On the existence of 0/1 polytopes
with high semidefinite extension complexity
Daniel Dadush
Centrum Wiskunde & Informatica (CWI)
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
1
Introduction
Joint Work with. . .
My coauthors:
• Sebastian Pokutta (Georgia Tech)
• Jop Briët (CWI)
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
2
Introduction
What is the expressive power of linear / semidefinite programming?
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
3
Introduction
What is the expressive power of linear / semidefinite programming?
For convex hulls such as
Matchings, Hamiltonian cycles, Graph Cuts, ...
what is the smallest linear / semidefinite program whose feasible
region captures them (even approximately)?
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
3
Introduction
What is the expressive power of linear / semidefinite programming?
For convex hulls such as
Matchings, Hamiltonian cycles, Graph Cuts, ...
what is the smallest linear / semidefinite program whose feasible
region captures them (even approximately)?
Alternative measure of complexity independent from P vs N P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
3
Introduction
Lower bounds on the size of Linear Programs:
1
2
3
4
Any symmetric LP that captures the TSP or Matching
polytope must have size 2Ω(n) [Yannakakis ’91]
There exists a convex hull of 0/1 points that cannot be
captured by an LP of size less than 2Ω(n) [ Rothvoss ’11]
Any LP that captures the TSP polytope must have size
1/2
2Ω(n ) [Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]
Any LP that ρ-approximates the Correlation polytope must
have size 2Ω(n/ρ) [Braun, Fiornini, Pokutta, Steurer ’12,
Braverman, Moitra ’13, Pokutta, Braun ’13]
5
Any LP of relaxation of size nr for the Correlation polytope
has integrality gap at least as large as O(r) levels of
Sherali-Adams [Chan, Lee, Raghavendra, Steurer ’13]
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
4
Introduction
Lower bounds on the size of Linear Programs:
1
2
3
4
Any symmetric LP that captures the TSP or Matching
polytope must have size 2Ω(n) [Yannakakis ’91]
There exists a convex hull of 0/1 points that cannot be
captured by an LP of size less than 2Ω(n) [ Rothvoss ’11]
Any LP that captures the TSP polytope must have size
1/2
2Ω(n ) [Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]
Any LP that ρ-approximates the Correlation polytope must
have size 2Ω(n/ρ) [Braun, Fiornini, Pokutta, Steurer ’12,
Braverman, Moitra ’13, Pokutta, Braun ’13]
5
Any LP of relaxation of size nr for the Correlation polytope
has integrality gap at least as large as O(r) levels of
Sherali-Adams [Chan, Lee, Raghavendra, Steurer ’13]
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
4
Introduction
Lower bounds on the size of Linear Programs:
1
2
3
4
Any symmetric LP that captures the TSP or Matching
polytope must have size 2Ω(n) [Yannakakis ’91]
There exists a convex hull of 0/1 points that cannot be
captured by an LP of size less than 2Ω(n) [ Rothvoss ’11]
Any LP that captures the TSP polytope must have size
1/2
2Ω(n ) [Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]
Any LP that ρ-approximates the Correlation polytope must
have size 2Ω(n/ρ) [Braun, Fiornini, Pokutta, Steurer ’12,
Braverman, Moitra ’13, Pokutta, Braun ’13]
5
Any LP of relaxation of size nr for the Correlation polytope
has integrality gap at least as large as O(r) levels of
Sherali-Adams [Chan, Lee, Raghavendra, Steurer ’13]
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
4
Introduction
Lower bounds on the size of Linear Programs:
1
2
3
4
Any symmetric LP that captures the TSP or Matching
polytope must have size 2Ω(n) [Yannakakis ’91]
There exists a convex hull of 0/1 points that cannot be
captured by an LP of size less than 2Ω(n) [ Rothvoss ’11]
Any LP that captures the TSP polytope must have size
1/2
2Ω(n ) [Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]
Any LP that ρ-approximates the Correlation polytope must
have size 2Ω(n/ρ) [Braun, Fiornini, Pokutta, Steurer ’12,
Braverman, Moitra ’13, Pokutta, Braun ’13]
5
Any LP of relaxation of size nr for the Correlation polytope
has integrality gap at least as large as O(r) levels of
Sherali-Adams [Chan, Lee, Raghavendra, Steurer ’13]
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
4
Introduction
Lower bounds on the size of Linear Programs:
1
2
3
4
Any symmetric LP that captures the TSP or Matching
polytope must have size 2Ω(n) [Yannakakis ’91]
There exists a convex hull of 0/1 points that cannot be
captured by an LP of size less than 2Ω(n) [ Rothvoss ’11]
Any LP that captures the TSP polytope must have size
1/2
2Ω(n ) [Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]
Any LP that ρ-approximates the Correlation polytope must
have size 2Ω(n/ρ) [Braun, Fiornini, Pokutta, Steurer ’12,
Braverman, Moitra ’13, Pokutta, Braun ’13]
5
Any LP of relaxation of size nr for the Correlation polytope
has integrality gap at least as large as O(r) levels of
Sherali-Adams [Chan, Lee, Raghavendra, Steurer ’13]
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
4
Introduction
What about Semidefinite Programs?
Theorem ([Briët, D., Pokutta ’13])
There exists a convex hull of 0/1 points that cannot be captured
by an SDP of size less than 2Ω(n) .
Extend framework of Rothvoss to SDP setting.
Baby step towards understanding lower bounds for SDPs.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
5
Introduction
What about Semidefinite Programs?
Theorem ([Briët, D., Pokutta ’13])
There exists a convex hull of 0/1 points that cannot be captured
by an SDP of size less than 2Ω(n) .
