A bound for the number of vertices of a polytope with applications – A. Barvinok
=
presented by Marek Krčál
Theorem 1. ∀α there is γ = γ(α): ∀m ≤ αn
n
n
∀u1 . . . um ∈ R where kui k ≤ 1, ∀P ⊆ R polytope
√
If {|hx, ui i| ≤ 1 for i = 1, . . . , m} ⊆ P ⊆ {kxk ≤ α n}
Lemma 1. Let y ∈ Rn be a random
|
|
{z
}
{z
} Pf of Thm 1. Choose β = β(α) “big enough’:’ 1)
2) . . .
Gaussian vector. Then
Q
∀n > 16
2
−n/16
αn
1. Pry kyk < n/2 ≤ e
β
n
y ∈ βQ
1 − e− 2
Pry maxhy, xi ≥
≥Pry
≥
n
2
n
kyk ≥ 2
x∈P
2β
2. For arbitrary u ∈ Rn holds
−e− 16
2
2
|
{z
}
√
Pry [hu, yi ≥ τ ] ≤ eτ /2kuk
α n
X
τ2
|W | − 2kvk
3. For u1 , . . . , um ∈ Rn , kui k ≤ 1 holds
2
Pry maxhy, vi ≥ τ ≤
Pry [hy, vi ≥ τ ] ≤
e
m
β 2 /2
v∈W
2
Pr
[hu
,
yi
≤
β]
≥
1
−
e
·(1/2)
y
i
v∈W
|W | − 8αn2 β2
≤
e
then P has at least 2γn vertices.
2
Pr (G) ⊆ R|E| - the convex hull of all indicators χH
where H are r-factors of a given G = (V, E)
is given by
X
xe
∈
xe
= r
for all v ∈ V
xe
≥ 1 − |F |
for all U ⊆ V, F ⊆ δ(U )
s. t. r|U | + |F | is odd
[0, 1]
for all e ∈ E
e∈δ(V )
X
xe −
e∈δ(V )\F
X
e∈F
Lemma 2. ∀k, r > 0 with k ≥ 2r + 1 there is Let G be a k-regular graph such that
Theorem 2. ∀k, r > 0 with k ≥ 2r + 1
there is γ = γ(k, r) s. t.:
Let G be a k-regular graph such that
k
|δ(U )| >
r
for every U ⊆ V such that
2 ≤ |U | ≤ |V | − 2.
Then for all y ∈ RE such that
X
ye
∈
[−, ] for all e ∈ E
ye
=
0 for all v ∈ V
e∈δ(v)
Pr
{ kr 1 + y}
holds
r
k1
+ y ∈ Pr (G)
|δ(U )| >
k
r
for every U ⊆ V such that
2 ≤ |U | ≤ |V | − 2.
Then the number of r-factors of G is at
least 2γ|V | .