectors of . +
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).
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).
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).
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).
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).
Lower Bound: Simplex Bound cannot be improved to . No interior lattice points. (interior of S) Pf: If and , then a contradiction.
Lower Bound: Simplex Bound cannot be improved to . Pf: For , then
Flatness Theorem Theorem*: For a convex body and lattice , if such that , then . By shift invariance of .
Flatness Theorem Theorem**: For a convex body and lattice , either1) , or2) .
Covering Radius Definition: Covering radius of with respect to .
Covering Radius Definition: Covering radius of with respect to .
Covering Radius Definition: Covering radius of with respect to . Condition from Flatness Theorem
Covering Radius Definition: Covering radius of with respect to . Condition from Flatness Theorem
Covering Radius Definition: Covering radius of with respect to . Must scale by factor about to hit . Therefore .
Covering Radius Definition: Covering radius of with respect to . contains a fundamental domain