On Approximating the Covering Radius and Finding Dense Lattice - PowerPoint PPT Presentation

On Approximating the Covering Radius and Finding Dense Lattice Subspaces Daniel Dadush Centrum Wiskunde & Informatica (CWI) ICERM April 2018

Outline 1. Integer Programming and the Kannan-Lovász (KL) Conjecture. 2. ℓ 2 KL Conjecture & the Reverse Minkowski Conjecture. 3. Finding dense lattice subspaces.

Integer Programming (IP) min𝑑𝑦 𝑑 s.t. 𝐵𝑦 ≤ 𝑐 𝐿 𝑦 𝜗 ℤ 𝑜 𝑜 variables, 𝑛 constraints Open Question: Is there a 2 𝑃(𝑜) time algorithm? First result: 2 𝑃(𝑜 2 ) [Lenstra `83] Best known complexity: 𝑜 𝑃(𝑜) [Kannan `87]

Main Dichotomy 𝜈 𝐿, ℤ 𝑜 ≔ smallest scaling 𝑡 such that every shift 𝑡𝐿 + 𝑢 contains an integer point. 𝐿 ℤ 2 Either covering radius 𝜈 𝐿, ℤ 𝑜 ≤ 1 .

Main Dichotomy 𝜈 𝐿, ℤ 𝑜 ≔ smallest scaling 𝑡 such that every shift 𝑡𝐿 + 𝑢 contains an integer point. 𝐿 ℤ 2 Either covering radius 𝜈 𝐿, ℤ 𝑜 ≤ 1 .

Main Dichotomy 𝜈 𝐿, ℤ 𝑜 ≔ smallest scaling 𝑡 such that every shift 𝑡𝐿 + 𝑢 contains an integer point. 𝐿 ℤ 2 Either covering radius 𝜈 𝐿, ℤ 𝑜 ≤ 1 .

Main Dichotomy 𝜈 𝐿, ℤ 𝑜 ≔ smallest scaling 𝑡 such that every shift 𝑡𝐿 + 𝑢 contains an integer point. 𝜈 𝐿, ℤ 2 = 2 2 3 𝐿 3 ℤ 2 Either covering radius 𝜈 𝐿, ℤ 𝑜 ≤ 1 .

Main Dichotomy Can find integer point in 2 𝑃(𝑜) time [D. 12] 𝐿 ℤ 2 Either covering radius 𝜈 𝐿, ℤ 𝑜 ≤ 1 .

Main Dichotomy Or 𝐿 is “flat”: Projection on 𝑧 -axis 𝐿 𝑄 = (0 1) ℤ 2 There exists rank 𝑙 ≥ 1 integer projection 𝑄 ∈ ℤ 𝑜×𝑙 1 𝑙 is small . such vol 𝑙 𝑄𝐿

Main Dichotomy Or 𝐿 is “flat”: Projection on 𝑧 -axis 𝐿 𝑄 = (0 1) ℤ 2 There exists rank 𝑙 ≥ 1 integer projection 𝑄 ∈ ℤ 𝑜×𝑙 1 𝑙 is small . such vol 𝑙 𝑄𝐿

Main Dichotomy Or 𝐿 is “flat”: Recurse on ≈ vol k PK subproblems [D. 12] 𝐿 ℤ 2 There exists rank 𝑙 ≥ 1 integer projection 𝑄 ∈ ℤ 𝑜×𝑙 1 𝑙 is small . such vol 𝑙 𝑄𝐿

Duality Relation 1 1 ≤ 𝜈 𝐿, ℤ 𝑜 𝑙 ≤ ? min vol 𝑙 𝑄𝐿 𝑄∈ℤ 𝑙×𝑜 𝑠𝑙 𝑄 =𝑙≥1 “Easy” side “Hard” side Either covering radius 𝜈(𝐿, ℤ 𝑜 ) is small or 𝐿 is “flat”.

