The matching polytope has exponential extension...
Transcript of The matching polytope has exponential extension...
The matching polytope has exponential
extension complexity
Thomas Rothvoß
Department of Mathematics, MIT
Guwahati, India — Dec 2013
Extended formulation
Extended formulation
Given polytope P = x ∈ Rn | Ax ≤ b
P
Extended formulation
Given polytope P = x ∈ Rn | Ax ≤ b
Write P = x ∈ Rn | ∃y : Bx+ Cy ≤ d
P
Q
linearprojection
Extended formulation
Given polytope P = x ∈ Rn | Ax ≤ b
→ many inequalities Write P = x ∈ R
n | ∃y : Bx+ Cy ≤ d→ few inequalities
P
Q
linearprojection
Extended formulation
Given polytope P = x ∈ Rn | Ax ≤ b
→ many inequalities Write P = x ∈ R
n | ∃y : Bx+ Cy ≤ d→ few inequalities
P
Q
linearprojection
Extension complexity:
xc(P ) := min
#facets of Q |
Q polyhedronp linear mapp(Q) = P
What’s known?
Compact formulations:
Spanning Tree Polytope [Kipp Martin ’91]
Perfect Matching in planar graphs [Barahona ’93]
Perfect Matching in bounded genus graphs[Gerards ’91]
O(n logn)-size for Permutahedron [Goemans ’10](→ tight)
nO(1/ε)-size ε-apx for Knapsack Polytope [Bienstock ’08]
. . .
What’s known?
Compact formulations:
Spanning Tree Polytope [Kipp Martin ’91]
Perfect Matching in planar graphs [Barahona ’93]
Perfect Matching in bounded genus graphs[Gerards ’91]
O(n logn)-size for Permutahedron [Goemans ’10](→ tight)
nO(1/ε)-size ε-apx for Knapsack Polytope [Bienstock ’08]
. . .
Here: When is the extension complexity super polynomial?
Lower bounds
Lower bounds
No symmetric compact form. for TSP [Yannakakis ’91]Compact formulation for logn size matchings, but nosymmetric one [Kaibel, Pashkovich & Theis ’10]
Lower bounds
No symmetric compact form. for TSP [Yannakakis ’91]Compact formulation for logn size matchings, but nosymmetric one [Kaibel, Pashkovich & Theis ’10]
xc(random 0/1 polytope) ≥ 2Ω(n) [R. ’11]
Lower bounds
No symmetric compact form. for TSP [Yannakakis ’91]Compact formulation for logn size matchings, but nosymmetric one [Kaibel, Pashkovich & Theis ’10]
xc(random 0/1 polytope) ≥ 2Ω(n) [R. ’11]
Breakthrough: xc(TSP) ≥ 2Ω(√n)
[Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]
Lower bounds
No symmetric compact form. for TSP [Yannakakis ’91]Compact formulation for logn size matchings, but nosymmetric one [Kaibel, Pashkovich & Theis ’10]
xc(random 0/1 polytope) ≥ 2Ω(n) [R. ’11]
Breakthrough: xc(TSP) ≥ 2Ω(√n)
[Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]
n1/2−ε-apx for clique polytope needs super-poly size[Braun, Fiorini, Pokutta, Steuer ’12]Improved to n1−ε [Braverman, Moitra ’13], [Braun, P. ’13]
Lower bounds
No symmetric compact form. for TSP [Yannakakis ’91]Compact formulation for logn size matchings, but nosymmetric one [Kaibel, Pashkovich & Theis ’10]
xc(random 0/1 polytope) ≥ 2Ω(n) [R. ’11]
Breakthrough: xc(TSP) ≥ 2Ω(√n)
[Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]
n1/2−ε-apx for clique polytope needs super-poly size[Braun, Fiorini, Pokutta, Steuer ’12]Improved to n1−ε [Braverman, Moitra ’13], [Braun, P. ’13]
(2− ε)-apx LPs for MaxCut have size nΩ(logn/ log logn)
[Chan, Lee, Raghavendra, Steurer ’13]
Lower bounds
No symmetric compact form. for TSP [Yannakakis ’91]Compact formulation for logn size matchings, but nosymmetric one [Kaibel, Pashkovich & Theis ’10]
xc(random 0/1 polytope) ≥ 2Ω(n) [R. ’11]
Breakthrough: xc(TSP) ≥ 2Ω(√n)
[Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]
n1/2−ε-apx for clique polytope needs super-poly size[Braun, Fiorini, Pokutta, Steuer ’12]Improved to n1−ε [Braverman, Moitra ’13], [Braun, P. ’13]
(2− ε)-apx LPs for MaxCut have size nΩ(logn/ log logn)
[Chan, Lee, Raghavendra, Steurer ’13]
Only NP-hard polytopes!!
What about poly-time problems?