Extend framework of Rothvoss to SDP setting.
Baby step towards understanding lower bounds for SDPs.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
5
Introduction
What about Semidefinite Programs?
Theorem ([Briët, D., Pokutta ’13])
There exists a convex hull of 0/1 points that cannot be captured
by an SDP of size less than 2Ω(n) .
Extend framework of Rothvoss to SDP setting.
Baby step towards understanding lower bounds for SDPs.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
5
Introduction
What about Semidefinite Programs?
Theorem ([Briët, D., Pokutta ’13])
There exists a convex hull of 0/1 points that cannot be captured
by an SDP of size less than 2Ω(n) .
Extend framework of Rothvoss to SDP setting.
Baby step towards understanding lower bounds for SDPs.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
5
Linear Extensions
Polytope P = {x ∈ Rn : Ax ≤ b} with m facets.
Question
Can we reduce the number of inequalities needed to define P by
adding auxilliary variables?
Definition (Linear Extension)
Q = {(z, y) : Cz + Dy = d, y ≥ 0, y ∈ Rr , z ∈ Rl }, is a
linear extension of P of size r if ∃ π : Rl+r → Rn such that
P = π(Q).
Definition (Linear Extension Complexity)
xc(P ) := minimum size of any linear extension of P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
6
Linear Extensions
Polytope P = {x ∈ Rn : Ax ≤ b} with m facets.
Question
Can we reduce the number of inequalities needed to define P by
adding auxilliary variables?
Definition (Linear Extension)
Q = {(z, y) : Cz + Dy = d, y ≥ 0, y ∈ Rr , z ∈ Rl }, is a
linear extension of P of size r if ∃ π : Rl+r → Rn such that
P = π(Q).
Definition (Linear Extension Complexity)
xc(P ) := minimum size of any linear extension of P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
6
Linear Extensions
Polytope P = {x ∈ Rn : Ax ≤ b} with m facets.
Question
Can we reduce the number of inequalities needed to define P by
adding auxilliary variables?
Definition (Linear Extension)
Q = {(z, y) : Cz + Dy = d, y ≥ 0, y ∈ Rr , z ∈ Rl }, is a
linear extension of P of size r if ∃ π : Rl+r → Rn such that
P = π(Q).
Definition (Linear Extension Complexity)
xc(P ) := minimum size of any linear extension of P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
6
Linear Extensions
Polytope P = {x ∈ Rn : Ax ≤ b} with m facets.
Question
Can we reduce the number of inequalities needed to define P by
adding auxilliary variables?
Definition (Linear Extension)
Q = {(z, y) : Cz + Dy = d, y ≥ 0, y ∈ Rr , z ∈ Rl }, is a
linear extension of P of size r if ∃ π : Rl+r → Rn such that
P = π(Q).
Definition (Linear Extension Complexity)
xc(P ) := minimum size of any linear extension of P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
6
Linear Extensions
Extension Complexity.
Some known results (constructions & lower bounds):
• xc(regular n-gon) = Θ(log n)
√
• xc(generic n-gon) = Ω( n)
[Ben-Tal, Nemirovski’01]
[Fironi, Rothvoss, Tiwary’11]
• xc(n-permutahedron) = Θ(n log n)
• xc(spanning tree polytope of Kn ) = O(n3 )
[Goemans’09]
[Kipp-Martin’87]
• xc(spanning tree polytope of planar graph G) = Θ(n)
[Williams’01]
• xc(stable set polytope of perfect graph G) = nO(log n)
[Yannakakis’91]
• ...
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
7
Linear Extensions
Slack Matrices.
Let A ∈ Rm×n ,
P
b ∈ Rm ,
V = {v1 , . . . , vN } ⊆ Rn
s.t.
n
= {x ∈ R | Ax 6 b} = conv(V)
vj
Sij
Ai x = bi
Definition (slack matrix)
Slack matrix S ∈ Rm×N
of P (w.r.t. Ax 6 b and V):
+
Sij := bi − Ai vj , ∀ i ∈ [m], j ∈ [N ]
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
8
Linear Extensions
Slack Matrices.
Let A ∈ Rm×n ,
P
b ∈ Rm ,
V = {v1 , . . . , vN } ⊆ Rn
s.t.
n
= {x ∈ R | Ax 6 b} = conv(V)
vj
Sij
Ai x = bi
Definition (slack matrix)
Slack matrix S ∈ Rm×N
of P (w.r.t. Ax 6 b and V):
+
Sij := bi − Ai vj , ∀ i ∈ [m], j ∈ [N ]
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
8
Linear Extensions
Nonnegative Factorizations and Extensions.
Definition (slack matrix)
Slack matrix S ∈ Rm×N
of P (w.r.t. Ax 6 b and V):
+
Sij := bi − Ai vj , ∀ i ∈ [m], j ∈ [N ]
Definition
A rank-r nonnegative factorization of S ∈ Rm×n
is
+
S = UV
where
U ∈ Rm×r
+
and V ∈ Rr×n
+
Proposition (Extensions from Factorizations)
Q = {(x, y) : Ax + U y = b, y ≥ 0}
is a linear extension of P of size r.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
9
Linear Extensions
Nonnegative Factorizations and Extensions.
Definition (slack matrix)
Slack matrix S ∈ Rm×N
of P (w.r.t. Ax 6 b and V):
+
Sij := bi − Ai vj , ∀ i ∈ [m], j ∈ [N ]
Definition
A rank-r nonnegative factorization of S ∈ Rm×n
is
+
S = UV
where
U ∈ Rm×r
+
and V ∈ Rr×n
+
Proposition (Extensions from Factorizations)
Q = {(x, y) : Ax + U y = b, y ≥ 0}
is a linear extension of P of size r.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
9
Linear Extensions
Nonnegative Factorizations and Extensions.