Khinchine Flatness Theorem 𝐿 vol 1 (𝑄𝐿) ≤ ෩ O(𝑜 Τ 1 ≤ 𝜈 𝐿, ℤ 𝑜 4 3 ) min 𝑄∈ℤ 1×𝑜 𝑠𝑙 𝑄 =1 [ Khinchine `48, Babai `86, Hastad `86, Lenstra-Lagarias- Schnorr `87, Kannan-Lovasz `88, Banaszczyk `93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00 ]

Kannan-Lovász Flatness Theorem 𝐿 1 1 ≤ 𝜈 𝐿, ℤ 𝑜 𝑙 ≤ 𝑜 min vol 𝑙 𝑄𝐿 𝑄∈ℤ 𝑙×𝑜 𝑠𝑙 𝑄 =𝑙≥1 [ Kannan `87, Kannan-Lovász `88 ]

Kannan-Lovász (KL) Conjecture 𝐿 1 1 ≤ 𝜈 𝐿, ℤ 𝑜 𝑙 ≤ 𝑃 log 𝑜 ‼ min vol 𝑙 𝑄𝐿 𝑄∈ℤ 𝑙×𝑜 𝑠𝑙 𝑄 =𝑙≥1

Faster Algorithm for IP? 1 1 ≤ 𝜈 𝐿, ℤ 𝑜 𝑙 ≤ 𝑃 log 𝑜 min vol 𝑙 𝑄𝐿 𝑄∈ℤ 𝑙×𝑜 𝑠𝑙 𝑄 =𝑙≥1 D. `12: Assuming KL conjecture + 𝑄 computable in (log 𝑜) 𝑃(𝑜) time then there is log 𝑜 𝑃(𝑜) time algorithm for IP.

ℓ 2 Kannan-Lovász Conjecture Does the conjecture hold for ellipsoids? 𝐹 0 ℤ 𝑜 𝑜 An ellipsoid is 𝐹 = 𝑈𝐶 2

ℓ 2 Kannan-Lovász Conjecture Answer: YES* [Regev-S.Davidowitz 17] 𝐹 0 ℤ 𝑜 * up to polylogarithmic factors

ℓ 2 Kannan-Lovász Conjecture Can we compute the projection P? 𝐹 0 ℤ 𝑜 THIS TALK: YES, in 2 𝑃(𝑜) time.

Outline 1. Integer Programming and the Kannan-Lovász (KL) Conjecture. 2. ℓ 2 KL Conjecture & the Reverse Minkowski Conjecture. 3. Finding dense lattice subspaces.

ℓ 2 Kannan-Lovász Conjecture Easier to think of Euclidean ball vs general lattice. 𝑜 𝑈𝐹 = 𝐶 2 0 ℒ = 𝑈ℤ 𝑜

Lattices A lattice ℒ ⊆ ℝ 𝑜 is 𝐶ℤ 𝑜 for a basis 𝐶 = 𝑐 1 , … , 𝑐 𝑜 . ℒ(𝐶) denotes the lattice generated by 𝐶 . 𝑐 2 𝑐 1 𝑐 1 Note: a lattice has many equivalent bases. 𝑐 2 𝑐 2 ℒ

Lattices A lattice ℒ ⊆ ℝ 𝑜 is 𝐶ℤ 𝑜 for a basis 𝐶 = 𝑐 1 , … , 𝑐 𝑜 . ℒ(𝐶) denotes the lattice generated by 𝐶 . 𝑐 1 The determinant of ℒ is | det 𝐶 | . 𝑐 2 ℒ

Lattices A lattice ℒ ⊆ ℝ 𝑜 is 𝐶ℤ 𝑜 for a basis 𝐶 = 𝑐 1 , … , 𝑐 𝑜 . ℒ(𝐶) denotes the lattice generated by 𝐶 . 𝑐 1 The determinant of ℒ is | det 𝐶 | . Equal to volume of 𝑐 2 any tiling set. ℒ

