Transference Theorems in the Geometry of Numbers

Transference Theorems in the Geometry of Numbers

Daniel DadushNew York University

Convex body . (convex, full dimensional and bounded).

Convex Bodies

𝐾𝑥

𝑦

Non convex set.

Convexity: Line between and in .Equivalently

𝑥𝑦

𝐾 1

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

Integer Programming Problem (IP)

ℤ𝑛𝐾 2

convex set

𝐾 1

Input: (integrality makes it hard)

LP Problem: Decide whether above system has a solution.

Polynomial Time Solvable: Khachiyan `79 (Ellipsoid Algorithm)

Integer Programming Problem (IP)Linear Programming (LP)

ℤ𝑛𝐾 2

convex set

Input: : Invertible Transformation Remark: can be restricted to any lattice .


𝐵𝐾1

𝐵ℤ𝑛

𝐵𝐾 2

Input: Remark: can be restricted to any lattice .


𝐾 1

𝐿𝐾 2

𝐾 1

1) When can we guarantee that a convex set contains a lattice point? (guarantee IP feasibility)

2) What do lattice free convex sets look like? (sets not containing integer points)

Central Geometric Questions

ℤ2𝐾 2

𝐾 1

Examples

ℤ2

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

“Hidden cube”

𝐾 1

Examples

ℤ2

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

Examples

ℤ2

Volume does NOT guarantee lattice points (in contrast with Minkowski’s theorem).

Infinite band

Examples

ℤ2

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

Lattice Width

ℤ2

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

𝑥1=1 𝑥1=8𝑥1=4 …

Lattice Width

ℤ2


𝑥2=1𝑥2=2

𝑥2=3

𝑥2=4

Lattice Width

ℤ2


Note: axis parallel hyperperplanes do NOT suffice.

𝑥1−𝑥2=1 𝑥1−𝑥2= 4

Lattice Width

ℤ2

Why is this useful? IP feasible regions: hyperplane decomposition enables reduction into dimensional sub-IPs.

# intersections # subproblems

𝑥2=1𝑥2=2

𝑥2=3

𝑥2=4

𝐾 2

𝐾 1

subproblems

subproblems

Lattice Width

ℤ2

Why is this useful? # intersections # subproblems

If # intersections is small, can solve IP via recursion.

𝑥2=1𝑥2=2

𝑥2=3

𝑥2=4

𝐾 2

𝐾 1

subproblems

subproblems

Lattice Width

Integer Hyperplane : Hyperplane where

Fact: is an integer hyperplane , ( called primitive if )

ℤ2

𝑥1−𝑥2=1 𝑥1−𝑥2= 4

𝐻 (1 ,−1)𝑎

Lattice Width

Hyperplane Decomposition of : For

(parallel hyperplanes)

ℤ2

𝑥1=1 𝑥1=8

(1 ,0)𝑎

…

Lattice Width

Hyperplane Decomposition of : For

(parallel hyperplanes)

Note: If is not primitive, decomposition is finer than necessary.

ℤ2

2 𝑥1=2 2 𝑥1=16

(2 ,0)2𝑎

2 𝑥1=5 2 𝑥1=9……

𝐾

Lattice Width

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

ℤ2

𝑥1=1 𝑥1=8

(1 ,0)𝑎𝑥𝑚𝑎𝑥

𝑥𝑚𝑖𝑛

2 3 4 6 71.9

7.2

Lattice Width

Width Norm of : for any Lattice Width: width

ℤ2

𝐾 𝑦width𝐾 ( 𝑦 )=1.2

Kinchine’s Flatness Theorem

Theorem: For a convex body , , .

[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).

Bounds improvements:Khinchine `48: Babai `86: Lenstra-Lagarias-Schnorr `87: Kannan-Lovasz `88: Banaszczyk et al `99: Rudelson `00:

𝐾

Properties of

Width Norm of : for any

𝑟 𝐵2𝑛

𝑡

𝑅𝐵2𝑛 0

𝐾 −𝐾

2𝑅𝐵2𝑛

Convex & Centrally Symmetric 2𝑟 𝐵2

𝑛

𝐾

Properties of

Width Norm of : for any Bounds:

𝑟 𝐵2𝑛

𝑡

𝑅𝐵2𝑛 0

2𝑟 𝐵2𝑛

2𝑅𝐵2𝑛

Convex & Centrally Symmetric

𝐾 −𝐾

𝐾

Properties of

Width Norm of : for any Symmetry: By symmetry of

𝑟 𝐵2𝑛

𝑡

𝑅𝐵2𝑛 0

2𝑟 𝐵2𝑛

2𝑅𝐵2𝑛


𝐾 −𝐾

𝐾

Properties of

Width Norm of : for any Symmetry: Therefore

𝑟 𝐵2𝑛

𝑡

𝑅𝐵2𝑛 0

2𝑟 𝐵2𝑛

2𝑅𝐵2𝑛


𝐾 −𝐾

𝐾

Properties of

Width Norm of : for any Homogeneity: For (Trivial)

𝑟 𝐵2𝑛

𝑡

𝑅𝐵2𝑛 0

2𝑟 𝐵2𝑛

2𝑅𝐵2𝑛


𝐾 −𝐾

𝐾

Properties of

Width Norm of : for any Triangle Inequality:

𝑟 𝐵2𝑛

𝑡

𝑅𝐵2𝑛 0

2𝑟 𝐵2𝑛

2𝑅𝐵2𝑛


𝐾 −𝐾

Properties of

is invariant under translations of .