Definition (Nonnegative Rank)
rk+ (S) := min{r | ∃ rank-r nonnegative factorization of S}
Theorem (Factorization Theorem [Yannakakis’91])
For every slack matrix S of P :
xc(P ) = rk+ (S)
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
10
PSD Extensions
PSD Matrices.
Definition (PSD matrix)
A matrix U ∈ Rr×r is PSD if U is symmetric and
xT U x ≥ 0 ∀x ∈ Rr .
Let Sr+ denote the set of r × r PSD matrices.
Definition (Spectral Decomposition)
U is r × r PSD iff U admits a Spectral Decomposition
U=
r
X
λi ui uT
i ,
i=1
λ1 , . . . , λr ≥ 0, u1 , . . . , ur an orthonormal basis.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
11
PSD Extensions
PSD Matrices.
Definition (PSD matrix)
A matrix U ∈ Rr×r is PSD if U is symmetric and
xT U x ≥ 0 ∀x ∈ Rr .
Let Sr+ denote the set of r × r PSD matrices.
Definition (Spectral Decomposition)
U is r × r PSD iff U admits a Spectral Decomposition
U=
r
X
λi ui uT
i ,
i=1
λ1 , . . . , λr ≥ 0, u1 , . . . , ur an orthonormal basis.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
11
PSD Extensions
PSD Matrices.
Definition (Operator norm)
For a matrix T ∈ Rr×r the operator norm of T is
kT kop = max kT xk2
kxk2 =1
For a PSD matrix U ∈ Rr×r
kU kop = max xT U x = largest eigenvalue of U .
kxk2 =1
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
12
PSD Extensions
PSD Matrices.
Definition (Trace)
For a matrix T ∈ Rr×r , we define Tr [T ] =
Pr
i=1 Tii .
Remark (Trace Inner Product)
For A, B ∈ Rr×r symmetric, Tr [AB] =
P
i,j∈[r] Aij Bij .
Proposition
For PSD matrices U, V ∈ Sr+ ,
Tr [U V ] =
X
λi γj hui , vj i2 ≥ 0 ,
i,j∈[r]
P
Pr
T
where U = ri=1 λi ui uT
i and V =
j=1 γj vj vj are the respective
spectral decompositions.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
13
PSD Extensions
PSD Matrices.
Definition (Trace)
For a matrix T ∈ Rr×r , we define Tr [T ] =
Pr
i=1 Tii .
Remark (Trace Inner Product)
For A, B ∈ Rr×r symmetric, Tr [AB] =
P
i,j∈[r] Aij Bij .
Proposition
For PSD matrices U, V ∈ Sr+ ,
Tr [U V ] =
X
λi γj hui , vj i2 ≥ 0 ,
i,j∈[r]
P
Pr
T
where U = ri=1 λi ui uT
i and V =
j=1 γj vj vj are the respective
spectral decompositions.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
13
PSD Extensions
PSD Matrices.
Definition (Trace)
For a matrix T ∈ Rr×r , we define Tr [T ] =
Pr
i=1 Tii .
Remark (Trace Inner Product)
For A, B ∈ Rr×r symmetric, Tr [AB] =
P
i,j∈[r] Aij Bij .
Proposition
For PSD matrices U, V ∈ Sr+ ,
Tr [U V ] =
X
λi γj hui , vj i2 ≥ 0 ,
i,j∈[r]
P
Pr
T
where U = ri=1 λi ui uT
i and V =
j=1 γj vj vj are the respective
spectral decompositions.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
13
PSD Extensions
Polytope P = {x ∈ Rn : Ax ≤ b} with m facets.
Definition (PSD Extension)
Q = {(z, Y ) : Ci z + Tr [Di Y ] = di , ∀ i ∈ [l], Y ∈ Sr+ , z ∈ Rl }, is a
PSD extension of P of size r if ∃ π : Rl × Sr+ → Rn such that
P = π(Q).
Definition (PSD Extension Complexity)
xcpsd (P ) := minimum size of any PSD extension of P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
14
PSD Extensions
Polytope P = {x ∈ Rn : Ax ≤ b} with m facets.
Definition (PSD Extension)
Q = {(z, Y ) : Ci z + Tr [Di Y ] = di , ∀ i ∈ [l], Y ∈ Sr+ , z ∈ Rl }, is a
PSD extension of P of size r if ∃ π : Rl × Sr+ → Rn such that
P = π(Q).
Definition (PSD Extension Complexity)
xcpsd (P ) := minimum size of any PSD extension of P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
14
PSD Extensions
Polytope P = {x ∈ Rn : Ax ≤ b} with m facets.
Definition (PSD Extension)
Q = {(z, Y ) : Ci z + Tr [Di Y ] = di , ∀ i ∈ [l], Y ∈ Sr+ , z ∈ Rl }, is a
PSD extension of P of size r if ∃ π : Rl × Sr+ → Rn such that
P = π(Q).
Definition (PSD Extension Complexity)
xcpsd (P ) := minimum size of any PSD extension of P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
14
PSD Extensions
PSD Factorizations and Extensions
Definition (Slack Matrix)
Slack matrix S ∈ Rm×N
of P (w.r.t. Ax 6 b and vertices V):
+
Sij := bi − Ai vj , ∀ i ∈ [m], j ∈ [N ]
Definition
A rank-r PSD factorization of S ∈ Rm×N
is
+
Sij = Tr [Ui Vj ]
where
Ui , Vj are r × r PSD ∀ i ∈ [m], j ∈ [N ].