ℓ 2 Covering Radius 𝑜 , ℒ 𝜈 ℒ ≔ 𝜈 𝐶 2 Distance of farthest point to the lattice ℒ . 𝜈 𝒲 ℒ Voronoi cell 𝒲 ≔ all points closer to 0

Volumetric Lower Bounds 𝑜 𝜈 ℒ vol 𝑜 𝐶 2 ≥ vol n 𝒲 = det(ℒ) 𝜈 𝒲 ℒ Voronoi cell 𝒲 ≔ all points closer to 0

Volumetric Lower Bounds 𝑜 −1 1 𝑜 det ℒ 𝜈(ℒ) ≥ vol n 𝐶 2 𝑜 𝜈 𝒲 ℒ Voronoi cell 𝒲 ≔ all points closer to 0

Volumetric Lower Bounds 1 𝜈(ℒ) ≿ 𝑜 det ℒ 𝑜 𝜈 𝒲 ℒ Voronoi cell 𝒲 ≔ all points closer to 0

Volumetric Lower Bounds 𝜈 ℒ ≥ 𝜈 ℒ ↓𝑋 𝜈 𝜈 ↓𝑋 𝑋 ℒ ℒ ↓𝑋 projection onto 𝑋

Volumetric Lower Bounds 1 𝜈 ℒ ≥ 𝜈 ℒ ↓𝑋 ≳ 𝑙 det ℒ ↓𝑋 𝑙 𝜈 𝜈 ↓𝑋 𝑋 ℒ ℒ ↓𝑋 projection onto 𝑋 dim(𝑋) = 𝑙 ≥ 1

Volumetric Lower Bounds 1 𝜈 ℒ ≳ max 𝑙 det ℒ ↓𝑋 𝑙 dim 𝑋 =𝑙≥1 𝜈 𝜈 ↓𝑋 𝑋 ℒ ℒ ↓𝑋 projection onto 𝑋 dim(𝑋) = 𝑙 ≥ 1

ℓ 2 Kannan-Lovász Conjecture Define 𝐷 𝐿𝑀,2 (𝑜) to be smallest number such that 1 𝜈 ℒ ≤ 𝐷 𝐿𝑀,2 (𝑜) max 𝑙 det ℒ ↓𝑋 𝑙 dim 𝑋 =𝑙≥1 for all lattices of dimension at most 𝑜 . 𝐷 𝐿𝑀,2 𝑜 = Ω( log 𝑜) 1 1 Lower bound for ℒ with basis 𝑓 1 , 2 𝑓 2 , … , 𝑜 𝑓 𝑜 .

KL Bounds 1 𝜈 ℒ ≤ 𝐷 𝐿𝑀,2 (𝑜) max 𝑙 det ℒ ↓𝑋 𝑙 dim 𝑋 =𝑙≥1 𝑜 Kannan-Lovász `88: D. Regev `16: log 𝑃(1) 𝑜 Assuming Reverse Minkowski Conjecture. 3 2 𝑜 Regev, S.Davidowitz `17: log Τ Reverse Minkowski Conjecture is proved!

Our Results 𝑜 dimensional lattice ℒ ≔ ℒ(𝐶) 1. Can compute subspace 𝑋 , dim 𝑋 = 𝑙 ≥ 1 1 𝜈 ℒ ≤ 𝑃(log 2.5 𝑜 ) 𝑙 det ℒ ↓𝑋 𝑙 in 2 𝑃(𝑜) time with high probability. Prior work: 𝑜 in 2 𝑃(𝑜) time. Kannan Lovász `88: D. Micciancio `13: best subspace in 𝑜 𝑃(𝑜 2 ) time.