Perfect matching polytope
Perfect matching polytope G = (V,E)(complete)
Perfect matching polytope G = (V,E)(complete)
Perfect matching polytope
x(δ(v)) = 1 ∀v ∈ V
xe ≥ 0 ∀e ∈ E
G = (V,E)(complete)
Perfect matching polytope
x(δ(v)) = 1 ∀v ∈ V
xe ≥ 0 ∀e ∈ E
12
12
12
12
12
12
G = (V,E)(complete)
Perfect matching polytope
x(δ(v)) = 1 ∀v ∈ V
xe ≥ 0 ∀e ∈ E
U
12
12
12
12
12
12
G = (V,E)(complete)
Perfect matching polytope
x(δ(v)) = 1 ∀v ∈ V
x(δ(U)) ≥ 1 ∀U ⊆ V : |U | odd
xe ≥ 0 ∀e ∈ E
U
12
12
12
12
12
12
G = (V,E)(complete)
Quick facts:
Description by [Edmonds ’65]
Perfect matching polytope
x(δ(v)) = 1 ∀v ∈ V
x(δ(U)) ≥ 1 ∀U ⊆ V : |U | odd
xe ≥ 0 ∀e ∈ E
U
12
12
12
12
12
12
G = (V,E)(complete)
Quick facts:
Description by [Edmonds ’65] Can optimize cTx in strongly poly-time [Edmonds ’65]
Perfect matching polytope
x(δ(v)) = 1 ∀v ∈ V
x(δ(U)) ≥ 1 ∀U ⊆ V : |U | odd
xe ≥ 0 ∀e ∈ E
U
12
12
12
12
12
12
G = (V,E)(complete)
Quick facts:
Description by [Edmonds ’65] Can optimize cTx in strongly poly-time [Edmonds ’65] Separation problem polytime [Padberg, Rao ’82]
Perfect matching polytope
x(δ(v)) = 1 ∀v ∈ V
x(δ(U)) ≥ 1 ∀U ⊆ V : |U | odd
xe ≥ 0 ∀e ∈ E
U
12
12
12
12
12
12
G = (V,E)(complete)
Quick facts:
Description by [Edmonds ’65] Can optimize cTx in strongly poly-time [Edmonds ’65] Separation problem polytime [Padberg, Rao ’82] 2Θ(n) facets
Perfect matching polytope
x(δ(v)) = 1 ∀v ∈ V
x(δ(U)) ≥ 1 ∀U ⊆ V : |U | odd
xe ≥ 0 ∀e ∈ E
U
12
12
12
12
12
12
G = (V,E)(complete)
Quick facts:
Description by [Edmonds ’65] Can optimize cTx in strongly poly-time [Edmonds ’65] Separation problem polytime [Padberg, Rao ’82] 2Θ(n) facets
Theorem (R.13)
xc(perfect matching polytope) ≥ 2Ω(n).
Previously known: xc(P ) ≥ Ω(n2)
Slack-matrix
Write: P = conv(x1, . . . , xv) = x ∈ Rn | Ax ≤ b
S# facets
# vertices
SijSij = bi −AT
i xj
slack-matrix
Pb
b b
b
b
Slack-matrix
Write: P = conv(x1, . . . , xv) = x ∈ Rn | Ax ≤ b
S# facets
# vertices
facet i
vertexj
SijSij = bi −AT
i xj
slack-matrix
Pb
b b
b
bAix = bi
bxj
Sij
Slack-matrix
Write: P = conv(x1, . . . , xv) = x ∈ Rn | Ax ≤ b
S# facets
# vertices
U≥0
V ≥ 0rr
SijSij = bi −AT
i xj
slack-matrix
Pb
b b
b
bAix = bi
bxj
Sij
Non-negative rank:
rk+(S) = minr | ∃U ∈ Rf×r≥0 , V ∈ R
r×v≥0 : S = UV
Yannakakis’ Theorem
Theorem (Yannakakis ’91)
If S is the slack-matrix for P = x ∈ Rn | Ax ≤ b, then
xc(P ) = rk+(S).
P
Yannakakis’ Theorem
Theorem (Yannakakis ’91)
If S is the slack-matrix for P = x ∈ Rn | Ax ≤ b, then
xc(P ) = rk+(S).
Factorization S = UV ⇒ extended formulation:
Let P = x ∈ Rn | ∃y ≥ 0 : Ax+ Uy = b
P
Yannakakis’ Theorem
Theorem (Yannakakis ’91)
If S is the slack-matrix for P = x ∈ Rn | Ax ≤ b, then
xc(P ) = rk+(S).
Factorization S = UV ⇒ extended formulation:
Let P = x ∈ Rn | ∃y ≥ 0 : Ax+ Uy = b
Extended form. ⇒ factorization:
Given an extensionQ = (x, y) | Bx+ Cy ≤ d
Qb
b b
b
b
b
b
b
b
b
b
b
P
Yannakakis’ Theorem
Theorem (Yannakakis ’91)
If S is the slack-matrix for P = x ∈ Rn | Ax ≤ b, then
xc(P ) = rk+(S).
Factorization S = UV ⇒ extended formulation:
Let P = x ∈ Rn | ∃y ≥ 0 : Ax+ Uy = b
Extended form. ⇒ factorization:
Given an extensionQ = (x, y) | Bx+ Cy ≤ d
Q
Aix+ 0y ≤ bi
b
b b
b
b
b
b
b
b
b
b
b
xjb
P
〈u(i), v(j)〉 = Sij
Yannakakis’ Theorem
Theorem (Yannakakis ’91)
If S is the slack-matrix for P = x ∈ Rn | Ax ≤ b, then
xc(P ) = rk+(S).