Proposition (Extensions from Factorizations)
Q = {(x, Y ) : Ai x + Tr [Ui Y ] = bi , ∀i ∈ [m], Y ∈ Sr+ }
is a PSD extension of P of size r.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
15
PSD Extensions
PSD Factorizations and Extensions
Definition (Slack Matrix)
Slack matrix S ∈ Rm×N
of P (w.r.t. Ax 6 b and vertices V):
+
Sij := bi − Ai vj , ∀ i ∈ [m], j ∈ [N ]
Definition
A rank-r PSD factorization of S ∈ Rm×N
is
+
Sij = Tr [Ui Vj ]
where
Ui , Vj are r × r PSD ∀ i ∈ [m], j ∈ [N ].
Proposition (Extensions from Factorizations)
Q = {(x, Y ) : Ai x + Tr [Ui Y ] = bi , ∀i ∈ [m], Y ∈ Sr+ }
is a PSD extension of P of size r.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
15
PSD Extensions
PSD Factorizations and Extensions.
Definition (PSD Rank)
rkpsd (S) := min{r | ∃ rank-r PSD factorization of S}
Theorem (Factorization Theorem [Gouviea, Thomas, Parillo ’11,
Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12])
For every slack matrix S of P :
xcpsd (P ) = rkpsd (S)
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
16
Counting Argument
Rothvoss’s Counting Argument
Goal:
Show existence of X ⊆ {0, 1}n with xc(conv(X)) = 2Ω(n) .
Let R = maxX⊆{0,1}n xc(conv(X)).
High level:
1
“Discretize” an optimal a linear EF for conv(X) to compress
the description of X.
2
Show that the number of discretized linear EFs of size R is
bounded by 2poly(R,n) .
n
Conclusion: 22 subsets of {0, 1}n means that R ≥ 2Ω(n) .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
17
Counting Argument
Rothvoss’s Counting Argument
Goal:
Show existence of X ⊆ {0, 1}n with xc(conv(X)) = 2Ω(n) .
Let R = maxX⊆{0,1}n xc(conv(X)).
High level:
1
“Discretize” an optimal a linear EF for conv(X) to compress
the description of X.
2
Show that the number of discretized linear EFs of size R is
bounded by 2poly(R,n) .
n
Conclusion: 22 subsets of {0, 1}n means that R ≥ 2Ω(n) .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
17
Counting Argument
Rothvoss’s Counting Argument
Goal:
Show existence of X ⊆ {0, 1}n with xc(conv(X)) = 2Ω(n) .
Let R = maxX⊆{0,1}n xc(conv(X)).
High level:
1
“Discretize” an optimal a linear EF for conv(X) to compress
the description of X.
2
Show that the number of discretized linear EFs of size R is
bounded by 2poly(R,n) .
n
Conclusion: 22 subsets of {0, 1}n means that R ≥ 2Ω(n) .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
17
Counting Argument
Rothvoss’s Counting Argument
Goal:
Show existence of X ⊆ {0, 1}n with xc(conv(X)) = 2Ω(n) .
Let R = maxX⊆{0,1}n xc(conv(X)).
High level:
1
“Discretize” an optimal a linear EF for conv(X) to compress
the description of X.
2
Show that the number of discretized linear EFs of size R is
bounded by 2poly(R,n) .
n
Conclusion: 22 subsets of {0, 1}n means that R ≥ 2Ω(n) .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
17
Counting Argument
Rothvoss’s Counting Argument
Goal:
Show existence of X ⊆ {0, 1}n with xc(conv(X)) = 2Ω(n) .
Let R = maxX⊆{0,1}n xc(conv(X)).
High level:
1
“Discretize” an optimal a linear EF for conv(X) to compress
the description of X.
2
Show that the number of discretized linear EFs of size R is
bounded by 2poly(R,n) .
n
Conclusion: 22 subsets of {0, 1}n means that R ≥ 2Ω(n) .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
17
Counting Argument
Rothvoss’s Counting Argument
Goal:
Show existence of X ⊆ {0, 1}n with xcpsd (conv(X)) = 2Ω(n) .
Let R = maxX⊆{0,1}n xcpsd (conv(X)).
High level:
1
“Discretize” an optimal a PSD EF for conv(X) to compress
the description of X.
2
Show that the number of discretized PSD EFs of size R is
bounded by 2poly(R,n) .
n
Conclusion: 22 subsets of {0, 1}n means that R ≥ 2Ω(n) .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
18
Discretizing EFs
Discretizing Linear EFs
X = {v1 , . . . , vN } ⊆ {0, 1}n . conv(X) = {x ∈ Rn : Ax ≤ b}.
By Hadamard can assume A ∈ Zm×n , b ∈ Zm
and max |Aij |, max |bi | ≤ ∆ ≈ nn/2 .
Let Sij = bi − Ai vj , ∀i ∈ [m], j ∈ [N ], be the slack matrix.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
19
Discretizing EFs
Discretizing Linear EFs
X = {v1 , . . . , vN } ⊆ {0, 1}n . conv(X) = {x ∈ Rn : Ax ≤ b}.
By Hadamard can assume A ∈ Zm×n , b ∈ Zm
and max |Aij |, max |bi | ≤ ∆ ≈ nn/2 .
Let Sij = bi − Ai vj , ∀i ∈ [m], j ∈ [N ], be the slack matrix.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
19
Discretizing EFs
Discretizing Linear EFs
X = {v1 , . . . , vN } ⊆ {0, 1}n . conv(X) = {x ∈ Rn : Ax ≤ b}.
