The matching polytope has exponential extension complexity Thomas - - PowerPoint PPT Presentation

The matching polytope has exponential extension complexity

Thomas Rothvoss

Department of Mathematics, UW Seattle

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 linear projection

Extended formulation

◮ Given polytope P = {x ∈ Rn | Ax ≤ b}

→ many inequalities

◮ Write P = {x ∈ Rn | ∃y : Bx + Cy ≤ d}

→ few inequalities P Q linear projection

Extended formulation

◮ Given polytope P = {x ∈ Rn | Ax ≤ b}

→ many inequalities

◮ Write P = {x ∈ Rn | ∃y : Bx + Cy ≤ d}

→ few inequalities P Q linear projection

◮ Extension complexity:

xc(P) := min   #facets of Q | Q polyhedron p linear map p(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 log n)-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 log n)-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 log n size matchings, but no symmetric one [Kaibel, Pashkovich & Theis ’10]

Lower bounds

◮ No symmetric compact form. for TSP [Yannakakis ’91]

Compact formulation for log n size matchings, but no symmetric 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 log n size matchings, but no symmetric 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 log n size matchings, but no symmetric 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 log n size matchings, but no symmetric 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(log n/ log log n)

[Chan, Lee, Raghavendra, Steurer ’13]

Lower bounds

◮ No symmetric compact form. for TSP [Yannakakis ’91]

Compact formulation for log n size matchings, but no symmetric 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(log n/ log log n)

[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 ≥ ∀e ∈ E G = (V, E) (complete)

Perfect matching polytope

x(δ(v)) = 1 ∀v ∈ V xe ≥ ∀e ∈ E

1 2 1 2 1 2 1 2 1 2 1 2

G = (V, E) (complete)

Perfect matching polytope

x(δ(v)) = 1 ∀v ∈ V xe ≥ ∀e ∈ E U

1 2 1 2 1 2 1 2 1 2 1 2

G = (V, E) (complete)

Perfect matching polytope

x(δ(v)) = 1 ∀v ∈ V x(δ(U)) ≥ 1 ∀U ⊆ V : |U| odd xe ≥ ∀e ∈ E U

1 2 1 2 1 2 1 2 1 2 1 2

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 ≥ ∀e ∈ E U

1 2 1 2 1 2 1 2 1 2 1 2

G = (V, E) (complete) Quick facts:

◮ Description by [Edmonds ’65] ◮ Can optimize cT x in strongly poly-time [Edmonds ’65]

Perfect matching polytope

x(δ(v)) = 1 ∀v ∈ V x(δ(U)) ≥ 1 ∀U ⊆ V : |U| odd xe ≥ ∀e ∈ E U

1 2 1 2 1 2 1 2 1 2 1 2

G = (V, E) (complete) Quick facts:

◮ Description by [Edmonds ’65] ◮ Can optimize cT x 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 ≥ ∀e ∈ E U

1 2 1 2 1 2 1 2 1 2 1 2

G = (V, E) (complete) Quick facts:

◮ Description by [Edmonds ’65] ◮ Can optimize cT x 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 ≥ ∀e ∈ E U

1 2 1 2 1 2 1 2 1 2 1 2

G = (V, E) (complete) Quick facts:

◮ Description by [Edmonds ’65] ◮ Can optimize cT x 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 Sij Sij = bi − AT

i xj

slack-matrix P

b b b b b

Slack-matrix

Write: P = conv({x1, . . . , xv}) = {x ∈ Rn | Ax ≤ b}

S

# facets # vertices facet i vertex j Sij Sij = bi − AT

i xj

slack-matrix P

b b b b b Aix = bi b

xj Sij

Slack-matrix

Write: P = conv({x1, . . . , xv}) = {x ∈ Rn | Ax ≤ b}

S

# facets # vertices

U ≥ V ≥ 0

r r Sij Sij = bi − AT

i xj

slack-matrix P

b b b b b Aix = bi b

xj Sij Non-negative rank: rk+(S) = min{r | ∃U ∈ Rf×r

≥0 , V ∈ Rr×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). Idea: Factor S = UV with U = (conic comb. to derive constraint i)i V = (slack vector of (xj, vj))j Q Aix + 0y ≤ bi

