Depth 4 lower bounds for elementary symmetric polynomials Nutan - - PowerPoint PPT Presentation

Depth 4 lower bounds for elementary symmetric polynomials Nutan Limaye

CSE, IITB Joint work with Herv´ e Fournier, Meena Mahajan and Srikanth Srinivasan Workshop on low depth complexity

• St. Petersburg, Russia, May, 2016

Elementary symmetric polynomials

Elemenraty symmetric polynomial of degree D on n variables. SD

n (X) :=

X

T⊆[n]:|T|=D

Y

i∈T

xi

Elementary symmetric polynomials

Elemenraty symmetric polynomial of degree D on n variables. SD

n (X) :=

X

T⊆[n]:|T|=D

Y

i∈T

xi where, X := {x1, x2, . . . , xn}.

Elementary symmetric polynomials

Elemenraty symmetric polynomial of degree D on n variables. SD

n (X) :=

X

T⊆[n]:|T|=D

Y

i∈T

xi where, X := {x1, x2, . . . , xn}. Polynomials which have polynomial sized circuits DET, IMM,

Elementary symmetric polynomials

Elemenraty symmetric polynomial of degree D on n variables. SD

n (X) :=

X

T⊆[n]:|T|=D

Y

i∈T

xi where, X := {x1, x2, . . . , xn}. Polynomials which have polynomial sized circuits DET, IMM, SD

n are all in VP

Elementary symmetric polynomials

Elemenraty symmetric polynomial of degree D on n variables. SD

n (X) :=

X

T⊆[n]:|T|=D

Y

i∈T

xi where, X := {x1, x2, . . . , xn}. Polynomials which have polynomial sized circuits DET, IMM, SD

n are all in VP, i.e. they have poly sized circuits.

Elementary symmetric polynomials

Elemenraty symmetric polynomial of degree D on n variables. SD

n (X) :=

X

T⊆[n]:|T|=D

Y

i∈T

xi where, X := {x1, x2, . . . , xn}. Polynomials which have polynomial sized circuits DET, IMM, SD

n are all in VP, i.e. they have poly sized circuits.

Recall: depth 3, depth 4 formulas

Depth 3 formulas X Y X

Recall: depth 3, depth 4 formulas

Depth 3 formulas X Y X Depth 4 formulas X Y X Y

Recall: depth 3, depth 4 formulas

Depth 3 formulas X Y X Depth 4 formulas X Y X Y Depth 4 formulas with fan-in bounds X Y [p] X Y [q]

Recall: depth 3, depth 4 formulas

Depth 3 formulas X Y X Depth 4 formulas X Y X Y Depth 4 formulas with fan-in bounds X Y [p] X Y [q] Homogeneous vs. inhomogeneous Degree of all the input polynomials to any P gate is the same.

Known results

Small inhomogeneous formulas exist For every D 2 N, there is a depth 3 inhomogeneous formula computing SD

n of size nO(1) [Ben-Or].

Known results

Small inhomogeneous formulas exist For every D 2 N, there is a depth 3 inhomogeneous formula computing SD

n of size nO(1) [Ben-Or].

Homogeneous depth 3 lower bound Any depth 3 homogeneous formula computing SD

n requires size

nΩ(D) [Nisan & Wigderson, 1997].

Known results

Small inhomogeneous formulas exist For every D 2 N, there is a depth 3 inhomogeneous formula computing SD

n of size nO(1) [Ben-Or].

Homogeneous depth 3 lower bound Any depth 3 homogeneous formula computing SD

n requires size

nΩ(D) [Nisan & Wigderson, 1997].

Natural questions

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)?

Natural questions

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)? Open.

Natural questions

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)? Open. Strong lower bounds for inhomonegeous depth 3 formulas

Natural questions

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)? Open. Strong lower bounds for inhomonegeous depth 3 formulas Does there exists an explicit f (X) 2 F[X] on n variables such that depth 3 inhomogeneous formula computing f requires size nω(1)?

Natural questions

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)? Open. Strong lower bounds for inhomonegeous depth 3 formulas Does there exists an explicit f (X) 2 F[X] on n variables such that depth 3 inhomogeneous formula computing f requires size nω(1)? Open.

Natural questions

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)? Open. Strong lower bounds for inhomonegeous depth 3 formulas Does there exists an explicit f (X) 2 F[X] on n variables such that depth 3 inhomogeneous formula computing f requires size nω(1)? Open.

Our result

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)?

Our result

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)? Open.

Our result

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)? Open. Our Result: Any depth 4 homogeneous ΣΠΣΠ[t] formula computing SD

n requires size nΩ(D/t).