By Hadamard can assume A ∈ Zm×n , b ∈ Zm
and max |Aij |, max |bi | ≤ ∆ ≈ nn/2 .
Let Sij = bi − Ai vj , ∀i ∈ [m], j ∈ [N ], be the slack matrix.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
19
Discretizing EFs
Discretizing Linear EFs
Let S = U V be rank r nonnegative factorization.
Examine linear EF Q = {(x, y) : Ax + U y = b, y ≥ 0}.
Let (AS , US ), S ⊆ [m], be any maximal set of linearly independent
set of rows of (A, U ).
Note that Q = {(x, y) : AS x + US y = bS , y ≥ 0}.
Can represent conv(X) using only at most (n + r) × (n + r + 1)
numbers!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
20
Discretizing EFs
Discretizing Linear EFs
Let S = U V be rank r nonnegative factorization.
Examine linear EF Q = {(x, y) : Ax + U y = b, y ≥ 0}.
Let (AS , US ), S ⊆ [m], be any maximal set of linearly independent
set of rows of (A, U ).
Note that Q = {(x, y) : AS x + US y = bS , y ≥ 0}.
Can represent conv(X) using only at most (n + r) × (n + r + 1)
numbers!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
20
Discretizing EFs
Discretizing Linear EFs
Let S = U V be rank r nonnegative factorization.
Examine linear EF Q = {(x, y) : Ax + U y = b, y ≥ 0}.
Let (AS , US ), S ⊆ [m], be any maximal set of linearly independent
set of rows of (A, U ).
Note that Q = {(x, y) : AS x + US y = bS , y ≥ 0}.
Can represent conv(X) using only at most (n + r) × (n + r + 1)
numbers!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
20
Discretizing EFs
Discretizing Linear EFs
Let S = U V be rank r nonnegative factorization.
Examine linear EF Q = {(x, y) : Ax + U y = b, y ≥ 0}.
Let (AS , US ), S ⊆ [m], be any maximal set of linearly independent
set of rows of (A, U ).
Note that Q = {(x, y) : AS x + US y = bS , y ≥ 0}.
Can represent conv(X) using only at most (n + r) × (n + r + 1)
numbers!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
20
Discretizing EFs
Discretizing Linear EFs
Let S = U V be rank r nonnegative factorization.
Examine linear EF Q = {(x, y) : Ax + U y = b, y ≥ 0}.
Let (AS , US ), S ⊆ [m], be any maximal set of linearly independent
set of rows of (A, U ).
Note that Q = {(x, y) : AS x + US y = bS , y ≥ 0}.
Can represent conv(X) using only at most (n + r) × (n + r + 1)
numbers!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
20
Discretizing EFs
Discretizing Linear EFs
Q = {(x, y) : AS x + US y = bS , y ≥ 0}.
Can represent conv(X) using only at most (n + r) × (n + r + 1)
numbers!
Issues with counting:
1
Numbers in US may be unbounded.
2
Numbers in US don’t fall in a discrete set.
Fix idea:
1
Round numbers in U to a grid.
2
Build “rounded” EF Q̄ that contains SAME 0/1 points as Q.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
21
Discretizing EFs
Discretizing Linear EFs
Q = {(x, y) : AS x + US y = bS , y ≥ 0}.
Can represent conv(X) using only at most (n + r) × (n + r + 1)
numbers!
Issues with counting:
1
Numbers in US may be unbounded.
2
Numbers in US don’t fall in a discrete set.
Fix idea:
1
Round numbers in U to a grid.
2
Build “rounded” EF Q̄ that contains SAME 0/1 points as Q.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
21
Discretizing EFs
Discretizing Linear EFs
Rounded Linear EF:
Q̄ = {(x, y) : kAS x + ŪS y − bS k∞ ≤ 1/poly(n, r),
√
y ≥ 0, kyk∞ ≤ ∆}.
High Level Fix:
1
Rescale factorization S = U V so that max entry U, V ≤
2
Round U → Ū to nearest multiple of 1/poly(n, r, ∆).
3
Choose rows S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
√
∆.
Can represent X using only at most (n + r) × (n + r + 1) numbers
in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
22
Discretizing EFs
Discretizing Linear EFs
Rounded Linear EF:
Q̄ = {(x, y) : kAS x + ŪS y − bS k∞ ≤ 1/poly(n, r),
√
y ≥ 0, kyk∞ ≤ ∆}.
High Level Fix:
1
Rescale factorization S = U V so that max entry U, V ≤
2
Round U → Ū to nearest multiple of 1/poly(n, r, ∆).
3
Choose rows S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
√
∆.
Can represent X using only at most (n + r) × (n + r + 1) numbers
in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
22
Discretizing EFs
Discretizing Linear EFs
Rounded Linear EF:
Q̄ = {(x, y) : kAS x + ŪS y − bS k∞ ≤ 1/poly(n, r),
√
y ≥ 0, kyk∞ ≤ ∆}.
High Level Fix:
1
Rescale factorization S = U V so that max entry U, V ≤
2
Round U → Ū to nearest multiple of 1/poly(n, r, ∆).
3
Choose rows S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
√
∆.
Can represent X using only at most (n + r) × (n + r + 1) numbers
in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
22
Discretizing EFs
Discretizing Linear EFs
Rounded Linear EF:
Q̄ = {(x, y) : kAS x + ŪS y − bS k∞ ≤ 1/poly(n, r),
√
y ≥ 0, kyk∞ ≤ ∆}.
High Level Fix:
1
Rescale factorization S = U V so that max entry U, V ≤
2
Round U → Ū to nearest multiple of 1/poly(n, r, ∆).