Our Results 𝑜 dimensional lattice ℒ ≔ ℒ(𝐶) 2. Can combine lower bounds over different subspaces to certify 𝜈 𝑀 up to the slicing constant 𝑀 𝑜 for “stable” Voronoi cells*. 𝒲 * If vol n 𝒲 = 1 𝐼 can find hyperplane 𝐼 s.t. vol n−1 𝒲 ∩ 𝐼 = Ω( 1 𝑀𝑜 )

Our Results 𝑜 dimensional lattice ℒ ≔ ℒ(𝐶) 2. Can combine lower bounds over different subspaces to certify 𝜈 𝑀 up to the slicing constant 𝑀 𝑜 for “stable” Voronoi cells*. 𝒲 Slicing Conjecture: 𝐼 𝑀 𝑜 = 𝑃(1) for all convex bodies! For “stable” Voronoi cells: 𝑀 𝑜 = 𝑃(log 𝑜) [RS `17]

Notation 𝑁 ⊆ ℒ sublattice of dimension 𝑙 Convention: 𝑁 = {0} then det 𝑁 ≔ 1 . Normalized Determinant: Τ 1 𝑙 nd 𝑁 ≔ det 𝑁 Projected Sublattice: ℒ 𝑁 ≔ ℒ projected onto span 𝑁 ⊥ Τ

Lower Bounds for Chains Theorem [D. 17]: For 0 = ℒ 0 ⊂ ℒ 1 ⊂ ⋯ ⊂ ℒ 𝑙 = ℒ then 𝜈 ℒ 2 ≿ σ 𝑗=1 𝑙 ℒ ℒ 𝑗−1 2 Τ Τ dim( ℒ 𝑗 ℒ 𝑗−1 ) nd Only “missing ingredient”: Combined with techniques from [R.S. `17] easily get tightness within slicing constant 𝑀 𝑜 .

Lower Bounds for Chains Theorem [D. 17]: For 0 = ℒ 0 ⊂ ℒ 1 ⊂ ⋯ ⊂ ℒ 𝑙 = ℒ then 𝜈 ℒ 2 ≿ σ 𝑗=1 𝑙 ℒ ℒ 𝑗−1 2 Τ Τ dim( ℒ 𝑗 ℒ 𝑗−1 ) nd Proof Idea: 1. Establish SDP based lower bound: [D.R. `16] 𝜈 ℒ 2 ≿ max σ 𝑗 rk 𝑄 𝑗 nd 𝑄 𝑗 ℒ 2 ∗ 𝑄 𝑗 ≼ 𝐽 𝑜 s.t. σ 𝑗 𝑄 𝑗 2. Build solution to above starting from any chain.

Lattice Density 𝑜 ℒ ∩ 𝑠𝐶 2 1 lim 𝑜 ) = vol 𝑜 (𝑠𝐶 2 det ℒ 𝑠→∞ 𝑠 ℒ

Lattice Density 𝑜 ℒ ∩ 𝑠𝐶 2 1 lim 𝑜 ) = vol 𝑜 (𝑠𝐶 2 det ℒ 𝑠→∞ 𝑠 Global density of lattice points per unit volume ℒ

Minkowski’s First Theorem 𝑜 ) 𝑜 ≥ 2 −𝑜 vol 𝑜 (𝑠𝐶 2 ℒ ∩ 𝑠𝐶 2 det ℒ 1889 𝑠 ℒ Global density implies “local density”

Reverse Minkowski Theorem Regev-S.Davidowitz `17: ℒ lattice dimension 𝑜 . If all sublattices of ℒ have determinant at least 1 then: ℒ has at most 2 𝑃(log 2 𝑜 𝑠 2 ) points at distance 𝑠 . Almost tight: ℤ 𝑜 has 𝑜 Ω(𝑙) points at distance 𝑠 for 𝑙 ≪ 𝑜 .

Outline 1. Integer Programming and the Kannan-Lovász (KL) Conjecture. 2. ℓ 2 KL Conjecture & the Reverse Minkowski Conjecture. 3. Finding dense lattice subspaces.