Factorization S = UV ⇒ extended formulation:
Let P = x ∈ Rn | ∃y ≥ 0 : Ax+ Uy = b
Extended form. ⇒ factorization:
Given an extensionQ = (x, y) | Bx+ Cy ≤ d
For facet i:u(i) := conic comb of i
Q
Aix+ 0y ≤ bi
b
b b
b
b
b
b
b
b
b
b
b
xjb
P
〈u(i), v(j)〉 = Sij
Yannakakis’ Theorem
Theorem (Yannakakis ’91)
If S is the slack-matrix for P = x ∈ Rn | Ax ≤ b, then
xc(P ) = rk+(S).
Factorization S = UV ⇒ extended formulation:
Let P = x ∈ Rn | ∃y ≥ 0 : Ax+ Uy = b
Extended form. ⇒ factorization:
Given an extensionQ = (x, y) | Bx+ Cy ≤ d
For facet i:u(i) := conic comb of i
For vertex xj :v(j) := d−Bxj − Cyj = slack of (xj , yj)
Q
Aix+ 0y ≤ bi
b
b b
b
b
b
b
b
b
b
b
b
xjb
(xj , yj)b
P
〈u(i), v(j)〉 = Sij
Yannakakis’ Theorem
Theorem (Yannakakis ’91)
If S is the slack-matrix for P = x ∈ Rn | Ax ≤ b, then
xc(P ) = rk+(S).
Factorization S = UV ⇒ extended formulation:
Let P = x ∈ Rn | ∃y ≥ 0 : Ax+ Uy = b
Extended form. ⇒ factorization:
Given an extensionQ = (x, y) | Bx+ Cy ≤ d
For facet i:u(i) := conic comb of i
For vertex xj :v(j) := d−Bxj − Cyj = slack of (xj , yj)
Q
Aix+ 0y ≤ bi
b
b b
b
b
b
b
b
b
b
b
b
xjb
(xj , yj)b
P
〈u(i), v(j)〉 = u(i)Td︸ ︷︷ ︸
=bi
−u(i)B︸ ︷︷ ︸
=Ai
xj − u(i)C︸ ︷︷ ︸
=0
yj = Sij
Rectangle covering lower bound
Observation
rk+(S) ≥ rectangle-covering-number(S).
Rectangle covering lower bound
U
V
S
3
1
0
0
2
0 0 2 1 0
2
1
2
0
0
0 2 2 0 3
0 4 10 3 5
0 2 4 1 3
0 4 4 0 6
0 0 0 0 0
0 0 4 2 0
Observation
rk+(S) ≥ rectangle-covering-number(S).
Rectangle covering lower bound
U
V
S
+
+
0
0+
0 0 + + 0
+
+
+
0
0
0 + + 0 +
0 + + + +
0 + + + +
0 + + 0 +
0 0 0 0 0
0 0 + + 0
Observation
rk+(S) ≥ rectangle-covering-number(S).
Rectangle covering lower bound
U
V
S
+
+
0
0+
0 0 + + 0
+
+
+
0
0
0 + + 0 +
0 + + + +
0 + + + +
0 + + 0 +
0 0 0 0 0
0 0 + + 0
Observation
rk+(S) ≥ rectangle-covering-number(S).
Rectangle covering lower bound
U
V
S
+
+
0
0+
0 0 + + 0
+
+
+
0
0
0 + + 0 +
0 + + + +
0 + + + +
0 + + 0 +
0 0 0 0 0
0 0 + + 0
Observation
rk+(S) ≥ rectangle-covering-number(S).
Rectangle covering for matching
Recall SU,M = |δ(U) ∩M | − 1
Observation
Rect-cov-num(matching polytope) ≤ O(n4).
Re1,e2
matchings
cuts
S
Rectangle covering for matching
Recall SU,M = |δ(U) ∩M | − 1
Observation
Rect-cov-num(matching polytope) ≤ O(n4).
e1
e2 Re1,e2
matchings
cuts
S
For e1, e2 ∈ E:
Rectangle covering for matching
Recall SU,M = |δ(U) ∩M | − 1
Observation
Rect-cov-num(matching polytope) ≤ O(n4).
U
e1
e2 Re1,e2
matchings
cuts
S
For e1, e2 ∈ E: take U | e1, e2 ∈ δ(U)
Rectangle covering for matching
Recall SU,M = |δ(U) ∩M | − 1
Observation
Rect-cov-num(matching polytope) ≤ O(n4).
U
e1
e2M
Re1,e2
matchings
cuts
S
For e1, e2 ∈ E: take U | e1, e2 ∈ δ(U) ×M | e1, e2 ∈ M
Rectangle covering for matching
Recall SU,M = |δ(U) ∩M | − 1
Observation
Rect-cov-num(matching polytope) ≤ O(n4).
U M
e1
e2...ek
Re1,e2
matchings
cuts
S
For e1, e2 ∈ E: take U | e1, e2 ∈ δ(U) ×M | e1, e2 ∈ M (U,M) with M ∩ δ(U) = e1, . . . , ek lies in
(k2
)rectangles
Rectangle covering for matching
Recall SU,M = |δ(U) ∩M | − 1
Observation
Rect-cov-num(matching polytope) ≤ O(n4).
U M
e1
e2...ek
Re1,e2
matchings
cuts
S
For e1, e2 ∈ E: take U | e1, e2 ∈ δ(U) ×M | e1, e2 ∈ M (U,M) with M ∩ δ(U) = e1, . . . , ek lies in
(k2
)rectangles
S?=
∑
e1,e2
0 1 1
0 1 1
0 0 0
Re1,e2
Rectangle covering for matching
Recall SU,M = |δ(U) ∩M | − 1
Observation
Rect-cov-num(matching polytope) ≤ O(n4).