𝐾

𝑦

𝑚𝑖𝑛 𝑚𝑎𝑥𝑤𝑖𝑑𝑡 h𝐾 (𝑦 )=𝑚𝑎𝑥−𝑚𝑖𝑛

Properties of

is invariant under translations of .

𝐾

𝑦

𝑚𝑖𝑛 𝑚𝑎𝑥

𝐾 + 𝑡

𝑦

𝑚𝑖𝑛+ ⟨𝑦 , 𝑡 ⟩ +

Properties of

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

𝐾

𝑦

𝑚𝑖𝑛 𝑚𝑎𝑥

𝐾 + 𝑡

𝑦

𝑚𝑖𝑛+ ⟨𝑦 , 𝑡 ⟩ +




Remark: Finding flatness direction is a general norm SVP!

Theorem: For a convex body , such that , .




Easy generalize to arbitrary lattices. (note )





Easy generalize to arbitrary lattices.

where is dual lattice.

Theorem: For a convex body and lattice , such that , .


Theorem: For a convex body and lattice , such that , .



Homegeneity of Lattice Width:

Lower Bound: Simplex

Bound cannot be improved to .

(interior of S)Pf: If and , then a contradiction.

𝑛𝑒1

𝑛𝑒20

No interior lattice points.

𝑆

Lower Bound: Simplex

Bound cannot be improved to .

Pf: For , then

𝑛𝑒1

𝑛𝑒20𝑆 ℤ2

Flatness Theorem

Theorem*: For a convex body and lattice , if such that , then .

By shift invariance of .

𝐾𝐾 + 𝑡

ℤ2

Flatness Theorem

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

𝐾𝐾 + 𝑡

ℤ2

Covering Radius

Definition: Covering radius of with respect to .

ℤ2𝐾

Covering Radius


ℤ2

Covering Radius


ℤ2𝐾 Condition from

Flatness Theorem

Covering Radius


ℤ2𝐾Condition fromFlatness Theorem𝑥

Covering Radius


ℤ22 (𝐾−𝑥 )+𝑥𝑥Must scale by factor about to hit .

Therefore .

Covering Radius


ℤ2𝐾 contains a fundamental domain

Covering Radius

Definition: Further Properties:

ℤ2𝐾

Flatness Theorem

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

𝐾𝐾 + 𝑡

ℤ2

𝐾𝐾 + 𝑡

Flatness Theorem

Theorem: For a convex body and lattice :

By homogeneity of and .

ℤ2

Flatness Theorem

Theorem [Banaszczyk `93]: For a lattice in :

(since -=)

Bound Improvements:Khinchine `48: Babai `86: Lenstra-Lagarias-Schnorr `87: Kannan-Lovasz `88: Banaszczyk `93: (asymptotically optimal)

Flatness Theorem

Theorem [Banaszczyk `93]: For a lattice in :

𝐿𝑥𝜇

is max distance to

Flatness Theorem

Theorem [Banaszczyk `96]: For a symmetric convex body :

Bound Improvements:Khinchine `48: Babai `86: Kannan-Lovasz `88: Banaszczyk `93: Banaszczyk `96:

ℤ2𝐾

Lower Bounds: Random Lattices

Theorem: For a convex body and ``random’’ lattice in :

(Minkowski Hlawka + Volume Product Bound)

ℤ2𝐾

Symmetric convex body (.

Gauge function:

0

𝑥𝐾 𝑠𝐾

Norms and Convex Bodies

-

1. (triangle inequality) 2. (homogeneity)3. (symmetry)

is unit ball of

Convex body containing origin in its interior.

Gauge function: 𝑥𝐾 𝑠𝐾


0

1. (triangle inequality) 2. (homogeneity)3. (symmetry)

is unit ball of

Gauge function:

Triangle Inequality:

Want to show

By definition . May assume Need to show .


convex combination

asymmetric norm in . Unit ball .

𝑥

𝐾


0

is convex: Take

𝑦-

𝐾

Symmetric convex body and lattice in .If , such that .

Minkowski’s Convex Body Theorem

0

𝐾

Symmetric convex body and lattice in .If , such that .