b b b b b b b b b b b b

xj

b

(xj, yj)

b

P

Hyperplane separation lower bound [Fiorini]

rk+(S) = min

• r : S =

r

• i=1

Ri and Ri ≥ 0 rank-1 matrix

• S

= + . . . + R1 Rr

1 1 1 1 2 2 1 2 2 3 0 3 0 0 0 3 0 3

Hyperplane separation lower bound [Fiorini]

rk+(S) = min

• r : S =

r

• i=1

λi

• ≤S∞

Ri and 0 ≤ Ri ≤ 1 rank-1 matrix

• S

= + . . . + R1 Rr

1 2 1 2 1 2 1 2

1 1

1 2

1 1

λ1 λr

1 0 1 0 0 0 1 0 1

Hyperplane separation lower bound [Fiorini]

rk+(S) min

• λ1 : S =

r

• i=1

λiRi and 0 ≤ Ri ≤ 1 rank-1 matrix

• S

= + . . . + R1 Rr

1 2 1 2 1 2 1 2

1 1

1 2

1 1

λ1 λr

1 0 1 0 0 0 1 0 1

Hyperplane separation lower bound [Fiorini]

rk+(S) min

• λ1 : S =

r

• i=1

λiRi and Ri ∈ {0, 1}f×v rank-1

• R

b b b b b

rectangles [0, 1]-rank-1 matrices

S

= + . . . + R1 Rr

0 0 0 0 1 1 0 1 1

λ1 λr

1 0 1 0 0 0 1 0 1

Hyperplane separation lower bound [Fiorini]

rk+(S) min

• λ1 : W, S =

r

• i=1

λi W, Ri and Ri rect.

• R

S

b b b b b

rectangles [0, 1]-rank-1 matrices W W, R ≤ α

S

= + . . . + R1 Rr

0 0 0 0 1 1 0 1 1

λ1 λr

1 0 1 0 0 0 1 0 1

Hyperplane separation lower bound [Fiorini]

rk+(S) min W, S W, R : R rectangle

• R

S

b b b b b

rectangles [0, 1]-rank-1 matrices W W, R ≤ α

S

= + . . . + R1 Rr

0 0 0 0 1 1 0 1 1

λ1 λr

1 0 1 0 0 0 1 0 1

Applying the Hyperplane bound

Goal: Find W with W,S

W,R large for each rectangle. ◮ Slack matrix SUM = |δ(U) ∩ M| − 1

matchings cuts

S

Applying the Hyperplane bound

Goal: Find W with W,S

W,R large for each rectangle. ◮ Slack matrix SUM = |δ(U) ∩ M| − 1

|δ(U) ∩ M| − 1 M U matchings cuts

S

Applying the Hyperplane bound

Goal: Find W with W,S

W,R large for each rectangle. ◮ Slack matrix SUM = |δ(U) ∩ M| − 1 ◮ Abbreviate Qℓ := {(U, M) : |δ(U) ∩ M| = ℓ} ◮ Uniform measure: µℓ(R) := |R∩Qℓ| |Qℓ|

2 2 2 Q1 Q3 matchings cuts

S

Applying the Hyperplane bound

Goal: Find W with W,S

W,R large for each rectangle. ◮ Slack matrix SUM = |δ(U) ∩ M| − 1 ◮ Abbreviate Qℓ := {(U, M) : |δ(U) ∩ M| = ℓ} ◮ Uniform measure: µℓ(R) := |R∩Qℓ| |Qℓ| ◮ Choose

WU,M =     0

• therwise.

2 2 2 Q1 Q3 matchings cuts

S

b

W

Applying the Hyperplane bound

Goal: Find W with W,S

W,R large for each rectangle. ◮ Slack matrix SUM = |δ(U) ∩ M| − 1 ◮ Abbreviate Qℓ := {(U, M) : |δ(U) ∩ M| = ℓ} ◮ Uniform measure: µℓ(R) := |R∩Qℓ| |Qℓ| ◮ Choose

WU,M =      − ∞ |δ(U) ∩ M| = 1

• therwise.