Our result

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)? Open. Our Result: Any depth 4 homogeneous ΣΠΣΠ[t] formula computing SD

n requires size nΩ(D/t). For D = O(log n/ log log n).

Our result

Depth 4 homogeneous formulas for SD

n

Does SD

n have a depth 4 homogeneous formula of size

poly(n, D)? Open. Our Result: Any depth 4 homogeneous ΣΠΣΠ[t] formula computing SD

n requires size nΩ(D/t). For D = O(log n/ log log n).

Proving lower bounds

Notation Let p(X) be a polynomial over a field F. Let C be an arithmetic circuit of size s.

Proving lower bounds

Notation Let p(X) be a polynomial over a field F. Let C be an arithmetic circuit of size s. Goal To prove that if C computes p then s L.

Proving lower bounds

Notation Let p(X) be a polynomial over a field F. Let C be an arithmetic circuit of size s. Goal To prove that if C computes p then s L. To prove this

Proving lower bounds

Notation Let p(X) be a polynomial over a field F. Let C be an arithmetic circuit of size s. Goal To prove that if C computes p then s L. To prove this Design a function µ : F[X] ! R, such that

Proving lower bounds

Notation Let p(X) be a polynomial over a field F. Let C be an arithmetic circuit of size s. Goal To prove that if C computes p then s L. To prove this Design a function µ : F[X] ! R, such that

µ(C)  U · s

Proving lower bounds

Notation Let p(X) be a polynomial over a field F. Let C be an arithmetic circuit of size s. Goal To prove that if C computes p then s L. To prove this Design a function µ : F[X] ! R, such that

µ(C)  U · s µ(p) > L

Proving lower bounds

Notation Let p(X) be a polynomial over a field F. Let C be an arithmetic circuit of size s. Goal To prove that if C computes p then s L. To prove this Design a function µ : F[X] ! R, such that

µ(C)  U · s µ(p) > L Conclude that s L/U

Partial derivatives of SD

n

Notation

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1 Let R = {1, 2}. Then, ∂R(x1 + x2 + x3) = 0

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1 Let R = {1, 2}. Then, ∂R(x1 + x2 + x3) = 0 Let R = {1}. Then, ∂R(SD

n ) := ∂(SD

n )

∂x1

= SD−1

n−1 (x2, . . . , xn)

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1 Let R = {1, 2}. Then, ∂R(x1 + x2 + x3) = 0 Let R = {1}. Then, ∂R(SD

n ) := ∂(SD

n )

∂x1

= SD−1

n−1 (x2, . . . , xn)

Let ∂k(p) := {∂R(p) | R ✓ [n], |R| = k}.

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1 Let R = {1, 2}. Then, ∂R(x1 + x2 + x3) = 0 Let R = {1}. Then, ∂R(SD

n ) := ∂(SD

n )

∂x1

= SD−1

n−1 (x2, . . . , xn)

Let ∂k(p) := {∂R(p) | R ✓ [n], |R| = k}. The partial derivative measure: µk(p) := dim {span ∂k(p)}

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1 Let R = {1, 2}. Then, ∂R(x1 + x2 + x3) = 0 Let R = {1}. Then, ∂R(SD

n ) := ∂(SD

n )

∂x1

= SD−1

n−1 (x2, . . . , xn)

Let ∂k(p) := {∂R(p) | R ✓ [n], |R| = k}. The partial derivative measure: µk(p) := dim {span ∂k(p)}

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1 Let R = {1, 2}. Then, ∂R(x1 + x2 + x3) = 0 Let R = {1}. Then, ∂R(SD

n ) := ∂(SD

n )

∂x1

= SD−1

n−1 (x2, . . . , xn)

Let ∂k(p) := {∂R(p) | R ✓ [n], |R| = k}. The partial derivative measure: µk(p) := dim {span ∂k(p)} Lower bound of [Nisan & Wigderson, 1997] C be a ΣΠΣ homogeneous formula of size s computing S2d

n .

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1 Let R = {1, 2}. Then, ∂R(x1 + x2 + x3) = 0 Let R = {1}. Then, ∂R(SD

n ) := ∂(SD

n )

∂x1

= SD−1

n−1 (x2, . . . , xn)

Let ∂k(p) := {∂R(p) | R ✓ [n], |R| = k}. The partial derivative measure: µk(p) := dim {span ∂k(p)} Lower bound of [Nisan & Wigderson, 1997] C be a ΣΠΣ homogeneous formula of size s computing S2d

n .