U M
e1
e2...ek
Re1,e2
matchings
cuts
S
For e1, e2 ∈ E: take U | e1, e2 ∈ δ(U) ×M | e1, e2 ∈ M (U,M) with M ∩ δ(U) = e1, . . . , ek lies in
(k2
)rectangles
S?=
∑
e1,e2
0 1 1
0 1 1
0 0 0
Re1,e2|M ∩ δ(U)| = k
SUM = k − 1∼ k2
Rectangle covering for matching
Recall SU,M = |δ(U) ∩M | − 1
Observation
Rect-cov-num(matching polytope) ≤ O(n4).
U M
e1
e2...ek
Re1,e2
matchings
cuts
S
For e1, e2 ∈ E: take U | e1, e2 ∈ δ(U) ×M | e1, e2 ∈ M (U,M) with M ∩ δ(U) = e1, . . . , ek lies in
(k2
)rectangles
Question
Does every rectangle coveringover-cover entries of large slack?
Rectangle covering for matching
Recall SU,M = |δ(U) ∩M | − 1
Observation
Rect-cov-num(matching polytope) ≤ O(n4).
U M
e1
e2...ek
Re1,e2
matchings
cuts
S
For e1, e2 ∈ E: take U | e1, e2 ∈ δ(U) ×M | e1, e2 ∈ M (U,M) with M ∩ δ(U) = e1, . . . , ek lies in
(k2
)rectangles
Question
Does every rectangle coveringover-cover entries of large slack? YES!!
Hyperplane separation lower bound [Fiorini]
Frobenius inner product: 〈W,S〉 :=∑
i
∑
j WijSij
Hyperplane separation lower bound [Fiorini]
Frobenius inner product: 〈W,S〉 :=∑
i
∑
j WijSij
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R.
R
0
b
b
b
b
b W
〈W,R〉 ≤ αrectangles
Hyperplane separation lower bound [Fiorini]
Frobenius inner product: 〈W,S〉 :=∑
i
∑
j WijSij
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R. Then rk+(S) ≥〈W,S〉
‖S‖∞ · α
R
S0
b
b
b
b
b W
〈W,R〉 ≤ αrectangles
Hyperplane separation lower bound [Fiorini]
Frobenius inner product: 〈W,S〉 :=∑
i
∑
j WijSij
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R. Then rk+(S) ≥〈W,S〉
‖S‖∞ · α
Proof: Write S =∑r
i=1Ri with rk+(Ri) = 1. Then
〈W,S〉 =r∑
i=1
‖Ri‖∞·
⟨
W,Ri
‖Ri‖∞
⟩
︸ ︷︷ ︸
≤α
≤ α·r∑
i=1
‖Ri‖∞︸ ︷︷ ︸
≤‖S‖∞
≤ α·r·‖S‖∞.
R
S0
b
b
b
b
b W
〈W,R〉 ≤ αrectangles
[0, 1]-rank-1matrices
Applying the Hyperplane bound
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R. Then rk+(S) ≥〈W,S〉
‖S‖∞ · α
Applying the Hyperplane bound
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R. Then rk+(S) ≥〈W,S〉
‖S‖∞ · α
Recall SUM = |δ(U) ∩M | − 1
Applying the Hyperplane bound
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R. Then rk+(S) ≥〈W,S〉
‖S‖∞ · α
Recall SUM = |δ(U) ∩M | − 1 Abbreviate Qℓ := (U,M) : |δ(U) ∩M | = ℓ
Applying the Hyperplane bound
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R. Then rk+(S) ≥〈W,S〉
‖S‖∞ · α
Recall SUM = |δ(U) ∩M | − 1 Abbreviate Qℓ := (U,M) : |δ(U) ∩M | = ℓ Choose
WU,M =
0 otherwise.
Applying the Hyperplane bound
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R. Then rk+(S) ≥〈W,S〉
‖S‖∞ · α
Recall SUM = |δ(U) ∩M | − 1 Abbreviate Qℓ := (U,M) : |δ(U) ∩M | = ℓ Choose
WU,M =
−∞ |δ(U) ∩M | = 1
0 otherwise.
Then 〈W,S〉 = 0
Applying the Hyperplane bound
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R. Then rk+(S) ≥〈W,S〉
‖S‖∞ · α
Recall SUM = |δ(U) ∩M | − 1 Abbreviate Qℓ := (U,M) : |δ(U) ∩M | = ℓ Choose
WU,M =
−∞ |δ(U) ∩M | = 11
|Q3| |δ(U) ∩M | = 3
0 otherwise.
Then 〈W,S〉 = 0 + 2
Applying the Hyperplane bound
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R. Then rk+(S) ≥〈W,S〉
‖S‖∞ · α
Recall SUM = |δ(U) ∩M | − 1 Abbreviate Qℓ := (U,M) : |δ(U) ∩M | = ℓ Choose
WU,M =
−∞ |δ(U) ∩M | = 11
|Q3| |δ(U) ∩M | = 3
− 1k−1 · 1
|Qk| |δ(U) ∩M | = k
0 otherwise.
