The document discusses linear programs (LPs) and extended formulations for solving problems like the traveling salesman problem (TSP). It presents the following key points: 1) A previous claim that the TSP could be solved in polynomial time via an LP was disproven, showing LPs require super-polynomial size. 2) The document shows that every extended formulation (EF) of the TSP polytope requires size 2Ω(n1/4), via reductions from EFs of quantum stabilizer problems and cut polytopes. 3) This establishes a new connection between semidefinite programs and quantum information, and generalizes prior work linking classical communication complexity to linear EFs. The

1. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Linear vs. Semidefinite Extended Formulations:
Exponential Separation and Strong Lower Bounds
Samuel Fiorini Serge Massar Sebastian Pokutta
ULB/Math ULB/Phys Erlangen U.
Hans Raj Tiwary Ronald de Wolf
ULB/Math CWI
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 1

2. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
History
In 86-87, Swart claimed he could prove P = NP
How?
By giving a poly-size linear program (LP) for the TSP
Theorem (Yannakakis’88/91)
Every symmetric LP for the TSP has size 2Ω(n)
Swart’s LP was symmetric and of size poly(n) =⇒ it was wrong
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 2

3. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
History
In 86-87, Swart claimed he could prove P = NP
How?
By giving a poly-size linear program (LP) for the TSP
Theorem (Yannakakis’88/91)
Every symmetric LP for the TSP has size 2Ω(n)
Swart’s LP was symmetric and of size poly(n) =⇒ it was wrong
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 2

4. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
History
In 86-87, Swart claimed he could prove P = NP
How?
By giving a poly-size linear program (LP) for the TSP
Theorem (Yannakakis’88/91)
Every symmetric LP for the TSP has size 2Ω(n)
Swart’s LP was symmetric and of size poly(n) =⇒ it was wrong
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 2

5. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
History
In 86-87, Swart claimed he could prove P = NP
How?
By giving a poly-size linear program (LP) for the TSP
Theorem (Yannakakis’88/91)
Every symmetric LP for the TSP has size 2Ω(n)
Swart’s LP was symmetric and of size poly(n) =⇒ it was wrong
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 2

6. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
History
In 86-87, Swart claimed he could prove P = NP
How?
By giving a poly-size linear program (LP) for the TSP
Theorem (Yannakakis’88/91)
Every symmetric LP for the TSP has size 2Ω(n)
Swart’s LP was symmetric and of size poly(n) =⇒ it was wrong
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 2

7. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
History
From Yannakakis’91:
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 3

8. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
History
From Yannakakis’11:
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 3

9. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Recent previous results
Definition
P, Q polytopes.
Q is an EF of P if ∃ linear π with π(Q) = P.
Size of Q := #facets of Q.
Q
π
P
• Kaibel, Pashkovich & Theis’10: some polytopes have no
poly-size symmetric EF but poly-size non-symmetric EFs.
• Rothvoß’11: there are 0/1-polytopes in Rd such that
every EF has size 2(1/2−o(1))d .
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 4

10. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Recent previous results
Definition
P, Q polytopes.
Q is an EF of P if ∃ linear π with π(Q) = P.
Size of Q := #facets of Q.
Q
π
P
• Kaibel, Pashkovich & Theis’10: some polytopes have no
poly-size symmetric EF but poly-size non-symmetric EFs.
• Rothvoß’11: there are 0/1-polytopes in Rd such that
every EF has size 2(1/2−o(1))d .
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 4

11. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Recent previous results
Definition
P, Q polytopes.
Q is an EF of P if ∃ linear π with π(Q) = P.
Size of Q := #facets of Q.
Q
π
P
• Kaibel, Pashkovich & Theis’10: some polytopes have no
poly-size symmetric EF but poly-size non-symmetric EFs.
• Rothvoß’11: there are 0/1-polytopes in Rd such that
every EF has size 2(1/2−o(1))d .
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 4

12. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Our results
We show that
Every LP for the TSP has super-polynomial size.
via the following sequence of reductions:
1/4
• every EF of the TSP polytope has size 2Ω(n )
⇑
1/2
• ∃ (Gn )n s.t. every EF of STAB(Gn ) has size 2Ω(n )
⇑
• every EF of the cut polytope has size 2Ω(n)
This is the 1st outcome of a new “SDP←→quantum” connection
Remark. Generalizes “linear EF ←→ classical CC” connection
(Faenza, Fiorini, Grappe, Tiwary’11)
Today: We focus on the TSP result!
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 5

13. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Our results
We show that
Every LP for the TSP has super-polynomial size.
via the following sequence of reductions:
1/4
• every EF of the TSP polytope has size 2Ω(n )
⇑
1/2
• ∃ (Gn )n s.t. every EF of STAB(Gn ) has size 2Ω(n )
⇑
• every EF of the cut polytope has size 2Ω(n)
This is the 1st outcome of a new “SDP←→quantum” connection
Remark. Generalizes “linear EF ←→ classical CC” connection
(Faenza, Fiorini, Grappe, Tiwary’11)
Today: We focus on the TSP result!
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 5

14. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Our results
We show that
Every LP for the TSP has super-polynomial size.
via the following sequence of reductions:
1/4
• every EF of the TSP polytope has size 2Ω(n )
⇑
1/2
• ∃ (Gn )n s.t. every EF of STAB(Gn ) has size 2Ω(n )
⇑
• every EF of the cut polytope has size 2Ω(n)
This is the 1st outcome of a new “SDP←→quantum” connection
Remark. Generalizes “linear EF ←→ classical CC” connection
(Faenza, Fiorini, Grappe, Tiwary’11)
Today: We focus on the TSP result!
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 5

15. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Our results
We show that
Every LP for the TSP has super-polynomial size.
via the following sequence of reductions:
1/4
• every EF of the TSP polytope has size 2Ω(n )
⇑
1/2
• ∃ (Gn )n s.t. every EF of STAB(Gn ) has size 2Ω(n )
⇑
• every EF of the cut polytope has size 2Ω(n)
This is the 1st outcome of a new “SDP←→quantum” connection
Remark. Generalizes “linear EF ←→ classical CC” connection
(Faenza, Fiorini, Grappe, Tiwary’11)
Today: We focus on the TSP result!
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 5

16. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Our results
We show that
Every LP for the TSP has super-polynomial size.
via the following sequence of reductions:
1/4
• every EF of the TSP polytope has size 2Ω(n )
⇑
1/2
• ∃ (Gn )n s.t. every EF of STAB(Gn ) has size 2Ω(n )
⇑
• every EF of the cut polytope has size 2Ω(n)
This is the 1st outcome of a new “SDP←→quantum” connection
Remark. Generalizes “linear EF ←→ classical CC” connection
(Faenza, Fiorini, Grappe, Tiwary’11)
Today: We focus on the TSP result!
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 5

17. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Our results
We show that
Every LP for the TSP has super-polynomial size.
via the following sequence of reductions:
1/4
• every EF of the TSP polytope has size 2Ω(n )
⇑
1/2
• ∃ (Gn )n s.t. every EF of STAB(Gn ) has size 2Ω(n )
⇑
• every EF of the cut polytope has size 2Ω(n)
This is the 1st outcome of a new “SDP←→quantum” connection
Remark. Generalizes “linear EF ←→ classical CC” connection
(Faenza, Fiorini, Grappe, Tiwary’11)
Today: We focus on the TSP result!
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 5

18. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Our results
We show that
Every LP for the TSP has super-polynomial size.
via the following sequence of reductions:
1/4
• every EF of the TSP polytope has size 2Ω(n )
⇑
1/2
• ∃ (Gn )n s.t. every EF of STAB(Gn ) has size 2Ω(n )
⇑
• every EF of the cut polytope has size 2Ω(n)
This is the 1st outcome of a new “SDP←→quantum” connection
Remark. Generalizes “linear EF ←→ classical CC” connection
(Faenza, Fiorini, Grappe, Tiwary’11)
Today: We focus on the TSP result!
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 5

19. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Extension complexity and slack matrices
Consider a polytope P (with dim(P) 1)
Definition (Extension complexity)
Extension complexity of P, xc(P) := minimum size of EF of P
Let A ∈ Rm×d , b ∈ Rm , V = {v1 , . . . , vn } ⊆ Rd s.t.
P = {x ∈ Rd | Ax b} = conv(V )
Definition (Slack matrix)
Slack matrix S ∈ Rm×n of P w.r.t. Ax
+ b and V :
Sij := bi − Ai vj
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 6

20. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Extension complexity and slack matrices
Consider a polytope P (with dim(P) 1)
Definition (Extension complexity)
Extension complexity of P, xc(P) := minimum size of EF of P
Let A ∈ Rm×d , b ∈ Rm , V = {v1 , . . . , vn } ⊆ Rd s.t.
P = {x ∈ Rd | Ax b} = conv(V )
Definition (Slack matrix)
Slack matrix S ∈ Rm×n of P w.r.t. Ax
+ b and V :
Sij := bi − Ai vj
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 6

21. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Extension complexity and slack matrices
Consider a polytope P (with dim(P) 1)
Definition (Extension complexity)
Extension complexity of P, xc(P) := minimum size of EF of P
Let A ∈ Rm×d , b ∈ Rm , V = {v1 , . . . , vn } ⊆ Rd s.t.
P = {x ∈ Rd | Ax b} = conv(V )
Definition (Slack matrix)
Slack matrix S ∈ Rm×n of P w.r.t. Ax
+ b and V :
Sij := bi − Ai vj
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 6

22. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Extension complexity and slack matrices
Consider a polytope P (with dim(P) 1)
Definition (Extension complexity)
Extension complexity of P, xc(P) := minimum size of EF of P
Let A ∈ Rm×d , b ∈ Rm , V = {v1 , . . . , vn } ⊆ Rd s.t.
P = {x ∈ Rd | Ax b} = conv(V )
Definition (Slack matrix)
Slack matrix S ∈ Rm×n of P w.r.t. Ax
+ b and V :
Sij := bi − Ai vj
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 6

23. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Nonnegative factorizations and the factorization theorem
Definition (Nonnegative factorization and rank)
A rank-r nonnegative factorization of S ∈ Rm×n is
S = TU where T ∈ Rm×r
+ and U ∈ Rr ×n
+
Nonnegative rank of S:
rk+ (S) := min{r | ∃ rank-r nonnegative factorization of S}
Theorem (Yannakakis’91)
For every slack matrix S of P:
xc(P) = rk+ (S)
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 7

24. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Nonnegative factorizations and the factorization theorem
Definition (Nonnegative factorization and rank)
A rank-r nonnegative factorization of S ∈ Rm×n is
S = TU where T ∈ Rm×r
+ and U ∈ Rr ×n
+
Nonnegative rank of S:
rk+ (S) := min{r | ∃ rank-r nonnegative factorization of S}
Theorem (Yannakakis’91)
For every slack matrix S of P:
xc(P) = rk+ (S)
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 7

25. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Nonnegative factorizations and the factorization theorem
Definition (Nonnegative factorization and rank)
A rank-r nonnegative factorization of S ∈ Rm×n is
S = TU where T ∈ Rm×r
+ and U ∈ Rr ×n
+
Nonnegative rank of S:
rk+ (S) := min{r | ∃ rank-r nonnegative factorization of S}
Theorem (Yannakakis’91)
For every slack matrix S of P:
xc(P) = rk+ (S)
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 7

26. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Bounding the extension complexity: the Rectangle Covering Bound
S = TU nonnegative factorization
= T k Uk sum of nonnegative rank-1 matrices
k∈[r ]
=⇒ supp(S) = supp(T k Uk )
k∈[r ]
= supp(T k ) × supp(Uk ) union of rectangles (as sets)
k∈[r ]
Definition (Rectangle covering number rc(S))
rc(S) := min{r | ∃ rectangle covering of size r }.
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 8

27. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Bounding the extension complexity: the Rectangle Covering Bound
S = TU nonnegative factorization
= T k Uk sum of nonnegative rank-1 matrices
k∈[r ]
=⇒ supp(S) = supp(T k Uk )
k∈[r ]
= supp(T k ) × supp(Uk ) union of rectangles (as sets)
k∈[r ]
Definition (Rectangle covering number rc(S))
rc(S) := min{r | ∃ rectangle covering of size r }.
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 8

28. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Bounding the extension complexity: the Rectangle Covering Bound
S = TU nonnegative factorization
= T k Uk sum of nonnegative rank-1 matrices
k∈[r ]
=⇒ supp(S) = supp(T k Uk )
k∈[r ]
= supp(T k ) × supp(Uk ) union of rectangles (as sets)
k∈[r ]
Definition (Rectangle covering number rc(S))
rc(S) := min{r | ∃ rectangle covering of size r }.
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 8

29. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Bounding the extension complexity: the Rectangle Covering Bound
0 1 1 1 1
1 0 1 1 1
1 1 0 1 1
1 1 1 0 1
1 1 1 1 0
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 9

30. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Bounding the extension complexity: the Rectangle Covering Bound
0 1 1 1 1
1 0 1 1 1
1 1 0 1 1
1 1 1 0 1
1 1 1 1 0
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 9

31. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Bounding the extension complexity: the Rectangle Covering Bound
Observation. rk+ (S) rc(S) (Yannakakis’91)
Observation. For every polytope P and slack matrix S of P:
xc(P) = rk+ (S) rc(S) = rc( suppmat(S) )
nonincidence matrix
(Fiorini, Kaibel, Pashkovich & Theis’11)
Remark. This is related to nondeterministic CC (= log rc(M) )
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 10

32. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Bounding the extension complexity: the Rectangle Covering Bound
Observation. rk+ (S) rc(S) (Yannakakis’91)
Observation. For every polytope P and slack matrix S of P:
xc(P) = rk+ (S) rc(S) = rc( suppmat(S) )
nonincidence matrix
(Fiorini, Kaibel, Pashkovich & Theis’11)
Remark. This is related to nondeterministic CC (= log rc(M) )
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 10

33. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Bounding the extension complexity: the Rectangle Covering Bound
Observation. rk+ (S) rc(S) (Yannakakis’91)
Observation. For every polytope P and slack matrix S of P:
xc(P) = rk+ (S) rc(S) = rc( suppmat(S) )
nonincidence matrix
(Fiorini, Kaibel, Pashkovich & Theis’11)
Remark. This is related to nondeterministic CC (= log rc(M) )
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 10

34. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the correlation polytope / cut polytope
Let M = M(n) be the 2n × 2n matrix with
Mab := (1 − aT b)2
for a, b ∈ {0, 1}n
Remark.
• Not a slack matrix, but can be embedded in a slack matrix
• Support matrix appears in de Wolf’03 for separating
classical vs. quantum nondeterministic complexity
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 11

35. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the correlation polytope / cut polytope
Let M = M(n) be the 2n × 2n matrix with
Mab := (1 − aT b)2
for a, b ∈ {0, 1}n
Remark.
• Not a slack matrix, but can be embedded in a slack matrix
• Support matrix appears in de Wolf’03 for separating
classical vs. quantum nondeterministic complexity
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 11

36. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the correlation polytope / cut polytope
Observation. For b ∈ {0, 1}n :
1 − 2diag(a) − aaT , bb T = 1 − 2 diag(a), bb T + aaT , bb T
= 1 − 2 diag(a), diag(b) + aaT , bb T
= 1 − 2 aT b + (aT b)2 = (1 − aT b)2 = Mab
Correlation polytope: COR(n) := conv{bb T ∈ Rn×n | b ∈ {0, 1}n }
Lemma (Key Lemma)
For every a ∈ {0, 1}n , the inequality
( ) 2diag(a) − aaT , x 1
is valid for COR(n). The slack of vertex bb T w.r.t. ( ) is Mab .
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 12

37. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the correlation polytope / cut polytope
Observation. For b ∈ {0, 1}n :
1 − 2diag(a) − aaT , bb T = 1 − 2 diag(a), bb T + aaT , bb T
= 1 − 2 diag(a), diag(b) + aaT , bb T
= 1 − 2 aT b + (aT b)2 = (1 − aT b)2 = Mab
Correlation polytope: COR(n) := conv{bb T ∈ Rn×n | b ∈ {0, 1}n }
Lemma (Key Lemma)
For every a ∈ {0, 1}n , the inequality
( ) 2diag(a) − aaT , x 1
is valid for COR(n). The slack of vertex bb T w.r.t. ( ) is Mab .
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 12

38. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the correlation polytope / cut polytope
Consider complete linear description for COR(n) starting with
2diag(a) − aaT , x 1 ∀a ∈ {0, 1}n
and corresponding slack matrix S
bb T
2diag(a) − aaT , x 1
M
S
xc(COR(n)) = rk+ (S)
rc(S)
rc(M)
= 2Ω(n) (de Wolf’03 using Razborov’92)
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 13

39. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the correlation polytope / cut polytope
Consider complete linear description for COR(n) starting with
2diag(a) − aaT , x 1 ∀a ∈ {0, 1}n
and corresponding slack matrix S
bb T
2diag(a) − aaT , x 1
M
S
xc(COR(n)) = rk+ (S)
rc(S)
rc(M)
= 2Ω(n) (de Wolf’03 using Razborov’92)
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 13

40. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the correlation polytope / cut polytope
Consider complete linear description for COR(n) starting with
2diag(a) − aaT , x 1 ∀a ∈ {0, 1}n
and corresponding slack matrix S
bb T
2diag(a) − aaT , x 1
M
S
xc(COR(n)) = rk+ (S)
rc(S)
rc(M)
= 2Ω(n) (de Wolf’03 using Razborov’92)
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 13

41. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the correlation polytope / cut polytope
As CUT(n) ∼ COR(n − 1) (affine) we obtain:
=
Theorem (Cut polytope)
xc(CUT(n)) = xc(COR(n − 1)) = 2Ω(n) .
This is our problem-0 and we do have a reduction mechanism:
Lemma (Monotonicity)
• Q is an EF of P =⇒ xc(Q) xc(P)
• P contains F as a face =⇒ xc(P) xc(F )
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 14

42. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the correlation polytope / cut polytope
As CUT(n) ∼ COR(n − 1) (affine) we obtain:
=
Theorem (Cut polytope)
xc(CUT(n)) = xc(COR(n − 1)) = 2Ω(n) .
This is our problem-0 and we do have a reduction mechanism:
Lemma (Monotonicity)
• Q is an EF of P =⇒ xc(Q) xc(P)
• P contains F as a face =⇒ xc(P) xc(F )
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 14

43. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the correlation polytope / cut polytope
As CUT(n) ∼ COR(n − 1) (affine) we obtain:
=
Theorem (Cut polytope)
xc(CUT(n)) = xc(COR(n − 1)) = 2Ω(n) .
This is our problem-0 and we do have a reduction mechanism:
Lemma (Monotonicity)
• Q is an EF of P =⇒ xc(Q) xc(P)
• P contains F as a face =⇒ xc(P) xc(F )
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 14

44. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the stable set polytope
Stable set polytope. ∀ k ∃ Hk with O(k 2 ) vertices s.t.
STAB(Hk ) has a face F = F (k) that is an EF of COR(k).
u u
e e e e
xc(STAB(Hk )) xc(F (k))
xc(COR(k))
= 2Ω(k)
v v
Theorem (Stable set polytope)
For all n there exists an n-vertex graph Gn s.t.
1/2 )
xc(STAB(Gn )) = 2Ω(n
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 15

45. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the stable set polytope
Stable set polytope. ∀ k ∃ Hk with O(k 2 ) vertices s.t.
STAB(Hk ) has a face F = F (k) that is an EF of COR(k).
u u
e e e e
xc(STAB(Hk )) xc(F (k))
xc(COR(k))
= 2Ω(k)
v v
Theorem (Stable set polytope)
For all n there exists an n-vertex graph Gn s.t.
1/2 )
xc(STAB(Gn )) = 2Ω(n
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 15

46. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the stable set polytope
Stable set polytope. ∀ k ∃ Hk with O(k 2 ) vertices s.t.
STAB(Hk ) has a face F = F (k) that is an EF of COR(k).
u u
e e e e
xc(STAB(Hk )) xc(F (k))
xc(COR(k))
= 2Ω(k)
v v
Theorem (Stable set polytope)
For all n there exists an n-vertex graph Gn s.t.
1/2 )
xc(STAB(Gn )) = 2Ω(n
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 15

47. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Lower bound for the TSP polytope
TSP polytope. ∀ k-vertex G , one can find face F = F (k) of
TSP(n) with n = O(k 2 ) such that F (k) is an EF of STAB(G ).
(Yannakakis’91).
Theorem (TSP polytope)
1/4 )
xc(TSP(n)) = 2Ω(n
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 16

48. Introduction and History Nonnegative factorizations and lower bounds Strong lower bounds
Thank you!
Sebastian Pokutta Linear vs. Semidefinite Extended Formulations 17