µd(C)  s · 22d

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1 Let R = {1, 2}. Then, ∂R(x1 + x2 + x3) = 0 Let R = {1}. Then, ∂R(SD

n ) := ∂(SD

n )

∂x1

= SD−1

n−1 (x2, . . . , xn)

Let ∂k(p) := {∂R(p) | R ✓ [n], |R| = k}. The partial derivative measure: µk(p) := dim {span ∂k(p)} Lower bound of [Nisan & Wigderson, 1997] C be a ΣΠΣ homogeneous formula of size s computing S2d

n .

µd(C)  s · 22d µd(S2d

n )

n

d

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1 Let R = {1, 2}. Then, ∂R(x1 + x2 + x3) = 0 Let R = {1}. Then, ∂R(SD

n ) := ∂(SD

n )

∂x1

= SD−1

n−1 (x2, . . . , xn)

Let ∂k(p) := {∂R(p) | R ✓ [n], |R| = k}. The partial derivative measure: µk(p) := dim {span ∂k(p)} Lower bound of [Nisan & Wigderson, 1997] C be a ΣΠΣ homogeneous formula of size s computing S2d

n .

µd(C)  s · 22d µd(S2d

n )

n

d

• Therefore, s

n

d

• /22d

Partial derivatives of SD

n

Notation For R = {i1, i2, . . . , ik} ✓ [n], let ∂R(p) :=

∂k(p) ∂xi1∂xi2...∂xik

Example

Let R = {1}. Then, ∂R(x1 + x2 + x3) = 1 Let R = {1, 2}. Then, ∂R(x1 + x2 + x3) = 0 Let R = {1}. Then, ∂R(SD

n ) := ∂(SD

n )

∂x1

= SD−1

n−1 (x2, . . . , xn)

Let ∂k(p) := {∂R(p) | R ✓ [n], |R| = k}. The partial derivative measure: µk(p) := dim {span ∂k(p)} Lower bound of [Nisan & Wigderson, 1997] C be a ΣΠΣ homogeneous formula of size s computing S2d

n .

µd(C)  s · 22d µd(S2d

n )

n

d

• Therefore, s

n

d

• /22d

Partial Derivatives of SD

n

For A ✓ [n], let XA denote Q

i∈A xi.

Partial Derivatives of SD

n

For A ✓ [n], let XA denote Q

i∈A xi.

Let rS(X) := ∂S(SD

n ) = P |A|=D−k, A∩S=∅ XA

Partial Derivatives of SD

n

For A ✓ [n], let XA denote Q

i∈A xi.

Let rS(X) := ∂S(SD

n ) = P |A|=D−k, A∩S=∅ XA

D:= {rS(X) | S ✓ [n], |S| = k}

Partial Derivatives of SD

n

For A ✓ [n], let XA denote Q

i∈A xi.

Let rS(X) := ∂S(SD

n ) = P |A|=D−k, A∩S=∅ XA

D:= {rS(X) | S ✓ [n], |S| = k} Underlying matrix M

Partial Derivatives of SD

n

For A ✓ [n], let XA denote Q

i∈A xi.

Let rS(X) := ∂S(SD

n ) = P |A|=D−k, A∩S=∅ XA

D:= {rS(X) | S ✓ [n], |S| = k} Underlying matrix M Rows labelled by degree D k monomials Columns labelled by r 2 D

Partial Derivatives of SD

n

For A ✓ [n], let XA denote Q

i∈A xi.

Let rS(X) := ∂S(SD

n ) = P |A|=D−k, A∩S=∅ XA

D:= {rS(X) | S ✓ [n], |S| = k} Underlying matrix M Rows labelled by degree D k monomials Columns labelled by r 2 D Relabelling the rows and columns of M

Partial Derivatives of SD

n

For A ✓ [n], let XA denote Q

i∈A xi.

Let rS(X) := ∂S(SD

n ) = P |A|=D−k, A∩S=∅ XA

D:= {rS(X) | S ✓ [n], |S| = k} Underlying matrix M Rows labelled by degree D k monomials Columns labelled by r 2 D Relabelling the rows and columns of M Rows labelled by sets R of size D k

Partial Derivatives of SD

n

For A ✓ [n], let XA denote Q

i∈A xi.

Let rS(X) := ∂S(SD

n ) = P |A|=D−k, A∩S=∅ XA

D:= {rS(X) | S ✓ [n], |S| = k} Underlying matrix M Rows labelled by degree D k monomials Columns labelled by r 2 D Relabelling the rows and columns of M Rows labelled by sets R of size D k Columns labelled by a set S of size k

Partial Derivatives of SD

n

For A ✓ [n], let XA denote Q

i∈A xi.