Then 〈W,S〉 = 0 + 2− 1 = 1
Applying the Hyperplane bound
Lemma
Pick W : 〈W,R〉 ≤ α ∀ rectangles R. Then rk+(S) ≥〈W,S〉
‖S‖∞ · α
Recall SUM = |δ(U) ∩M | − 1 Abbreviate Qℓ := (U,M) : |δ(U) ∩M | = ℓ Choose
WU,M =
−∞ |δ(U) ∩M | = 11
|Q3| |δ(U) ∩M | = 3
− 1k−1 · 1
|Qk| |δ(U) ∩M | = k
0 otherwise.
Then 〈W,S〉 = 0 + 2− 1 = 1
Lemma
For k large, any rectangle R has 〈W,R〉 ≤ 2−Ω(n).
Applying the Hyperplane bound (II)
Uniform measure: µℓ(R) := |R∩Qℓ||Qℓ|
Applying the Hyperplane bound (II)
Uniform measure: µℓ(R) := |R∩Qℓ||Qℓ|
Main lemma
µ1(R) = 0 =⇒ µ3(R) ≤ O( 1k2) · µk(R) + 2−Ω(n)
matchings
cuts
S
Applying the Hyperplane bound (II)
Uniform measure: µℓ(R) := |R∩Qℓ||Qℓ|
Main lemma
µ1(R) = 0 =⇒ µ3(R) ≤ O( 1k2) · µk(R) + 2−Ω(n)
R
matchings
cuts
S
Applying the Hyperplane bound (II)
Uniform measure: µℓ(R) := |R∩Qℓ||Qℓ|
Main lemma
µ1(R) = 0 =⇒ µ3(R) ≤ O( 1k2) · µk(R) + 2−Ω(n)
R
matchings
cuts
S
Applying the Hyperplane bound (II)
Uniform measure: µℓ(R) := |R∩Qℓ||Qℓ|
Main lemma
µ1(R) = 0 =⇒ µ3(R) ≤ O( 1k2) · µk(R) + 2−Ω(n)
R
matchings
cuts
S
Applying the Hyperplane bound (II)
Uniform measure: µℓ(R) := |R∩Qℓ||Qℓ|
Main lemma
µ1(R) = 0 =⇒ µ3(R) ≤ O( 1k2) · µk(R) + 2−Ω(n)
R
matchings
cuts
S
Applying the Hyperplane bound (II)
Uniform measure: µℓ(R) := |R∩Qℓ||Qℓ|
Main lemma
µ1(R) = 0 =⇒ µ3(R) ≤ O( 1k2) · µk(R) + 2−Ω(n)
R
matchings
cuts
S
Technique: Partition scheme [Razborov ’91]
Applying the Hyperplane bound (II)
Uniform measure: µℓ(R) := |R∩Qℓ||Qℓ|
Main lemma
µ1(R) = 0 =⇒ µ3(R) ≤ O( 1k2) · µk(R) + 2−Ω(n)
RT
matchings
cuts
S
Technique: Partition scheme [Razborov ’91]
Partitions
RT
matchings
cuts
S
Partition T = (A,C,D,B)
Partitions
RT
matchings
cuts
S
Partition T = (A,C,D,B)
A
Partitions
RT
matchings
cuts
S
Partition T = (A,C,D,B)
A B
Partitions
RT
matchings
cuts
S
Partition T = (A,C,D,B)
A C B
k
Partitions
RT
matchings
cuts
S
Partition T = (A,C,D,B)
A C D B
k k
Partitions
RT
matchings
cuts
S
Partition T = (A,C,D,B)
A C D B
A1
. . .
Amk − 3nodes
k k
Partitions
RT
matchings
cuts
S
Partition T = (A,C,D,B)
A C D B
B1. . . BmA1
. . .
Amk − 3nodes
k k 2(k − 3)nodes
Partitions
RT
matchings
cuts
S
Partition T = (A,C,D,B)
Edges E(T )
A C D B
B1. . . BmA1
. . .
Amk − 3nodes
k k 2(k − 3)nodes
Partitions
RT
matchings
cuts
S
Partition T = (A,C,D,B)
Edges E(T )
A C D B
B1. . . BmA1
. . .
Amk − 3nodes
k k 2(k − 3)nodes
Partitions
RT
matchings
cuts
S
Partition T = (A,C,D,B)
Edges E(T )
A C D B
B1. . . BmA1
. . .
Amk − 3nodes
k k 2(k − 3)nodes
U
Rewriting µ3(R)
RRT
matchings
cuts
S
Randomly generate (U,M) ∼ Q3:
µ3(R) =
Rewriting µ3(R)
RRT
matchings
cuts
S
A C D B
B1. . . BmA1
. . .
Am
Randomly generate (U,M) ∼ Q3:
1. Choose T
µ3(R) = ET
[ ]
Rewriting µ3(R)
RRT
matchings
cuts
S
A C D B
B1. . . BmA1
. . .
Am
H
Randomly generate (U,M) ∼ Q3:
1. Choose T2. Choose 3 edges H ⊆ C ×D
µ3(R) = ET
[
E|H|=3
[ ]]
Rewriting µ3(R)
RRT
matchings
cuts
S
A C D B
B1. . . BmA1
. . .
Am
H
Randomly generate (U,M) ∼ Q3:
1. Choose T2. Choose 3 edges H ⊆ C ×D3. Choose M ⊇ H (not cutting any other edge in C ×D)
µ3(R) = ET
[
E|H|=3
[ ]]
Rewriting µ3(R)
RRT
matchings
cuts
S
A C D B
B1. . . Bm
U
A1
. . .