0 𝑦

𝐾

Pf: basis for with parallelepiped .


0𝑏1

𝑏22𝑏2

2𝑏1𝑃

2𝑃

𝐾

Pf: basis for . Let be parallelepiped. Tile space with using .


0

2𝑃

Pf: basis for . Let be parallelepiped. Tile space with using .


0𝐾

2𝑃

Pf: basis for . Let be parallelepiped. Shift tiles intersecting into using .


0𝐾

2𝑃

Pf: basis for . Let be parallelepiped. Since , must have intersections.


0𝐾

2𝑃𝑐

Pf: basis for . Let be parallelepiped. Since , must have intersections.


0𝐾

2𝑃

+

+ 𝑐

Here (a) , (b) , (c)


0𝐾

2𝑃

+

+ 𝑐

Then (by symmetry of ) and .


0𝐾

2𝑃𝑐

+

+

2(𝑦−𝑥)

Then (by symmetry of ) and .


𝐾0

2(𝑦−𝑥)

2𝐾

So and


𝐾0𝑦−𝑥

Symmetric convex body and lattice in . Successive Minima

0𝐾

𝜆1𝐾- 𝑦 1


0

𝑦 1-𝜆1𝐾


𝑦 2𝜆2𝐾

-

0

𝑦 1-𝜆1𝐾

Symmetric convex body and lattice in .Minkowski’s First Theorem

0

𝑠𝐾

Symmetric convex body and lattice in .Pf: Let . For

Minkowski’s First Theorem

0

𝑦

Symmetric convex body and lattice in .Pf: By Minkowski’s convex body theorem, .

Since this holds as , .

Minkowski’s First Theorem

0𝑠𝐾

𝑦 2𝜆2𝐾

-

𝑦 1-𝜆1𝐾

Symmetric convex body and lattice in .Minkowski’s Second Theorem

0

Theorem [Kannan-Lovasz 88]:

For symmetric becomes

Covering Radius vs Successive Minima

“Naïve” Babai rounding

For a closed convex set and , there exists such that

Separator Theorem

Can use where is the closest point in to .

𝑝𝐾𝑦

𝑥∗

For a closed convex set and , there exists such that

𝑝𝐾

Separator Theorem

𝑦

If not separator can get closer to on line segment .

𝑧𝑥∗∗𝑥∗

For compact convex with in relative interior,the polar is

For , the dual norm

Remark:

Polar Bodies and Dual Norms

0(1,1)

(1,-1)

(-1,1)

(-1,-1) 0 (1,0)(-1,0)(0,1)

(0,-1)

Theorem: compact convex with in rel. int., then .In particular .

Polar Bodies and Dual NormsDef:

𝐾0

Theorem: compact convex with in rel. int., then .In particular .


Pf: Easy to check that.

Hence .

Must show .

𝐾0

Theorem: compact convex with in rel. int., .In particular .


Pf: Take and in .

By separator theorem such that

Scale such that

𝑝𝐾𝑦0

Then and therefore .

Theorem: compact convex with in rel. int., .In particular .

𝑝𝐾

Polar Bodies and Dual Norms

𝑦0

Def:

Theorem [Banaszczyk `95,`96]: For a symmetric convex body and lattice in

Pf of lower bnd: Take linearly independent vectors where , and where

Since , s.t. . Hence

Banaszczyk’s Transference Theorem

Theorem [Blashke `18, Santalo `49] : For a symmetric convex body

Theorem [Bourgain-Milman `87, Kuperberg `08]:For a symmetric convex body

Mahler Conjecture: minimized when is cube.

Volume Product Bounds

Theorem [Kannan-Lovasz `88] : For a symmetric convex body and lattice

Pf: By Minkowski’s first theorem

(Bourgain-Milman)

Minkowski Transference

a symmetric convex body and lattice L. Let the orthogonal projection onto a subspace .

Then for any

The following identities hold:

and .

Projected Norms and Lattices

𝐾𝐾 + 𝑡

Flatness Theorem

We will follow the proof of Kannan and Lovasz `88:Theorem: For a convex body and lattice :

ℤ2

Flatness Theorem

We will follow the proof of Kannan and Lovasz `88:Theorem: For a convex body and lattice :

Proof of lower bound:

𝐾𝐾 + 𝑡

ℤ2

Generalized Babai Rounding

Lemma [Kannan-Lovasz `88]: Let denote a basis of . Let be orthogonal projection onto . Then

where (Gram-Schmidt Orth.) and Pf: Let . Can write , .

Note that since .

By shifting , can assume that and (all quantities are invariant under shifts of )



where (Gram-Schmidt Orth.) and Pf: By shifting , have and

𝐾ℤ2

𝑏1𝜆 𝑧 1

𝜆 𝑧 2

𝐾



where (Gram-Schmidt Orth.) and Pf: By shifting , have and

ℤ2𝜆𝐾𝑏1



where (Gram-Schmidt Orth.) and Pf: Let . Note that .Suffices to show that that

Let , By induction , hence such .