3
Choose rows S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
√
∆.
Can represent X using only at most (n + r) × (n + r + 1) numbers
in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
22
Discretizing EFs
Discretizing Linear EFs
Rounded Linear EF:
Q̄ = {(x, y) : kAS x + ŪS y − bS k∞ ≤ 1/poly(n, r),
√
y ≥ 0, kyk∞ ≤ ∆}.
High Level Fix:
1
Rescale factorization S = U V so that max entry U, V ≤
2
Round U → Ū to nearest multiple of 1/poly(n, r, ∆).
3
Choose rows S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
√
∆.
Can represent X using only at most (n + r) × (n + r + 1) numbers
in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
22
Discretizing EFs
Discretizing PSD EFs
Question: What changes for PSD factorizations?
Rounded PSD EF:
Q̄ = {(x, y) : kAi x + Tr Ūi Y − bi k∞ ≤ 1/poly(n, r), ∀i ∈ S,
kY kop ≤???, Y ∈ Sr+ }.
Solution steps:
1
Rescale factorization Sij = Tr [Ui Vj ] so that
maxi kUi kop , maxj kVj kop ≤???.
2
Round each Ui ⇒ Ūi to nearest multiple of 1/poly(n, r, ???).
3
Choose rows of S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
Can represent X using only at most (n + r2 ) × (n + r2 + 1)
numbers in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
23
Discretizing EFs
Discretizing PSD EFs
Question: What changes for PSD factorizations?
Rounded PSD EF:
Q̄ = {(x, y) : kAi x + Tr Ūi Y − bi k∞ ≤ 1/poly(n, r), ∀i ∈ S,
kY kop ≤???, Y ∈ Sr+ }.
Solution steps:
1
Rescale factorization Sij = Tr [Ui Vj ] so that
maxi kUi kop , maxj kVj kop ≤???.
2
Round each Ui ⇒ Ūi to nearest multiple of 1/poly(n, r, ???).
3
Choose rows of S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
Can represent X using only at most (n + r2 ) × (n + r2 + 1)
numbers in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
23
Discretizing EFs
Discretizing PSD EFs
Question: What changes for PSD factorizations?
Rounded PSD EF:
Q̄ = {(x, y) : kAi x + Tr Ūi Y − bi k∞ ≤ 1/poly(n, r), ∀i ∈ S,
kY kop ≤???, Y ∈ Sr+ }.
Solution steps:
1
Rescale factorization Sij = Tr [Ui Vj ] so that
maxi kUi kop , maxj kVj kop ≤???.
2
Round each Ui ⇒ Ūi to nearest multiple of 1/poly(n, r, ???).
3
Choose rows of S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
Can represent X using only at most (n + r2 ) × (n + r2 + 1)
numbers in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
23
Discretizing EFs
Discretizing PSD EFs
Question: What changes for PSD factorizations?
Rounded PSD EF:
Q̄ = {(x, y) : kAi x + Tr Ūi Y − bi k∞ ≤ 1/poly(n, r), ∀i ∈ S,
kY kop ≤???, Y ∈ Sr+ }.
Solution steps:
1
Rescale factorization Sij = Tr [Ui Vj ] so that
maxi kUi kop , maxj kVj kop ≤???.
2
Round each Ui ⇒ Ūi to nearest multiple of 1/poly(n, r, ???).
3
Choose rows of S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
Can represent X using only at most (n + r2 ) × (n + r2 + 1)
numbers in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
23
Discretizing EFs
Discretizing PSD EFs
Question: What changes for PSD factorizations?
Rounded PSD EF:
Q̄ = {(x, y) : kAi x + Tr Ūi Y − bi k∞ ≤ 1/poly(n, r), ∀i ∈ S,
kY kop ≤???, Y ∈ Sr+ }.
Solution steps:
1
Rescale factorization Sij = Tr [Ui Vj ] so that
maxi kUi kop , maxj kVj kop ≤???.
2
Round each Ui ⇒ Ūi to nearest multiple of 1/poly(n, r, ???).
3
Choose rows of S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
Can represent X using only at most (n + r2 ) × (n + r2 + 1)
numbers in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
23
Discretizing EFs
Discretizing PSD EFs
Question: What changes for PSD factorizations?
Rounded PSD EF:
Q̄ = {(x, y) : kAi x + Tr Ūi Y − bi k∞ ≤ 1/poly(n, r), ∀i ∈ S,
kY kop ≤???, Y ∈ Sr+ }.
Solution steps:
1
Rescale factorization Sij = Tr [Ui Vj ] so that
maxi kUi kop , maxj kVj kop ≤???.
2
Round each Ui ⇒ Ūi to nearest multiple of 1/poly(n, r, ???).
3
Choose rows of S “carefully”.
4
Show that πx (Q̄) ∩ {0, 1}n = X.
Can represent X using only at most (n + r2 ) × (n + r2 + 1)
numbers in a bounded set!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
23
Rescaling Factorizations
Rescaling PSD Factorizations
Theorem ([Briët, D., Pokutta])
Let S ∈ Rm×N
with rkpsd (S) = r and kSk∞ = ∆.
+
Then ∃ rank r PSD factorization Sij = Tr [Ui Vj ], ∀ i, j such that
√
max kUi kop , max kVj kop ≤ r∆
i
Daniel Dadush
j
On the existence of 0/1 polytopes with high SDP rank
24
Rescaling Factorizations
Rescaling Nonnegative Factorizations
Lemma
Let S ∈ Rm×N
with rk+ (S) = r and kSk∞ = ∆.
