0/1 Polytopes with Quadratic
Chvátal Rank
Thomas Rothvoß
and
Laura Sanità
3rd Cargese Workshop on Combinatorial Optimization
Gomory Chvátal Cuts
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Given: Polytope P ⊆ Rn
P
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Gomory Chvátal Cuts
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b
Given: Polytope P ⊆ Rn
Want: Integral hull
PI := conv{P ∩ Zn }
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P
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PI
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Gomory Chvátal Cuts
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Given: Polytope P ⊆ Rn
Want: Integral hull
PI := conv{P ∩ Zn }
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P
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Idea: Let cx ≤ β valid
inequality for P (c ∈ Zn )
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cx ≤ β
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PI
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Gomory Chvátal Cuts
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◮
b
Given: Polytope P ⊆ Rn
Want: Integral hull
PI := conv{P ∩ Zn }
◮
Idea: Let cx ≤ β valid
inequality for P (c ∈ Zn )
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The Gomory Chvátal
cut cx ≤ ⌊β⌋ valid for PI
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P
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cx ≤ β
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PI
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cx ≤ ⌊β⌋
Gomory Chvátal Cuts
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◮
b
Given: Polytope P ⊆ Rn
Want: Integral hull
PI := conv{P ∩ Zn }
◮
Idea: Let cx ≤ β valid
inequality for P (c ∈ Zn )
◮
The Gomory Chvátal
cut cx ≤ ⌊β⌋ valid for PI
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P
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P′
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cx ≤ β
PI
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cx ≤ ⌊β⌋
Gomory Chvátal closure
\
\
{x | cx ≤ ⌊max{cy | y ∈ P }⌋}
P ′ = {all GC cuts for P } =
◮
c∈Zn
Gomory Chvátal Cuts
◮
◮
b
Given: Polytope P ⊆ Rn
Want: Integral hull
PI := conv{P ∩ Zn }
◮
Idea: Let cx ≤ β valid
inequality for P (c ∈ Zn )
◮
The Gomory Chvátal
cut cx ≤ ⌊β⌋ valid for PI
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P
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P′
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cx ≤ β
PI
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cx ≤ ⌊β⌋
Gomory Chvátal closure
\
\
{x | cx ≤ ⌊max{cy | y ∈ P }⌋}
P ′ = {all GC cuts for P } =
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c∈Zn
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kth closure P (k) := |P ′′′{z. . .}′
k times
Gomory Chvátal Cuts
◮
◮
b
Given: Polytope P ⊆ Rn
Want: Integral hull
PI := conv{P ∩ Zn }
◮
Idea: Let cx ≤ β valid
inequality for P (c ∈ Zn )
◮
The Gomory Chvátal
cut cx ≤ ⌊β⌋ valid for PI
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P
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P′
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cx ≤ β
PI
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cx ≤ ⌊β⌋
Gomory Chvátal closure
\
\
{x | cx ≤ ⌊max{cy | y ∈ P }⌋}
P ′ = {all GC cuts for P } =
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c∈Zn
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kth closure P (k) := |P ′′′{z. . .}′
k times
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Chvátal rank: P (rk(P )) = PI