2 2 2 Q1 Q3 matchings cuts

S

b

W −∞ −∞ −∞

Applying the Hyperplane bound

Goal: Find W with W,S

W,R large for each rectangle. ◮ Slack matrix SUM = |δ(U) ∩ M| − 1 ◮ Abbreviate Qℓ := {(U, M) : |δ(U) ∩ M| = ℓ} ◮ Uniform measure: µℓ(R) := |R∩Qℓ| |Qℓ| ◮ Choose

WU,M =      − ∞ |δ(U) ∩ M| = 1

1 |Q3|

|δ(U) ∩ M| = 3

• therwise.

2 2 2 Q1 Q3 matchings cuts

S

b

W −∞ −∞ −∞

1 |Q3| 1 |Q3| 1 |Q3|

Applying the Hyperplane bound

Goal: Find W with W,S

W,R large for each rectangle. ◮ Slack matrix SUM = |δ(U) ∩ M| − 1 ◮ Abbreviate Qℓ := {(U, M) : |δ(U) ∩ M| = ℓ} ◮ Uniform measure: µℓ(R) := |R∩Qℓ| |Qℓ| ◮ Choose

WU,M =      − ∞ |δ(U) ∩ M| = 1

1 |Q3|

|δ(U) ∩ M| = 3

• therwise.

2 2 2 Q1 Q3 matchings cuts

S

b

W −∞ −∞ −∞

1 |Q3| 1 |Q3| 1 |Q3|

R

Rectangle covering for matching

Claim: There is a rectangle with W, R = Θ( 1

n4 ).

Rectangle covering for matching

Claim: There is a rectangle with W, R = Θ( 1

n4 ).

e1 e2

◮ For e1, e2 ∈ E:

Rectangle covering for matching

Claim: There is a rectangle with W, R = Θ( 1

n4 ).

U

e1 e2

◮ For e1, e2 ∈ E: take {U | e1, e2 ∈ δ(U)}

Rectangle covering for matching

Claim: There is a rectangle with W, R = Θ( 1

n4 ).

U

e1 e2 M

◮ For e1, e2 ∈ E: take {U | e1, e2 ∈ δ(U)} ×{M | e1, e2 ∈ M}

Rectangle covering for matching

Claim: There is a rectangle with W, R = Θ( 1

n4 ).

U

M e1 e2

◮ For e1, e2 ∈ E: take {U | e1, e2 ∈ δ(U)} ×{M | e1, e2 ∈ M} ◮ But µk(R) = Θ( k2 n4 )

Applying the Hyperplane bound (II)

Goal: Find W with W,S

W,R large for each rectangle. ◮ Choose

WU,M =            − ∞ |δ(U) ∩ M| = 1

1 |Q3|

|δ(U) ∩ M| = 3

• therwise.

2 2 2 Q1 Q3 matchings cuts

S

b

W −∞ −∞ −∞

1 |Q3| 1 |Q3| 1 |Q3|

Applying the Hyperplane bound (II)

Goal: Find W with W,S

W,R large for each rectangle. ◮ Choose

WU,M =            − ∞ |δ(U) ∩ M| = 1

1 |Q3|

|δ(U) ∩ M| = 3

• therwise.

2 2 2 Q1 Q3 Qk k − 1 k − 1 k − 1 matchings cuts

S

b

W −∞ −∞ −∞

1 |Q3| 1 |Q3| 1 |Q3|

Applying the Hyperplane bound (II)

Goal: Find W with W,S

W,R large for each rectangle. ◮ Choose

WU,M =            − ∞ |δ(U) ∩ M| = 1

1 |Q3|

|δ(U) ∩ M| = 3 −

1 k−1 · 1 |Qk|

|δ(U) ∩ M| = k

• therwise.

2 2 2 Q1 Q3 Qk k − 1 k − 1 k − 1 matchings cuts

S

b

W −∞ −∞ −∞

1 |Q3| 1 |Q3| 1 |Q3|

−

1 k−1 1 |Qk|

−

1 k−1 1 |Qk|

−

1 k−1 1 |Qk|

Applying the Hyperplane bound (II)

Goal: Find W with W,S

W,R large for each rectangle. ◮ Choose

WU,M =            − ∞ |δ(U) ∩ M| = 1

1 |Q3|

|δ(U) ∩ M| = 3 −

1 k−1 · 1 |Qk|

|δ(U) ∩ M| = k

• therwise.