+
Then ∃ rank r nonnegative factorization S = U V such that
√
kU k∞ , kV k∞ ≤ ∆
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
25
Rescaling Factorizations
Rescaling Nonnegative Factorizations

v1T
 
. . . ur , V =  ... 
vrT

Proof: U = u1
S=
Pr
T
i=1 ui vi .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
26
Rescaling Factorizations
Rescaling Nonnegative Factorizations

v1T
 
. . . ur , V =  ... 
vrT

Proof: U = u1
S=
Pr
T
i=1 ui vi .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
26
Rescaling Factorizations
Rescaling Nonnegative Factorizations
Choose λ1 , . . . , λr > 0 such that kλi ui k∞ = k1/λi vi k∞ .


1/λ1 v1T
Let U 0 = λ1 u1 . . . λr ur , V 0 =  . . . .
1/λr vrT
Note that U 0 V 0 =
Daniel Dadush
Pr
T
i=1 (λi ui )(1/λi vi )
=
Pr
T
i=1 ui vi
On the existence of 0/1 polytopes with high SDP rank
= S.
27
Rescaling Factorizations
Rescaling Nonnegative Factorizations
Choose λ1 , . . . , λr > 0 such that kλi ui k∞ = k1/λi vi k∞ .


1/λ1 v1T
Let U 0 = λ1 u1 . . . λr ur , V 0 =  . . . .
1/λr vrT
Note that U 0 V 0 =
Daniel Dadush
Pr
T
i=1 (λi ui )(1/λi vi )
=
Pr
T
i=1 ui vi
On the existence of 0/1 polytopes with high SDP rank
= S.
27
Rescaling Factorizations
Rescaling Nonnegative Factorizations
Choose λ1 , . . . , λr > 0 such that kλi ui k∞ = k1/λi vi k∞ .


1/λ1 v1T
Let U 0 = λ1 u1 . . . λr ur , V 0 =  . . . .
1/λr vrT
Note that U 0 V 0 =
Daniel Dadush
Pr
T
i=1 (λi ui )(1/λi vi )
=
Pr
T
i=1 ui vi
On the existence of 0/1 polytopes with high SDP rank
= S.
27
Rescaling Factorizations
Rescaling Nonnegative Factorizations
def
M = kU 0 k∞ = maxi kλi ui k∞ = maxi k1/λvi k∞ = kV 0 k∞ .
Pick j ∈ [r] such that M = kλj uj k∞ = k1/λj vj k∞ .
∆ = kSk∞ = kU 0 V 0 k∞ ≥ k(λj uj )(1/λj vjT )k∞
= kλj uj k∞ k1/λj vj k∞ = M 2 .
Therefore kU 0 k∞ = kV 0 k∞ = M ≤
Daniel Dadush
√
∆ as needed. On the existence of 0/1 polytopes with high SDP rank
28
Rescaling Factorizations
Rescaling Nonnegative Factorizations
def
M = kU 0 k∞ = maxi kλi ui k∞ = maxi k1/λvi k∞ = kV 0 k∞ .
Pick j ∈ [r] such that M = kλj uj k∞ = k1/λj vj k∞ .
∆ = kSk∞ = kU 0 V 0 k∞ ≥ k(λj uj )(1/λj vjT )k∞
= kλj uj k∞ k1/λj vj k∞ = M 2 .
Therefore kU 0 k∞ = kV 0 k∞ = M ≤
Daniel Dadush
√
∆ as needed. On the existence of 0/1 polytopes with high SDP rank
28
Rescaling Factorizations
Rescaling Nonnegative Factorizations
def
M = kU 0 k∞ = maxi kλi ui k∞ = maxi k1/λvi k∞ = kV 0 k∞ .
Pick j ∈ [r] such that M = kλj uj k∞ = k1/λj vj k∞ .
∆ = kSk∞ = kU 0 V 0 k∞ ≥ k(λj uj )(1/λj vjT )k∞
= kλj uj k∞ k1/λj vj k∞ = M 2 .
Therefore kU 0 k∞ = kV 0 k∞ = M ≤
Daniel Dadush
√
∆ as needed. On the existence of 0/1 polytopes with high SDP rank
28
Rescaling Factorizations
Rescaling Nonnegative Factorizations
def
M = kU 0 k∞ = maxi kλi ui k∞ = maxi k1/λvi k∞ = kV 0 k∞ .
Pick j ∈ [r] such that M = kλj uj k∞ = k1/λj vj k∞ .
∆ = kSk∞ = kU 0 V 0 k∞ ≥ k(λj uj )(1/λj vjT )k∞
= kλj uj k∞ k1/λj vj k∞ = M 2 .
Therefore kU 0 k∞ = kV 0 k∞ = M ≤
Daniel Dadush
√
∆ as needed. On the existence of 0/1 polytopes with high SDP rank
28
Rescaling Factorizations
Rescaling PSD Factorizations
Theorem ([Briët, D., Pokutta])
Let S ∈ Rm×N
with rkpsd (S) = r and kSk∞ = ∆.
+
Then ∃ rank r PSD factorization Sij = Tr [Ui Vj ], ∀ i, j such that
√
max kUi kop , max kVj kop ≤ r∆
i
j
Comparison with Nonnegative Factorizations:
• Nonnegative setting: can rescale entries of non-negative
vector independently and maintain non-negativity.
• PSD setting: entries can be NEGATIVE.
CANNOT rescale entries of PSD matrix independently while
maintaining PSD property.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
29
Rescaling Factorizations
Rescaling PSD Factorizations
Theorem ([Briët, D., Pokutta])
Let S ∈ Rm×N
with rkpsd (S) = r and kSk∞ = ∆.