What’s known
What’s known
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For each rational polyhedron P , rk(P ) < ∞
What’s known
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For each rational polyhedron P , rk(P ) < ∞
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But for every k, there is P ⊆ R2 with rk(P ) ≥ k
(0, 1)
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(k,
P
(0, 0)
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1
2)
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What’s known
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For each rational polyhedron P , rk(P ) < ∞
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But for every k, there is P ⊆ R2 with rk(P ) ≥ k
(0, 1)
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(k,
P
(0, 0)
◮
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For the rest of the talk assume P ⊆ [0, 1]n
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1
2)
b
What’s known — if P ⊆ [0, 1]n
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rk(P ) ≤ O(n3 log n)
[Bockmayer, Eisenbrand, Hartmann, Schulz ’98]
What’s known — if P ⊆ [0, 1]n
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rk(P ) ≤ O(n3 log n)
[Bockmayer, Eisenbrand, Hartmann, Schulz ’98]
rk(P ) ≤ O(n2 log n) [Eisenbrand, Schulz ’99]
What’s known — if P ⊆ [0, 1]n
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rk(P ) ≤ O(n3 log n)
[Bockmayer, Eisenbrand, Hartmann, Schulz ’98]
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rk(P ) ≤ O(n2 log n) [Eisenbrand, Schulz ’99]
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For some P , rk(P ) ≥ (1 + ε)n [Eisenbrand, Schulz ’99]
What’s known — if P ⊆ [0, 1]n
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rk(P ) ≤ O(n3 log n)
[Bockmayer, Eisenbrand, Hartmann, Schulz ’98]
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rk(P ) ≤ O(n2 log n) [Eisenbrand, Schulz ’99]
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For some P , rk(P ) ≥ (1 + ε)n [Eisenbrand, Schulz ’99]
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For some P , rk(P ) ≥ 1.36n [Pokutta, Stauffer ’11]
What’s known — if P ⊆ [0, 1]n
◮
rk(P ) ≤ O(n3 log n)
[Bockmayer, Eisenbrand, Hartmann, Schulz ’98]
◮
rk(P ) ≤ O(n2 log n) [Eisenbrand, Schulz ’99]
◮
For some P , rk(P ) ≥ (1 + ε)n [Eisenbrand, Schulz ’99]
◮
For some P , rk(P ) ≥ 1.36n [Pokutta, Stauffer ’11]
What’s known — if P ⊆ [0, 1]n
◮
rk(P ) ≤ O(n3 log n)
[Bockmayer, Eisenbrand, Hartmann, Schulz ’98]
◮
rk(P ) ≤ O(n2 log n) [Eisenbrand, Schulz ’99]
◮
For some P , rk(P ) ≥ (1 + ε)n [Eisenbrand, Schulz ’99]
◮
For some P , rk(P ) ≥ 1.36n [Pokutta, Stauffer ’11]
Theorem (Sanità, R. ’12)
There exists a family of polytopes P ⊆ [0, 1]n with Chvátal rank
rk(P ) ≥ Ω(n2 ).
The polytope
cx = 12 kck1
The polytope
◮
Let c ∈ Zn≥0 be a vector
n
kck1 o
P (c) := conv x ∈ {0, 1}n : cx ≤
2 }
|
{z
Knapsack solutions
cx = 12 kck1
The polytope
◮
Let c ∈ Zn≥0 be a vector
n
kck1 o
P (c) := conv x ∈ {0, 1}n : cx ≤
2 }
|
{z
Knapsack solutions
( 12 , . . . , 12 )
cx = 12 kck1
The polytope
◮
Let c ∈ Zn≥0 be a vector
P (c, ε) := conv
nn
x ∈ {0, 1}n : cx ≤
|
{z
o
kck1 o
∪ {x∗ (ε)}
| {z }
2 }
Knapsack solutions
special vertex
x∗ (ε) = ( 21 + ε, . . . , 12 + ε)
( 12 , . . . , 12 )
P
PI
cx = 12 kck1
Critical vectors
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Call c̃ critical ⇔ c̃ maximized at x∗
x∗ (ε)
P
Critical vectors
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Call c̃ critical ⇔ c̃ maximized at x∗
c̃
x∗ (ε)
P
Critical vectors