◮ Then

W, S = 0 + 2 − 1 = 1

Lemma

For k large, any rectangle R has W, R ≤ 2−Ω(n). matchings cuts W −∞ −∞ −∞

1 |Q3| 1 |Q3| 1 |Q3|

−

1 k−1 1 |Qk|

−

1 k−1 1 |Qk|

−

1 k−1 1 |Qk|

R

Applying the Hyperplane bound (III)

Applying the Hyperplane bound (III)

Main lemma

µ1(R) = 0 = ⇒ µ3(R) ≤ O( 1

k2 ) · µk(R) + 2−Ω(n)

matchings cuts

S

Applying the Hyperplane bound (III)

Main lemma

µ1(R) = 0 = ⇒ µ3(R) ≤ O( 1

k2 ) · µk(R) + 2−Ω(n)

R matchings cuts

S

Applying the Hyperplane bound (III)

Main lemma

µ1(R) = 0 = ⇒ µ3(R) ≤ O( 1

k2 ) · µk(R) + 2−Ω(n)

R matchings cuts

S

Applying the Hyperplane bound (III)

Main lemma

µ1(R) = 0 = ⇒ µ3(R) ≤ O( 1

k2 ) · µk(R) + 2−Ω(n)

R matchings cuts

S

Applying the Hyperplane bound (III)

Main lemma

µ1(R) = 0 = ⇒ µ3(R) ≤ O( 1

k2 ) · µk(R) + 2−Ω(n)

R matchings cuts

S

Applying the Hyperplane bound (III)

Main lemma

µ1(R) = 0 = ⇒ µ3(R) ≤ O( 1

k2 ) · µk(R) + 2−Ω(n)

R matchings cuts

S

◮ Technique: Partition scheme [Razborov ’91]

Applying the Hyperplane bound (III)

Main lemma

µ1(R) = 0 = ⇒ µ3(R) ≤ O( 1

k2 ) · µk(R) + 2−Ω(n)

R

T

matchings cuts

S

◮ Technique: Partition scheme [Razborov ’91]

Partitions

R

T

matchings cuts

S

◮ Partition T = (A, C, D, B)

Partitions

R

T

matchings cuts

S

◮ Partition T = (A, C, D, B)

A

Partitions

R

T

matchings cuts

S

◮ Partition T = (A, C, D, B)

A B

Partitions

R

T

matchings cuts

S

◮ Partition T = (A, C, D, B)

A C B

k

Partitions

R

T

matchings cuts

S

◮ Partition T = (A, C, D, B)

A C D B

k k

Partitions

R

T

matchings cuts

S

◮ Partition T = (A, C, D, B)

A C D B A1 . . . Am

k − 3 nodes k k

Partitions

R

T

matchings cuts

S

◮ Partition T = (A, C, D, B)

A C D B B1 . . . Bm A1 . . . Am

k − 3 nodes k k 2(k − 3) nodes

Partitions

R

T

matchings cuts

S

◮ Partition T = (A, C, D, B) ◮ Edges E(T)

A C D B B1 . . . Bm A1 . . . Am

k − 3 nodes k k 2(k − 3) nodes

Partitions

R

T

matchings cuts

S

◮ Partition T = (A, C, D, B) ◮ Edges E(T)

A C D B B1 . . . Bm A1 . . . Am

k − 3 nodes k k 2(k − 3) nodes

Partitions

R

T

matchings cuts

S

◮ Partition T = (A, C, D, B) ◮ Edges E(T)

A C D B B1 . . . Bm A1 . . . Am

k − 3 nodes k k 2(k − 3) nodes U

Pseudo-random behaviour of large set systems

Imagine the following setting:

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements

b b b b b b

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Questions:

◮ Is it possible that ≥ 1% of elements are in no set at all?

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Questions:

◮ Is it possible that ≥ 1% of elements are in no set at all?

NO! The 0.99n active elements form at most 20.99n sets

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Questions:

◮ Is it possible that ≥ 1% of elements are in no set at all?

