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

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 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). 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 extension

Q = {(x, y) | Bx + Cy ≤ d} Q

b 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 extension

Q = {(x, y) | Bx + Cy ≤ d} Q Aix + 0y ≤ bi

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

xj

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 extension

Q = {(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

xj

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 extension

Q = {(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

xj

b

(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 extension

Q = {(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

xj

b

(xj, yj)

b

P u(i), v(j) = u(i)T d

=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 2 0 0 2 1 0 2 1 2 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

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

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

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 e2 M 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

k

2

• 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

k

2

• 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

k

2

• 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

k

2

• rectangles

Question

Does every rectangle covering

• ver-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

k

2

• rectangles

Question

Does every rectangle covering

• ver-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

b b b b b

W W, R ≤ α rectangles

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 S

b b b b b

W W, R ≤ α rectangles

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=1 Ri 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 S

b b b b b

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

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

• therwise.

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

• therwise.

◮ 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| = 1

1 |Q3|

|δ(U) ∩ M| = 3

• therwise.

◮ 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| = 1

1 |Q3|

|δ(U) ∩ M| = 3 −

1 k−1 · 1 |Qk|

|δ(U) ∩ M| = k

• therwise.

◮ 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| = 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).

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( 1

k2 ) · 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( 1

k2 ) · 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( 1

k2 ) · 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( 1

k2 ) · 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( 1

k2 ) · 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( 1

k2 ) · 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( 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

Rewriting µ3(R)

R

T

matchings cuts

S

Randomly generate (U, M) ∼ Q3: µ3(R) =

Rewriting µ3(R)

R

T

matchings cuts

S

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

Randomly generate (U, M) ∼ Q3:

• 1. Choose T

µ3(R) = E

T

Rewriting µ3(R)

R

T

matchings cuts

S

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

Randomly generate (U, M) ∼ Q3:

• 1. Choose T
• 2. Choose 3 edges H ⊆ C × D

µ3(R) = E

T

• E

|H|=3

Rewriting µ3(R)

R

T

matchings cuts

S

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

Randomly generate (U, M) ∼ Q3:

• 1. Choose T
• 2. Choose 3 edges H ⊆ C × D
• 3. Choose M ⊇ H (not cutting any other edge in C × D)

µ3(R) = E

T

• E

|H|=3

Rewriting µ3(R)

R

T

matchings cuts

S

A C D B B1 . . . Bm

U

A1 . . . Am H

Randomly generate (U, M) ∼ Q3:

• 1. Choose T
• 2. Choose 3 edges H ⊆ C × D
• 3. Choose M ⊇ H (not cutting any other edge in C × D)
• 4. Choose U cutting H (not cutting any Ai)

µ3(R) = E

T

• E

|H|=3

• Pr[(U, M) ∈ R | T, H]

Rewriting µ3(R)

R

T

matchings cuts

S

A C D B B1 . . . Bm

U

A1 . . . Am H

Randomly generate (U, M) ∼ Q3:

• 1. Choose T
• 2. Choose 3 edges H ⊆ C × D
• 3. Choose M ⊇ H (not cutting any other edge in C × D)
• 4. Choose U cutting H (not cutting any Ai)

µ3(R) = E

T

• E

|H|=3

• Pr[U ∈ R | T, H] · Pr[M ∈ R | T, H]

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]

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 1 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 1 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 Pr

x∼[q]n[x ∈ X] ≈

Pr

x∼[q]n[x ∈ X | xi = j]

M-good

Definition

(T, H) M-good if M ∼ {M ∈ R | H ⊆ M ⊆ E(T)} is ε-uniform

• n (C ∪ D)\V (H).

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

M-good

Definition

(T, H) M-good if M ∼ {M ∈ R | H ⊆ M ⊆ E(T)} is ε-uniform

• n (C ∪ D)\V (H).

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

M-good

Definition

(T, H) M-good if M ∼ {M ∈ R | H ⊆ M ⊆ E(T)} is ε-uniform

• n (C ∪ D)\V (H).

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

U-good A C D B B1 . . . Bm A1 . . . Am H

U-good

c

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

U-good

Definition

(T, H) U-good if U ∼ {U ∈ R | c ⊆ U; doesn’t cut any Ai} has Pr[U ∩ C = c] ≈ 1

2 ≈ Pr[U ∩ C = C].

c

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

U-good

Definition

(T, H) U-good if U ∼ {U ∈ R | c ⊆ U; doesn’t cut any Ai} has Pr[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} has Pr[U ∩ C = c] ≈ 1

2 ≈ Pr[U ∩ C = C].

c

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

Splitting µ3(R)

µ3(R)

Splitting µ3(R)

µ3(R) = E

T

• E

|H|=3

• Pr[(U, M) ∈ R | T, H]

Splitting µ3(R)