𝐾



where (Gram-Schmidt Orth.) and Pf: By induction , hence such .

ℤ2

𝑏1

𝑏2𝑥𝑥

𝑦

𝜋 2

𝐾



where (Gram-Schmidt Orth.) and Pf: Have such .By definition can find s.t. .

Since , we have , for some shift . Therefore , such that .

𝐾



where (Gram-Schmidt Orth.) and Pf: By definition can find s.t. .

ℤ2

𝛼𝑏1𝑥𝑥

𝑦

𝜋 2

𝐾

�̂�

𝑏1



where (Gram-Schmidt Orth.) and Pf: Since , we have , for some shift .

Therefore , such that .

Since , note that Hence

Generalized HKZ Basis

For a symmetric convex body and lattice in

A generalized HKZ basis for with respect to satisfies where is orthogonal projection onto .

Flatness Theorem


Pf: Let be a HKZ basis with respect to . Pick j such that .

By generalized Babai,

By definition , hence

Flatness Theorem


For , we have and .

By the Minkowski transference

Flatness Theorem


By the Minkowski transference

By inclusion .Hence

𝐾

𝐾𝐾

Theorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .

Subspace Flatness Theorem

ℤ2

𝐾𝐾



ℤ2

𝐾



ℤ2

𝐾

: r=2

0

𝐻={0 }

ℤ2

Subspace Flatness TheoremTheorem [Kannan-Lovasz `88, D. 12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .

𝐾

ℤ2

: r=2

0

Integral shifts of


𝐾

ℤ2

: r=2

Integral shifts of intersecting


𝐾

ℤ2

𝐻={( x , y ) : y=0 }

0

: r=1


𝐾

ℤ2

Integral shifts of

0


𝐾

ℤ2

Integral shifts of intersecting


𝐾

If , is a hyperplane.Forcing corresponds to Classical Flatness Theorem.

ℤ2


𝐾

ℤ2

Subspace Flatness TheoremConjecture [KL `88, D.12]: For convex, either1. contains an integer point in every translation, or2. subspace of dimension s.t. every translation of intersects at most integral shifts of .

Generalized HKZ Basis

For a symmetric convex body and lattice in

A generalized HKZ basis for with respect to satisfies where is orthogonal projection onto .

Inhomogeneous Minkowski

Theorem [Kannan-Lovasz `88]: Convex body and lattice . a subspace , of some dimension , such that

Hence having ``large’’ volume (i.e. relative to determinant) in every projection implies ``always’’ contains lattice points.

In this sense, we get a generalization of Minkowski’s theorem for arbitrary convex bodies.



Pf: Let be a HKZ basis with respect to with satisfying as before.



Pf: Let and . By Generalized Babai and Minkowski’s first theorem

.

Brunn-Minkowski: +

Gaussian Heuristic

Lemma: For a convex body and lattice , and , then for any

𝐾 +𝑥

Gaussian HeuristicWant to bound .By shifting may assume .

𝐿

𝐾

Gaussian HeuristicSince covers space, exists fundamental domain .

𝐹𝐿

𝐿

𝐾

Gaussian HeuristicPlace around each point in .

𝐹

𝐿

𝐾

Gaussian HeuristicPlace around each point in .

Hence

𝐹


Corollary: Convex body and lattice . Assume for . a subspace , of some dimension , such that

Pf: May assume , since this is worst case.Picking from inhomogeneous Minkowski theorem, we have

By Gaussian Heuristic, for any , .

𝐿

12 𝐾

Subspace FlatnessHave .

𝐾

𝐿

Subspace FlatnessHence can find projection such that

is small.

𝐾

𝐿

𝐾

Subspace FlatnessCan find projection such that

is small.

𝑊

𝜋𝑊 (𝐾 )

𝜋𝑊 (𝐿)

𝐿

𝐾

Subspace FlatnessCorresponds to small number of shifts

of intersecting .

𝑊

𝜋𝑊 (𝐾 )

𝜋𝑊 (𝐿)

+ +…


Conjecture [Kannan-Lovasz `88]: Convex body and lattice . a subspace , of some dimension , such that

Corollary of Conjecture: Convex body and lattice . Assume for . a subspace , of some dimension , such that

Subspace Flatness for norm

Conjecture [Kannan-Lovasz `88, D `12]:



Lower bound valid for all . Given lower bound is polytime computable.



Promise problem: -coGapCRP (Covering Radius Problem) where

YES instances:

No instances:



Promise problem: -coGapCRP (Covering Radius Problem)

Conjecture implies -coGapCRP NP.

Current best: -coGapCRP NP. (Exponential Improvement!)

Transference Theorems in the Geometry of Numbers

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