Am
H
Randomly generate (U,M) ∼ Q3:
1. Choose T2. Choose 3 edges H ⊆ C ×D3. Choose M ⊇ H (not cutting any other edge in C ×D)4. Choose U cutting H (not cutting any Ai)
µ3(R) = ET
[
E|H|=3
[
Pr[(U,M) ∈ R | T,H]]]
Rewriting µ3(R)
RRT
matchings
cuts
S
A C D B
B1. . . Bm
U
A1
. . .
Am
H
Randomly generate (U,M) ∼ Q3:
1. Choose T2. Choose 3 edges H ⊆ C ×D3. Choose M ⊇ H (not cutting any other edge in C ×D)4. Choose U cutting H (not cutting any Ai)
µ3(R) = ET
[
E|H|=3
[
Pr[U ∈ R | T,H] · Pr[M ∈ R | T,H]]]
Rewriting µk(R)
RRT
matchings
cuts
S
Randomly generate (U,M) ∼ Qk:
µk(R) =
Rewriting µk(R)
RRT
matchings
cuts
S
A C D B
B1. . . BmA1
. . .
Am
Randomly generate (U,M) ∼ Qk:
1. Choose T
µk(R) = ET
[ ]
Rewriting µk(R)
RRT
matchings
cuts
S
A C D B
B1. . . BmA1
. . .
Am
F
Randomly generate (U,M) ∼ Qk:
1. Choose T2. Choose k edges F ⊆ C ×D
µk(R) = ET
[
E|F |=k
[ ]]
Rewriting µk(R)
RRT
matchings
cuts
S
A C D B
B1. . . BmA1
. . .
Am
F
Randomly generate (U,M) ∼ Qk:
1. Choose T2. Choose k edges F ⊆ C ×D3. Choose M ⊇ F
µk(R) = ET
[
E|F |=k
[
Pr[M ∈ R | T,H]]]
Rewriting µk(R)
RRT
matchings
cuts
S
A C D B
B1. . . Bm
U
A1
. . .
Am
F
Randomly generate (U,M) ∼ Qk:
1. Choose T2. Choose k edges F ⊆ C ×D3. Choose M ⊇ F4. Choose U ⊇ C (not cutting any Ai)
µk(R) = ET
[
E|F |=k
[
Pr[M ∈ R | T,H] · Pr[U ∈ R | T,H]]]
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n.
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n.
Draw x ∼ X.
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n.
Draw x ∼ X.
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n.
Draw x ∼ X.
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
2
..
1
..
q
1
..
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n.
Draw x ∼ X.
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
1
..
q
2
..
q
1
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n.
Draw x ∼ X.
Lemma
|X| large ⇒ for most indices xi is approx. uniform
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
1
..
q
2
..
q
1
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n.
Draw x ∼ X.
Lemma
εn biased indices ⇒ |X|qn ≤ 2−Ω(n).
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
1
..
q
2
..
q
1
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n.
Draw x ∼ X.
Lemma
εn biased indices ⇒ |X|qn ≤ 2−Ω(n).
log2(|X|) = H(x)
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
1
..
q
2
..
q
1
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n. Draw x ∼ X.
Lemma
εn biased indices ⇒ |X|qn ≤ 2−Ω(n).
log2(|X|) = H(x) ≤n∑
i=1
H(xi)
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
1
..
q
2
..
q
1
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n.
Draw x ∼ X.
Lemma
εn biased indices ⇒ |X|qn ≤ 2−Ω(n).
log2(|X|) = H(x) ≤∑
i biased
H(xi) +∑
i unbiased
H(xi)
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
1
..
q
2
..
q
1
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n. Draw x ∼ X.
Lemma
εn biased indices ⇒ |X|qn ≤ 2−Ω(n).
log2(|X|) = H(x) ≤∑
i biased
H(xi)︸ ︷︷ ︸
≤log2(q)−Θ(1)
+∑
i unbiased
H(xi)︸ ︷︷ ︸
≤log2(q)
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
1
..
q
2
..
q
1
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n. Draw x ∼ X.
Lemma
εn biased indices ⇒ |X|qn ≤ 2−Ω(n).
log2(|X|) = H(x) ≤∑
i biased
H(xi)︸ ︷︷ ︸
≤log2(q)−Θ(1)
+∑
i unbiased
H(xi)︸ ︷︷ ︸
≤log2(q)
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
1
..
q
2
..
q
1
0
1
0 0.5 1.0
Entropy for q = 2
p
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n. Draw x ∼ X.
Lemma
εn biased indices ⇒ |X|qn ≤ 2−Ω(n).
log2(|X|) = H(x) ≤∑
i biased
H(xi)︸ ︷︷ ︸
≤log2(q)−Θ(1)
+∑
i unbiased
H(xi)︸ ︷︷ ︸
≤log2(q)
≤ n log2(q)−Ω(n)
i
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1
2
..
q
1 2 . . . n
1
..
q
2
..
q
1
0
1
0 0.5 1.0
Entropy for q = 2
p
Pseudorandom-behaviour of large sets
Consider vectors X ⊆ [q]n.
Draw x ∼ X.