µ3(R) = E

T

• E

|H|=3

• Pr[(U, M) ∈ R | T, H]
• ≤

E

T

• E

|H|=3

• GOOD(T, H) · Pr[(U, M) ∈ R | T, H]
• + E

T

• E

|H|=3

• M−BAD(T, H) · Pr[(U, M) ∈ R | T, H]
• + E

T

• E

|H|=3

• U−BAD(T, H) · Pr[(U, M) ∈ R | T, H]

Splitting µ3(R)

µ3(R) = E

T

• E

|H|=3

• Pr[(U, M) ∈ R | T, H]
• ≤

E

T

• E

|H|=3

• GOOD(T, H) · Pr[(U, M) ∈ R | T, H]
• + E

T

• E

|H|=3

• M−BAD(T, H) · Pr[(U, M) ∈ R | T, H]
• + E

T

• E

|H|=3

• U−BAD(T, H) · Pr[(U, M) ∈ R | T, H]
• ≤

O ( 1

k2 )µk(R)

Splitting µ3(R)

µ3(R) = E

T

• E

|H|=3

• Pr[(U, M) ∈ R | T, H]
• ≤

E

T

• E

|H|=3

• GOOD(T, H) · Pr[(U, M) ∈ R | T, H]
• + E

T

• E

|H|=3

• M−BAD(T, H) · Pr[(U, M) ∈ R | T, H]
• + E

T

• E

|H|=3

• U−BAD(T, H) · Pr[(U, M) ∈ R | T, H]
• ≤

O ( 1

k2 )µk(R)

≤ ε · µ3 ( R ) + 2−Ω(n)

Splitting µ3(R)

µ3(R) = E

T

• E

|H|=3

• Pr[(U, M) ∈ R | T, H]
• ≤

E

T

• E

|H|=3

• GOOD(T, H) · Pr[(U, M) ∈ R | T, H]
• + E

T

• E

|H|=3

• M−BAD(T, H) · Pr[(U, M) ∈ R | T, H]
• + E

T

• E

|H|=3

• U−BAD(T, H) · Pr[(U, M) ∈ R | T, H]
• ≤

O ( 1

k2 )µk(R)

≤ ε · µ3 ( R ) + 2−Ω(n) ≤ ε · µ3 ( R ) + 2−Ω(n)

Contribution of good partitions

For T

B1 . . . Bm A1 . . . Am

Contribution of good partitions

For T and F ⊆ C × D with |F| = k compare:

◮ Contribution to µk(R):

B1 . . . Bm A1 . . . Am F

Contribution of good partitions

For T and F ⊆ C × D with |F| = k compare:

◮ Contribution to µk(R):

B1 . . . Bm A1 . . . Am F

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 . . . Bm A1 . . . Am F

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 . . . Bm A1 . . . Am F

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∼(F

3)

[GOOD(T, H)·Pr[(U, M) ∈ R | T, H]]

B1 . . . Bm A1 . . . Am F 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∼(F

3)

[GOOD(T, H)·Pr[(U, M) ∈ R | T, H]] Pr[. . . | T, F]· Pr

H∼(F

3)

[GOOD(T, H)]

• =O(1/k2)

B1 . . . Bm A1 . . . Am F 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∼(F

3)

[GOOD(T, H)·Pr[(U, M) ∈ R | T, H]] Pr[. . . | T, F]· Pr

H∼(F

3)

[GOOD(T, H)]

• =O(1/k2)

B1 . . . Bm A1 . . . Am F 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∼(F

3)

[GOOD(T, H)·Pr[(U, M) ∈ R | T, H]] Pr[. . . | T, F]· Pr

H∼(F

3)

[GOOD(T, H)]

• =O(1/k2)

◮ Suffices to show: H, H∗ ⊆ F good ⇒ |H ∩ H∗| ≥ 2

B1 . . . Bm A1 . . . Am F 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∼(F

3)

[GOOD(T, H)·Pr[(U, M) ∈ R | T, H]] Pr[. . . | T, F]· Pr

H∼(F

3)

[GOOD(T, H)]

• =O(1/k2)

◮ Suffices to show: H, H∗ ⊆ F good ⇒ |H ∩ H∗| ≥ 2 ◮ Suppose |H ∩ H∗| ≤ 1

B1 . . . Bm A1 . . . 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∼(F

3)

[GOOD(T, H)·Pr[(U, M) ∈ R | T, H]] Pr[. . . | T, F]· Pr

H∼(F

3)

[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 . . . Bm A1 . . . 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∼(F

3)

[GOOD(T, H)·Pr[(U, M) ∈ R | T, H]] Pr[. . . | T, F]· Pr

H∼(F

3)

[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 . . . Bm A1 . . . 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∼(F

3)

[GOOD(T, H)·Pr[(U, M) ∈ R | T, H]] Pr[. . . | T, F]· Pr

H∼(F

3)

[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| = 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