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Call c̃ critical ⇔ c̃ maximized at x∗
c̃x ≤ ⌊β⌋ cuts of x∗ =⇒ c̃ critical
c̃
x∗ (ε)
P
Critical vectors
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Call c̃ critical ⇔ c̃ maximized at x∗
c̃x ≤ ⌊β⌋ cuts of x∗ =⇒ c̃ critical
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0
P
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Critical vectors
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Call c̃ critical ⇔ c̃ maximized at x∗
c̃x ≤ ⌊β⌋ cuts of x∗ =⇒ c̃ critical
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c̃
b
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0
P
b
b
Critical vectors
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Call c̃ critical ⇔ c̃ maximized at x∗
c̃x ≤ ⌊β⌋ cuts of x∗ =⇒ c̃ critical
b
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c̃
b
∼ε
P
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0
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b
Overview
Overview
critical vectors are long =⇒ Ω(n2 ) rank
Overview
critical vectors are good
Simultaneous Diophantine Approximations to c
critical vectors are long =⇒ Ω(n2 ) rank
Overview
random vector has no short, good SDA
critical vectors are good
Simultaneous Diophantine Approximations to c
critical vectors are long =⇒ Ω(n2 ) rank
A lower bound strategy
Theorem
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
A lower bound strategy
Theorem
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
b
ε0
P (0)
A lower bound strategy
Theorem
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
ε0
ε1
b
b
P (1)
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
ε0
ε1
b
b
b
ε2
P (2)
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
ε0
ε1
b
b
b
b
ε2
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
ε0
ε1
b
b
b
b
b
ε2
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
ε0
ε1
b
b
b
b
b
b
εk
ε2
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
b
b
b
b
b
b
εk
ε0
x∗ (εi )
x∗ (εi+1 )
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
b
b
b
b
b
b
ε0
x∗ (εi )
x∗ (εi+1 )
εk
c̃x ≤ β
◮
Let c̃x ≤ β be the GC cut “cutting furthest” in it. i
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
b
b
b
b
b
b
ε0
x∗ (εi )
x∗ (εi+1 )
εk
c̃x ≤ β
c̃x ≤ ⌊β⌋
◮
Let c̃x ≤ β be the GC cut “cutting furthest” in it. i
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
b
b
b
b
b
b
ε0
x∗ (εi )
x∗ (εi+1 )
εk
c̃x ≤ β
c̃x ≤ ⌊β⌋
◮
Let c̃x ≤ β be the GC cut “cutting furthest” in it. i
c̃x∗ (εi )−c̃x∗ (εi+1 )
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
b
b
b
b
b
b
ε0
x∗ (εi )
x∗ (εi+1 )
εk
c̃x ≤ β
c̃x ≤ ⌊β⌋
◮
Let c̃x ≤ β be the GC cut “cutting furthest” in it. i
1 ≥ c̃x∗ (εi )−c̃x∗ (εi+1 )
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
b
b
b
b
b
b
ε0
x∗ (εi )
x∗ (εi+1 )
εk
c̃x ≤ β
c̃x ≤ ⌊β⌋
◮
Let c̃x ≤ β be the GC cut “cutting furthest” in it. i
1 ≥ c̃x∗ (εi )−c̃x∗ (εi+1 ) = kc̃k1 ·(εi −εi+1 )
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
b
b
b
b
b
b
ε0
x∗ (εi )
x∗ (εi+1 )
εk
c̃x ≤ β
c̃x ≤ ⌊β⌋
◮
Let c̃x ≤ β be the GC cut “cutting furthest” in it. i
n
1 ≥ c̃x∗ (εi )−c̃x∗ (εi+1 ) = kc̃k1 ·(εi −εi+1 ) ≥ Ω( )·(εi −εi+1 )
εi
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
b
b
b
b
b
b
ε0
x∗ (εi )
x∗ (εi+1 )
εk
c̃x ≤ β
c̃x ≤ ⌊β⌋
◮
◮
Let c̃x ≤ β be the GC cut “cutting furthest” in it. i
n
1 ≥ c̃x∗ (εi )−c̃x∗ (εi+1 ) = kc̃k1 ·(εi −εi+1 ) ≥ Ω( )·(εi −εi+1 )
εi
εi+1
Θ(1)
Then εi ≥ 1 − n
A lower bound strategy
Theorem
c̃ critical =⇒ kc̃k1 ≥ Ω( nε ).
Assume: ∀ε ∈ [( 21 )Θ(n) , Θ(1)] :
Then rk(P ) ≥ Ω(n2 ).