+
Then ∃ rank r PSD factorization Sij = Tr [Ui Vj ], ∀ i, j such that
√
max kUi kop , max kVj kop ≤ r∆
i
j
Comparison with Nonnegative Factorizations:
• Nonnegative setting: can rescale entries of non-negative
vector independently and maintain non-negativity.
• PSD setting: entries can be NEGATIVE.
CANNOT rescale entries of PSD matrix independently while
maintaining PSD property.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
29
Rescaling Factorizations
Rescaling PSD Factorizations
Theorem ([Briët, D., Pokutta])
Let S ∈ Rm×N
with rkpsd (S) = r and kSk∞ = ∆.
+
Then ∃ rank r PSD factorization Sij = Tr [Ui Vj ], ∀ i, j such that
√
max kUi kop , max kVj kop ≤ r∆
i
j
Comparison with Nonnegative Factorizations:
• Nonnegative setting: can rescale entries of non-negative
vector independently and maintain non-negativity.
• PSD setting: entries can be NEGATIVE.
CANNOT rescale entries of PSD matrix independently while
maintaining PSD property.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
29
Rescaling Factorizations
Rescaling PSD Factorizations
Theorem ([Briët, D., Pokutta])
Let S ∈ Rm×N
with rkpsd (S) = r and kSk∞ = ∆.
+
Then ∃ rank r PSD factorization Sij = Tr [Ui Vj ], ∀ i, j such that
√
max kUi kop , max kVj kop ≤ r∆
i
j
Admissible PSD Rescalings: For A ∈ Rr×r invertible, send
Ui → AT Ui A and Vj → A−1 Vi A−T .
Preserves
inner product:
Tr AT Ui AA−1 Vj A−T = Tr Ui Vj A−T AT = Tr [Ui Vj ] = Sij .
Map U → AT U A is a symmetry of PSD cone.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
30
Rescaling Factorizations
Rescaling PSD Factorizations
Theorem ([Briët, D., Pokutta])
Let S ∈ Rm×N
with rkpsd (S) = r and kSk∞ = ∆.
+
Then ∃ rank r PSD factorization Sij = Tr [Ui Vj ], ∀ i, j such that
√
max kUi kop , max kVj kop ≤ r∆
i
j
Admissible PSD Rescalings: For A ∈ Rr×r invertible, send
Ui → AT Ui A and Vj → A−1 Vi A−T .
Preserves
inner product:
Tr AT Ui AA−1 Vj A−T = Tr Ui Vj A−T AT = Tr [Ui Vj ] = Sij .
Map U → AT U A is a symmetry of PSD cone.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
30
Rescaling Factorizations
Rescaling PSD Factorizations
Theorem ([Briët, D., Pokutta])
Let S ∈ Rm×N
with rkpsd (S) = r and kSk∞ = ∆.
+
Then ∃ rank r PSD factorization Sij = Tr [Ui Vj ], ∀ i, j such that
√
max kUi kop , max kVj kop ≤ r∆
i
j
Admissible PSD Rescalings: For A ∈ Rr×r invertible, send
Ui → AT Ui A and Vj → A−1 Vi A−T .
Preserves
inner product:
Tr AT Ui AA−1 Vj A−T = Tr Ui Vj A−T AT = Tr [Ui Vj ] = Sij .
Map U → AT U A is a symmetry of PSD cone.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
30
Rescaling Factorizations
Rescaling PSD Factorizations
Proof Idea: Variational Argument
1
Choose rescaling A ∈ Sr+ such that potential
maxi kAUi Akop × maxj kA−1 Vj A−1 kop is minimized.
2
Show that if potential > r∆, can find infinitessimal
pertubation A → (I + P )A(I + P ) which decreases
potential.
Require convex geometric tools
(John’s decomposition of the identity) to build perturbation P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
31
Rescaling Factorizations
Rescaling PSD Factorizations
Proof Idea: Variational Argument
1
Choose rescaling A ∈ Sr+ such that potential
maxi kAUi Akop × maxj kA−1 Vj A−1 kop is minimized.
2
Show that if potential > r∆, can find infinitessimal
pertubation A → (I + P )A(I + P ) which decreases
potential.
Require convex geometric tools
(John’s decomposition of the identity) to build perturbation P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
31
Rescaling Factorizations
Rescaling PSD Factorizations
Proof Idea: Variational Argument
1
Choose rescaling A ∈ Sr+ such that potential
maxi kAUi Akop × maxj kA−1 Vj A−1 kop is minimized.
2
Show that if potential > r∆, can find infinitessimal
pertubation A → (I + P )A(I + P ) which decreases
potential.
Require convex geometric tools
(John’s decomposition of the identity) to build perturbation P .
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
31
Open problems and future work.
1
Is the rescaling theorem tight? Do we need dependance on r?
2
Show that any nr PSD extended formulation for Correlation
polytope has integrality gap as large as O(r) levels of Lasserre.
3
Extension complexity of the matching polytope.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
32
Open problems and future work.
1
Is the rescaling theorem tight? Do we need dependance on r?
2
Show that any nr PSD extended formulation for Correlation
polytope has integrality gap as large as O(r) levels of Lasserre.
3
Extension complexity of the matching polytope.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
32
Open problems and future work.
1
Is the rescaling theorem tight? Do we need dependance on r?
2
Show that any nr PSD extended formulation for Correlation
polytope has integrality gap as large as O(r) levels of Lasserre.
3
Extension complexity of the matching polytope.
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
32
Thank you!
Daniel Dadush
On the existence of 0/1 polytopes with high SDP rank
33