NO! The 0.99n active elements form at most 20.99n sets

◮ Is it possible that ≥ 1% elements are in ≤ 49% of sets?

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Questions:

◮ Is it possible that ≥ 1% of elements are in no set at all?

NO! The 0.99n active elements form at most 20.99n sets

◮ Is it possible that ≥ 1% elements are in ≤ 49% of sets?

NO!

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Questions:

◮ Is it possible that ≥ 1% of elements are in no set at all?

NO! The 0.99n active elements form at most 20.99n sets

◮ Is it possible that ≥ 1% elements are in ≤ 49% of sets?

NO! Proof:

◮ Take a random set from S

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Questions:

◮ Is it possible that ≥ 1% of elements are in no set at all?

NO! The 0.99n active elements form at most 20.99n sets

◮ Is it possible that ≥ 1% elements are in ≤ 49% of sets?

NO! Proof:

◮ Take a random set from S ◮ Denote char. vector as x ∈ {0, 1}n

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Questions:

◮ Is it possible that ≥ 1% of elements are in no set at all?

NO! The 0.99n active elements form at most 20.99n sets

◮ Is it possible that ≥ 1% elements are in ≤ 49% of sets?

NO! Proof:

◮ Take a random set from S ◮ Denote char. vector as x ∈ {0, 1}n

log |S| = H(x)

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Questions:

◮ Is it possible that ≥ 1% of elements are in no set at all?

NO! The 0.99n active elements form at most 20.99n sets

◮ Is it possible that ≥ 1% elements are in ≤ 49% of sets?

NO! Proof:

◮ Take a random set from S ◮ Denote char. vector as x ∈ {0, 1}n

log |S| = H(x)

subadd

≤

n

• i=1

H(xi)

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Questions:

◮ Is it possible that ≥ 1% of elements are in no set at all?

NO! The 0.99n active elements form at most 20.99n sets

◮ Is it possible that ≥ 1% elements are in ≤ 49% of sets?

NO! Proof:

◮ Take a random set from S ◮ Denote char. vector as x ∈ {0, 1}n

log |S| = H(x)

subadd

≤

n

• i=1

H(xi) ≤ n − Ω(n) 1 0.5 1.0 entropy p

Pseudo-random behaviour of large set systems

Imagine the following setting:

◮ n elements ◮ set system S with 2(1−o(1))n sets

b b b b b b

Questions:

◮ Is it possible that ≥ 1% of elements are in no set at all?

NO! The 0.99n active elements form at most 20.99n sets

◮ Is it possible that ≥ 1% elements are in ≤ 49% of sets?

NO!

Lemma

If S large, for most elements i, Pr

S⊆[n][S ∈ S] ≈ Pr S⊆[n][S ∈ S | i ∈ S]

Rewriting µk(R)

R

T

matchings cuts

S

Randomly generate (U, M) ∼ Qk: µk(R) =

Rewriting µk(R)

R

T

matchings cuts

S

A C D B B1 . . . Bm A1 . . . Am

Randomly generate (U, M) ∼ Qk:

• 1. Choose T

µk(R) = E

T

Rewriting µk(R)

R

T

matchings cuts

S

A C D B B1 . . . Bm A1 . . . Am F

Randomly generate (U, M) ∼ Qk:

• 1. Choose T
• 2. Choose k edges F ⊆ C × D

µk(R) = E

T

• E

|F|=k

Rewriting µk(R)

R

T

matchings cuts

S

A C D B B1 . . . Bm A1 . . . Am F

Randomly generate (U, M) ∼ Qk:

• 1. Choose T
• 2. Choose k edges F ⊆ C × D
• 3. Choose M ⊇ F

µk(R) = E

T

• E

|F|=k

• Pr[M ∈ R | T, H]

Rewriting µk(R)

R

T

matchings cuts

S

A C D B B1 . . . Bm

U

A1 . . . Am F

Randomly generate (U, M) ∼ Qk:

• 1. Choose T
• 2. Choose k edges F ⊆ C × D
• 3. Choose M ⊇ F
• 4. Choose U ⊇ C (not cutting any Ai)