b
b
b
b
b
b
ε0
x∗ (εi )
x∗ (εi+1 )
εk
c̃x ≤ β
c̃x ≤ ⌊β⌋
◮
◮
Let c̃x ≤ β be the GC cut “cutting furthest” in it. i
n
1 ≥ c̃x∗ (εi )−c̃x∗ (εi+1 ) = kc̃k1 ·(εi −εi+1 ) ≥ Ω( )·(εi −εi+1 )
εi
εi+1
Θ(1)
2
Then εi ≥ 1 − n =⇒ k ≥ Ω(n )
Overview
critical vectors are long =⇒ Ω(n2 ) rank
Overview
critical vectors are good
Simultaneous Diophantine Approximations to c
critical vectors are long =⇒ Ω(n2 ) rank
Simultaneous Diophantine Approximation
Lemma
Under magical assumptions
c̃ critical =⇒ kλc̃ − ck1 ≤ O(ε) · kck1
◮
(for some λ > 0)
Intuition: c̃ critical =⇒ c̃ well-approximates c
c
x∗ (ε)
P
c̃
Simultaneous Diophantine Approximation
Lemma
Under magical assumptions
c̃ critical =⇒ kλc̃ − ck1 ≤ O(ε) · kck1
◮
(for some λ > 0)
Intuition: c̃ critical =⇒ c̃ well-approximates c
c
x∗ (ε)
P
c̃
λc̃
Simultaneous Diophantine Approximation
Lemma
Under magical assumptions
c̃ critical =⇒ kλc̃ − ck1 ≤ O(ε) · kck1
◮
(for some λ > 0)
Intuition: c̃ critical =⇒ c̃ well-approximates c
c
x∗ (ε)
λc̃
c̃
P
◮
Lemma follows from:
critical
1
1
c
+ ε kc̃k1 ≥ max{c̃x | x ∈ PI } ≥ kc̃k1 + Ω c̃ − 2
2
λ 1
Simultaneous Diophantine Approximation
Lemma
Under magical assumptions
c̃ critical =⇒ kλc̃ − ck1 ≤ O(ε) · kck1
◮
(for some λ > 0)
Intuition: c̃ critical =⇒ c̃ well-approximates c
c
x∗ (ε)
λc̃
c̃
P
◮
Lemma follows from:
to show:
critical
1
1
c
+ ε kc̃k1 ≥ max{c̃x | x ∈ PI } ≥ kc̃k1 + Ω c̃ − 2
2
λ 1
Finding good knapsack solutions (the ideal case)
◮
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
c̃i
ci
c1
c2
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
c̃i
ci
c1
c2
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
∈J
c1
c2
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
∈J
c1
c2
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
∈J
c1
c2
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
∈J
c1
c2
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
∈J
c1
c2
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
∈J
c1
c2
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
∈J
c1
c2
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
P
kck1
Pick λ > 0 s.t.
i:c̃i /ci >1/λ ci = 2
∈J
1
λ
c1
c2
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
P
kck1
Pick λ > 0 s.t.
i:c̃i /ci >1/λ ci = 2
∈J
1
λ
c1 c2
X
c̃i
i∈J
1
2 kck1
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
P
kck1
Pick λ > 0 s.t.
i:c̃i /ci >1/λ ci = 2
∈J
1
λ
c1 c2
X
1X
1
c̃i
c̃i = kc̃k1 +
2
2
i∈J
i∈J
1
2 kck1
1X
c̃i
−
2
i∈J
/
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
P
kck1
Pick λ > 0 s.t.
i:c̃i /ci >1/λ ci = 2
∈J
1
λ
1
c1 c2
2 kck1
X
X
ci
1
1 X
ci 1
c̃i −
−
c̃i −
c̃i = kc̃k1 +
2
2
λ
2
λ
i∈J
i∈J
i∈J
/
kck1
Finding good knapsack solutions (the ideal case)
◮
◮
◮
c̃i
ci
We want: max{c̃x | x ∈ PI } ≥ 12 kc̃k1 + Ω c̃ − λc 1
Sort cc̃11 > . . . > cc̃nn (i.e. according to profit
cost ratio).
