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