Lemma
εn biased indices ⇒ |X|qn ≤ 2−Ω(n).
log2(|X|) = H(x) ≤∑
i biased
H(xi)︸ ︷︷ ︸
≤log2(q)−Θ(1)
+∑
i unbiased
H(xi)︸ ︷︷ ︸
≤log2(q)
≤ n log2(q)−Ω(n)
Corollary
If X large, then for most i
Prx∼[q]n
[x ∈ X] ≈ Prx∼[q]n
[x ∈ X | xi = j]
M-good
Definition
(T,H) M-good if M ∼ M ∈ R | H ⊆ M ⊆ E(T ) is ε-uniformon (C ∪D)\V (H).
A C D B
B1. . . BmA1
. . .
Am
H
M-good
Definition
(T,H) M-good if M ∼ M ∈ R | H ⊆ M ⊆ E(T ) is ε-uniformon (C ∪D)\V (H).
A C D B
B1. . . BmA1
. . .
Am
H
M-good
Definition
(T,H) M-good if M ∼ M ∈ R | H ⊆ M ⊆ E(T ) is ε-uniformon (C ∪D)\V (H).
A C D B
B1. . . BmA1
. . .
Am
H
U-good
A C D B
B1. . . BmA1
. . .
Am
H
U-good
c
A C D B
B1. . . BmA1
. . .
Am
H
U-good
Definition
(T,H) U-good if U ∼ U ∈ R | c ⊆ U ; doesn’t cut any Ai hasPr[U ∩ C = c] ≈ 1
2 ≈ Pr[U ∩ C = C].
c
A C D B
B1. . . BmA1
. . .
Am
H
U-good
Definition
(T,H) U-good if U ∼ U ∈ R | c ⊆ U ; doesn’t cut any Ai hasPr[U ∩ C = c] ≈ 1
2 ≈ Pr[U ∩ C = C].
c
A C D B
B1. . . Bm
U
A1
. . .
Am
H
U-good
Definition
(T,H) U-good if U ∼ U ∈ R | c ⊆ U ; doesn’t cut any Ai hasPr[U ∩ C = c] ≈ 1
2 ≈ Pr[U ∩ C = C].
c
A C D B
B1. . . BmA1
. . .
Am
H
Splitting µ3(R)
µ3(R)
Splitting µ3(R)
µ3(R) = ET
[
E|H|=3
[Pr[(U,M) ∈ R | T,H]
]]
Splitting µ3(R)
µ3(R) = ET
[
E|H|=3
[Pr[(U,M) ∈ R | T,H]
]]
≤ ET
[
E|H|=3
[
GOOD(T,H) · Pr[(U,M) ∈ R | T,H]]]
+ ET
[
E|H|=3
[
M−BAD(T,H) · Pr[(U,M) ∈ R | T,H]]]
+ ET
[
E|H|=3
[
U−BAD(T,H) · Pr[(U,M) ∈ R | T,H]]]
Splitting µ3(R)
µ3(R) = ET
[
E|H|=3
[Pr[(U,M) ∈ R | T,H]
]]
≤ ET
[
E|H|=3
[
GOOD(T,H) · Pr[(U,M) ∈ R | T,H]]]
+ ET
[
E|H|=3
[
M−BAD(T,H) · Pr[(U,M) ∈ R | T,H]]]
+ ET
[
E|H|=3
[
U−BAD(T,H) · Pr[(U,M) ∈ R | T,H]]]
≤ O( 1k2)µk(R)
Splitting µ3(R)
µ3(R) = ET
[
E|H|=3
[Pr[(U,M) ∈ R | T,H]
]]
≤ ET
[
E|H|=3
[
GOOD(T,H) · Pr[(U,M) ∈ R | T,H]]]
+ ET
[
E|H|=3
[
M−BAD(T,H) · Pr[(U,M) ∈ R | T,H]]]
+ ET
[
E|H|=3
[
U−BAD(T,H) · Pr[(U,M) ∈ R | T,H]]]
≤ O( 1k2)µk(R)
≤ ε · µ3(R) + 2−Ω(n)
Splitting µ3(R)
µ3(R) = ET
[
E|H|=3
[Pr[(U,M) ∈ R | T,H]
]]
≤ ET
[
E|H|=3
[
GOOD(T,H) · Pr[(U,M) ∈ R | T,H]]]
+ ET
[
E|H|=3
[
M−BAD(T,H) · Pr[(U,M) ∈ R | T,H]]]
+ ET
[
E|H|=3
[
U−BAD(T,H) · Pr[(U,M) ∈ R | T,H]]]
≤ O( 1k2)µk(R)
≤ ε · µ3(R) + 2−Ω(n)
≤ ε · µ3(R) + 2−Ω(n)
Contribution of good partitions
For T
B1. . . BmA1
. . .
Am
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R):
B1. . . BmA1
. . .
AmF
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R):
B1. . . BmA1
. . .
AmF
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R): Pr[(U,M) ∈ R | T, F ]
B1. . . BmA1
. . .
AmF
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R): Pr[(U,M) ∈ R | T, F ] Contribution to µ3(R):
B1. . . BmA1
. . .
AmF
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R): Pr[(U,M) ∈ R | T, F ] Contribution to µ3(R):
E
H∼(F3)[GOOD(T,H)·Pr[(U,M) ∈ R | T,H]]
B1. . . BmA1
. . .
AmF
H
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R): Pr[(U,M) ∈ R | T, F ] Contribution to µ3(R):
E
H∼(F3)[GOOD(T,H)·Pr[(U,M) ∈ R | T,H]] . Pr[. . . | T, F ]· Pr
H∼(F3)[GOOD(T,H)]
︸ ︷︷ ︸
=O(1/k2)
B1. . . BmA1
. . .