P
kck1
Pick λ > 0 s.t.
i:c̃i /ci >1/λ ci = 2
∈J
1
λ
1
kck1
c1 c2
2 kck1
X
X
ci
1
1 X
ci 1
1
1
c
c̃i −
−
c̃i −
= kc̃k1 + c̃− c̃i = kc̃k1 +
2
2
λ
2
λ
2
2
λ 1
i∈J
i∈J
i∈J
/
. . . now more realistic
c̃i
ci
c1
c2
1
2 kck1
kck1
. . . now more realistic
◮
Trick: Let c contain numbers 20 , 21 , 22 , . . . , kck∞ (3 copies)
c̃i
ci
c1
◮
1
2 kck1
c2
kck1
Knapsack capacity:
0
1
2 kck1
. . . now more realistic
◮
Trick: Let c contain numbers 20 , 21 , 22 , . . . , kck∞ (3 copies)
c̃i
ci
c1
◮
1
2 kck1
c2
kck1
Knapsack capacity:
0
1
2 kck1
. . . now more realistic
◮
Trick: Let c contain numbers 20 , 21 , 22 , . . . , kck∞ (3 copies)
c̃i
ci
c1
◮
1
2 kck1
c2
kck1
Knapsack capacity:
0
1
2 kck1
. . . now more realistic
◮
Trick: Let c contain numbers 20 , 21 , 22 , . . . , kck∞ (3 copies)
c̃i
ci
c1
◮
1
2 kck1
c2
kck1
Knapsack capacity:
0
1
2 kck1
. . . now more realistic
◮
Trick: Let c contain numbers 20 , 21 , 22 , . . . , kck∞ (3 copies)
c̃i
ci
c1
◮
1
2 kck1
c2
kck1
Knapsack capacity:
0
1
2 kck1
. . . now more realistic
◮
Trick: Let c contain numbers 20 , 21 , 22 , . . . , kck∞ (3 copies)
c̃i
ci
c1
◮
1
2 kck1
c2
kck1
Knapsack capacity:
0
1
2 kck1
. . . now more realistic
◮
Trick: Let c contain numbers 20 , 21 , 22 , . . . , kck∞ (3 copies)
c̃i
ci
c1
◮
1
2 kck1
c2
kck1
Knapsack capacity:
0
1
2 kck1
. . . now more realistic
◮
Trick: Let c contain numbers 20 , 21 , 22 , . . . , kck∞ (3 copies)
c̃i
ci
c1
◮
1
2 kck1
c2
kck1
Knapsack capacity:
0
1
2 kck1
. . . now more realistic
◮
Trick: Let c contain numbers 20 , 21 , 22 , . . . , kck∞ (3 copies)
c̃i
ci
1
λ
c1
◮
1
2 kck1
c2
kck1
Knapsack capacity:
0
1
2 kck1
Overview
critical vectors are good
Simultaneous Diophantine Approximations to c
critical vectors are long =⇒ Ω(n2 ) rank
Overview
random vector has no short, good SDA
critical vectors are good
Simultaneous Diophantine Approximations to c
critical vectors are long =⇒ Ω(n2 ) rank
Lemma
For D := 2n/8 , pick ci ∈ {1, . . . , D} at random.
Lemma
For D := 2n/8 , pick ci ∈ {1, . . . , D} at random. W.h.p.
n
kλc̃ − ck1 ≥ εkck1 ∀ε ∀λ > 0 ∀kc̃k1 ≤ o
ε
Lemma
For D := 2n/8 , pick ci ∈ {1, . . . , D} at random. W.h.p.
n
kλc̃ − ck1 ≥ εkck1 ∀ε ∀λ > 0 ∀kc̃k1 ≤ o
ε
◮
Suffices: For fixed λ, ε,
h
1
i
: kλc̃ − ck∞ ≤ εD ≤ o(1)n
(∗) Pr ∃kc̃k∞ ≤ o
ε
Lemma
For D := 2n/8 , pick ci ∈ {1, . . . , D} at random. W.h.p.
n
kλc̃ − ck1 ≥ εkck1 ∀ε ∀λ > 0 ∀kc̃k1 ≤ o
ε
◮
◮
Suffices: For fixed λ, ε,
h
1
i
: kλc̃ − ck∞ ≤ εD ≤ o(1)n
(∗) Pr ∃kc̃k∞ ≤ o
ε
Reason: Number of λ’s and ε’s is 2O(n) ; there must be a
Ω(n)-size subset of indices satisfying (∗)
Lemma
For D := 2n/8 , pick ci ∈ {1, . . . , D} at random. W.h.p.
n
kλc̃ − ck1 ≥ εkck1 ∀ε ∀λ > 0 ∀kc̃k1 ≤ o
ε
D
◮
◮
...