µk(R) = E

T

• E

|F|=k

• Pr[M ∈ R | T, H] · Pr[U ∈ R | T, H]

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

B1 . . . Bm A1 . . . Am F

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

B1 . . . Bm A1 . . . Am F

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

B1 . . . Bm A1 . . . Am F

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

◮ Then

µ3(R) ≈ E

H∼(F

3)

[

≤O(1/k2)

Pr[(U, M) ∈ R | T, H]

• ≈p

]

B1 . . . Bm A1 . . . Am F H

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

◮ Then

µ3(R) ≈ E

H∼(F

3)

[GOOD(T, H)

• ≤O(1/k2)

· Pr[(U, M) ∈ R | T, H]

• ≈p

]

B1 . . . Bm A1 . . . Am F H

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

◮ Then

µ3(R) ≈ E

H∼(F

3)

[GOOD(T, H)

• ≤O(1/k2)

· Pr[(U, M) ∈ R | T, H]

• ≈p

]

◮ GOOD means it doesn’t matter what condition on here

B1 . . . Bm A1 . . . Am F H

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

◮ Then

µ3(R) ≈ E

H∼(F

3)

[GOOD(T, H)

• ≤O(1/k2)

· Pr[(U, M) ∈ R | T, H]

• ≈p

]

◮ GOOD means it doesn’t matter what condition on here

B1 . . . Bm A1 . . . Am F H

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

◮ Then

µ3(R) ≈ E

H∼(F

3)

[GOOD(T, H)

• ≤O(1/k2)

· Pr[(U, M) ∈ R | T, H]

• ≈p

]

◮ GOOD means it doesn’t matter what condition on here

B1 . . . Bm A1 . . . Am F H

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

◮ Then

µ3(R) ≈ E

H∼(F

3)

[GOOD(T, H)

• ≤O(1/k2)

· Pr[(U, M) ∈ R | T, H]

• ≈p

]

◮ GOOD means it doesn’t matter what condition on here ◮ Suffices to show: H, H∗ ⊆ F good ⇒ |H ∩ H∗| ≥ 2

B1 . . . Bm A1 . . . Am F H

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

◮ Then

µ3(R) ≈ E

H∼(F

3)

[GOOD(T, H)

• ≤O(1/k2)

· Pr[(U, M) ∈ R | T, H]

• ≈p

]

◮ GOOD means it doesn’t matter what condition on here ◮ Suffices to show: H, H∗ ⊆ F good ⇒ |H ∩ H∗| ≥ 2 ◮ Suppose |H ∩ H∗| ≤ 1

B1 . . . Bm A1 . . . Am H H∗

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

◮ Then

µ3(R) ≈ E

H∼(F

3)

[GOOD(T, H)

• ≤O(1/k2)

· Pr[(U, M) ∈ R | T, H]

• ≈p

]

◮ GOOD means it doesn’t matter what condition on here ◮ Suffices to show: H, H∗ ⊆ F good ⇒ |H ∩ H∗| ≥ 2 ◮ Suppose |H ∩ H∗| ≤ 1 ◮ (T, H) good

⇒ ∃M : {u, v} ∈ M

B1 . . . Bm A1 . . . Am H H∗

u v

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

◮ Then

µ3(R) ≈ E

H∼(F

3)

[GOOD(T, H)

• ≤O(1/k2)

· Pr[(U, M) ∈ R | T, H]

• ≈p

]

◮ GOOD means it doesn’t matter what condition on here ◮ 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 . . . Bm A1 . . . Am H H∗

u v

How does an average partition look like

◮ Suppose for a fixed (T, F):

µk(R) ≈ Pr[(U, M) ∈ R | T, F] =: p

◮ Then

µ3(R) ≈ E

H∼(F

3)

[GOOD(T, H)

• ≤O(1/k2)

· Pr[(U, M) ∈ R | T, H]

• ≈p

]

◮ GOOD means it doesn’t matter what condition on here ◮ 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| = 1

Contradiction!

B1 . . . Bm A1 . . . 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 the Correlation/TSP/matching polytope!

Open problems

Open problem

Show that there is no small SDP representing the Correlation/TSP/matching polytope!

Thanks for your attention