AmF
H
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R): Pr[(U,M) ∈ R | T, F ] Contribution to µ3(R):
E
H∼(F3)[GOOD(T,H)·Pr[(U,M) ∈ R | T,H]] . Pr[. . . | T, F ]· Pr
H∼(F3)[GOOD(T,H)]
︸ ︷︷ ︸
=O(1/k2)
B1. . . BmA1
. . .
AmF
H
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R): Pr[(U,M) ∈ R | T, F ] Contribution to µ3(R):
E
H∼(F3)[GOOD(T,H)·Pr[(U,M) ∈ R | T,H]] . Pr[. . . | T, F ]· Pr
H∼(F3)[GOOD(T,H)]
︸ ︷︷ ︸
=O(1/k2)
Suffices to show: H,H∗ ⊆ F good ⇒ |H ∩H∗| ≥ 2
B1. . . BmA1
. . .
AmF
H
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R): Pr[(U,M) ∈ R | T, F ] Contribution to µ3(R):
E
H∼(F3)[GOOD(T,H)·Pr[(U,M) ∈ R | T,H]] . Pr[. . . | T, F ]· Pr
H∼(F3)[GOOD(T,H)]
︸ ︷︷ ︸
=O(1/k2)
Suffices to show: H,H∗ ⊆ F good ⇒ |H ∩H∗| ≥ 2
Suppose |H ∩H∗| ≤ 1B1
. . . BmA1
. . .
Am
H
H∗
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R): Pr[(U,M) ∈ R | T, F ] Contribution to µ3(R):
E
H∼(F3)[GOOD(T,H)·Pr[(U,M) ∈ R | T,H]] . Pr[. . . | T, F ]· Pr
H∼(F3)[GOOD(T,H)]
︸ ︷︷ ︸
=O(1/k2)
Suffices to show: H,H∗ ⊆ F good ⇒ |H ∩H∗| ≥ 2
Suppose |H ∩H∗| ≤ 1
(T,H) good⇒ ∃M : u, v ∈ M
B1. . . BmA1
. . .
Am
H
H∗u
v
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R): Pr[(U,M) ∈ R | T, F ] Contribution to µ3(R):
E
H∼(F3)[GOOD(T,H)·Pr[(U,M) ∈ R | T,H]] . Pr[. . . | T, F ]· Pr
H∼(F3)[GOOD(T,H)]
︸ ︷︷ ︸
=O(1/k2)
Suffices to show: H,H∗ ⊆ F good ⇒ |H ∩H∗| ≥ 2
Suppose |H ∩H∗| ≤ 1
(T,H) good⇒ ∃M : u, v ∈ M
(T,H∗) good⇒ ∃U : u, v ∈ U
B1. . . BmA1
. . .
Am
H
H∗u
v
Contribution of good partitions
For T and F ⊆ C ×D with |F | = k compare: Contribution to µk(R): Pr[(U,M) ∈ R | T, F ] Contribution to µ3(R):
E
H∼(F3)[GOOD(T,H)·Pr[(U,M) ∈ R | T,H]] . Pr[. . . | T, F ]· Pr
H∼(F3)[GOOD(T,H)]
︸ ︷︷ ︸
=O(1/k2)
Suffices to show: H,H∗ ⊆ F good ⇒ |H ∩H∗| ≥ 2
Suppose |H ∩H∗| ≤ 1
(T,H) good⇒ ∃M : u, v ∈ M
(T,H∗) good⇒ ∃U : u, v ∈ U
|δ(U) ∩M | = 1Contradiction!
B1. . . BmA1
. . .
Am
H
u
v
Most partitions are good
Lemma
Pr[(T,H) is M -bad] ≤ ε
Most partitions are good
Lemma
Pr[(T,H) is M -bad] ≤ ε
Pick H
H
Most partitions are good
Lemma
Pr[(T,H) is M -bad] ≤ ε
Pick H, A
A
A1
. . .
Am
H
Most partitions are good
Lemma
Pr[(T,H) is M -bad] ≤ ε
Pick H, A, B1, . . . , Bm+1.
A B1 B2. . . Bm+1
A1
. . .
Am
H
Most partitions are good
Lemma
Pr[(T,H) is M -bad] ≤ ε
Pick H, A, B1, . . . , Bm+1. Split Bi = Ci∪Di.
A B1 B2. . . Bm+1
C2
D2
. . .
. . .
Cm+1
Dm+1
A1
. . .
Am
H
Most partitions are good
Lemma
Pr[(T,H) is M -bad] ≤ ε
Pick H, A, B1, . . . , Bm+1. Split Bi = Ci∪Di. Pick randomly i ∈ 1, . . . ,m
A B1 B2. . . Bm+1
C2
D2
. . .
. . .
Cm+1
Dm+1
A1
. . .
Am
H
i
Most partitions are good
Lemma
Pr[(T,H) is M -bad] ≤ ε
Pick H, A, B1, . . . , Bm+1. Split Bi = Ci∪Di. Pick randomly i ∈ 1, . . . ,m and let C := Ci, D := Di
A C D B1. . . Bm
A1
. . .
Am
H
Open problems
Open problem
Show that there is no small SDP representing theCorrelation/TSP/matching polytope!
Open problems
Open problem
Show that there is no small SDP representing theCorrelation/TSP/matching polytope!
Thanks for your attention