...
cn
c1
c2
Suffices: For fixed λ, ε,
h
1
i
: kλc̃ − ck∞ ≤ εD ≤ o(1)n
(∗) Pr ∃kc̃k∞ ≤ o
ε
Reason: Number of λ’s and ε’s is 2O(n) ; there must be a
Ω(n)-size subset of indices satisfying (∗)
Lemma
For D := 2n/8 , pick ci ∈ {1, . . . , D} at random. W.h.p.
n
kλc̃ − ck1 ≥ εkck1 ∀ε ∀λ > 0 ∀kc̃k1 ≤ o
ε
o( 1ε ) many
◮
◮
λZ
D
...
...
cn
c1
c2
Suffices: For fixed λ, ε,
h
1
i
: kλc̃ − ck∞ ≤ εD ≤ o(1)n
(∗) Pr ∃kc̃k∞ ≤ o
ε
Reason: Number of λ’s and ε’s is 2O(n) ; there must be a
Ω(n)-size subset of indices satisfying (∗)
Lemma
For D := 2n/8 , pick ci ∈ {1, . . . , D} at random. W.h.p.
n
kλc̃ − ck1 ≥ εkck1 ∀ε ∀λ > 0 ∀kc̃k1 ≤ o
ε
2εD λZ
◮
◮
D
...
...
cn
c1
c2
Suffices: For fixed λ, ε,
h
1
i
: kλc̃ − ck∞ ≤ εD ≤ o(1)n
(∗) Pr ∃kc̃k∞ ≤ o
ε
Reason: Number of λ’s and ε’s is 2O(n) ; there must be a
Ω(n)-size subset of indices satisfying (∗)
Lemma
For D := 2n/8 , pick ci ∈ {1, . . . , D} at random. W.h.p.
n
kλc̃ − ck1 ≥ εkck1 ∀ε ∀λ > 0 ∀kc̃k1 ≤ o
ε
2εD λZ
D
o(D)
◮
◮
...
...
cn
c1
c2
Suffices: For fixed λ, ε,
h
1
i
: kλc̃ − ck∞ ≤ εD ≤ o(1)n
(∗) Pr ∃kc̃k∞ ≤ o
ε
Reason: Number of λ’s and ε’s is 2O(n) ; there must be a
Ω(n)-size subset of indices satisfying (∗)
Lemma
For D := 2n/8 , pick ci ∈ {1, . . . , D} at random. W.h.p.
n
kλc̃ − ck1 ≥ εkck1 ∀ε ∀λ > 0 ∀kc̃k1 ≤ o
ε
2εD λZ
D
o(D)
◮
◮
...
...
cn
c1
c2
Suffices: For fixed λ, ε,
h
1
i
: kλc̃ − ck∞ ≤ εD ≤ o(1)n
(∗) Pr ∃kc̃k∞ ≤ o
ε
Reason: Number of λ’s and ε’s is 2O(n) ; there must be a
Ω(n)-size subset of indices satisfying (∗)
Overview
random vector has no short, good SDA
critical vectors are good
Simultaneous Diophantine Approximations to c
critical vectors are long =⇒ Ω(n2 ) rank
The end
Thanks for your attention
Where is the bottleneck for ω(n2 ) bound?
◮
◮
◮
Problem 1: Our proof technique does not extent!
n
n
2 random numbers in {1, . . . , D} + 2 “fill numbers”
cannot work for D ≫ 2n
Problem 2: Set of normal vectors with ci ≥ 2Ω(n log n) is
extremely sparse!
2
2
(2O(n ) potential normal vectors, but 2Ω(n log n) vectors
with n log n bits per coefficient)
Problem 3: For coefficients > 2ω(n) , better SDAs exist!
For c ∈ [0, 1]n and N ∈ N. Find Q ∈ {1, . . . , N } s.t.
n
minimize kc − ZQ k∞ .
◮
◮
For Q := N , error ≤ N1
Dirichlet’s Theorem: error ≤
1
Q·N 1/n