Let rS(X) := ∂S(SD

n ) = P |A|=D−k, A∩S=∅ XA

D:= {rS(X) | S ✓ [n], |S| = k} Underlying matrix M Rows labelled by degree D k monomials Columns labelled by r 2 D Relabelling the rows and columns of M Rows labelled by sets R of size D k Columns labelled by a set S of size k M[R, S] = 1 if R \ S = ; and 0 otherwise.

Partial Derivatives of SD

n

For A ✓ [n], let XA denote Q

i∈A xi.

Let rS(X) := ∂S(SD

n ) = P |A|=D−k, A∩S=∅ XA

D:= {rS(X) | S ✓ [n], |S| = k} Underlying matrix M Rows labelled by degree D k monomials Columns labelled by r 2 D Relabelling the rows and columns of M Rows labelled by sets R of size D k Columns labelled by a set S of size k M[R, S] = 1 if R \ S = ; and 0 otherwise. µd(S2d

n ) =

n

d

• [Gottlieb, 1966]

The almost high rank matrix

Rows labelled by a tuple of sets (R1, . . . , Rτ)

The almost high rank matrix

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

The almost high rank matrix

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n],

The almost high rank matrix

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n], I (R1, . . . , Rτ) partition [n], and (S1, . . . , Sτ, T) partition [n],

The almost high rank matrix

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n], I (R1, . . . , Rτ) partition [n], and (S1, . . . , Sτ, T) partition [n], I |Si| |Si+1| for all i 2 [τ],

The almost high rank matrix

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n], I (R1, . . . , Rτ) partition [n], and (S1, . . . , Sτ, T) partition [n], I |Si| |Si+1| for all i 2 [τ], I |T| = k, 8i : 1  i < τ |Ri| = |Si| and |Rτ| = |Sτ| + k.

The almost high rank matrix

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n], I (R1, . . . , Rτ) partition [n], and (S1, . . . , Sτ, T) partition [n], I |Si| |Si+1| for all i 2 [τ], I |T| = k, 8i : 1  i < τ |Ri| = |Si| and |Rτ| = |Sτ| + k.

M[(R1, . . . , Rτ), (S1, . . . , Sτ, T)] = 1 if 8 > > > > > > > < > > > > > > > : T ✓ R1 S1 ✓ R1 [ R2 S2 ✓ R2 [ R3 . . . Sτ−1 ✓ Rτ−1 [ Rτ Sτ ✓ Rτ

The almost high rank matrix

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n], I (R1, . . . , Rτ) partition [n], and (S1, . . . , Sτ, T) partition [n], I |Si| |Si+1| for all i 2 [τ], I |T| = k, 8i : 1  i < τ |Ri| = |Si| and |Rτ| = |Sτ| + k.

M[(R1, . . . , Rτ), (S1, . . . , Sτ, T)] = 1 if 8 > > > > > > > < > > > > > > > : T ✓ R1 S1 ✓ R1 [ R2 S2 ✓ R2 [ R3 . . . Sτ−1 ✓ Rτ−1 [ Rτ Sτ ✓ Rτ

Lemma (Main technical lemma)

Rank(M) = min{#cols, #rows}(1 o(1))

Open problems

Some problems arising from the work For all D 2 [n], any depth 4 homogeneous ΣΠΣΠ[t] formula computing SD

n requires size nΩ(D/t).

Open problems

Some problems arising from the work For all D 2 [n], any depth 4 homogeneous ΣΠΣΠ[t] formula computing SD

n requires size nΩ(D/t).

Give an explicit f (X) 2 F[X] on n variables such that depth 3 inhomogeneous formula computing f require size nω(1)?

Open problems

Rows labelled by a tuple of sets (R1, . . . , Rτ)

Open problems

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

Open problems

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n],

Open problems

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n], I (R1, . . . , Rτ) partition [n], and (S1, . . . , Sτ, T) partition [n],

Open problems

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n], I (R1, . . . , Rτ) partition [n], and (S1, . . . , Sτ, T) partition [n], I |Si| = [n]/τ for all i 2 [τ], I |T| = k, 8i : 1  i < τ |Ri| = |Si| and |Rτ| = |Sτ| + k.

Open problems

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n], I (R1, . . . , Rτ) partition [n], and (S1, . . . , Sτ, T) partition [n], I |Si| = [n]/τ for all i 2 [τ], I |T| = k, 8i : 1  i < τ |Ri| = |Si| and |Rτ| = |Sτ| + k.

M[(R1, . . . , Rτ), (S1, . . . , Sτ, T)] = 1 if 8 > > > > > > > < > > > > > > > : T ✓ R1 S1 ✓ R1 [ R2 S2 ✓ R2 [ R3 . . . Sτ−1 ✓ Rτ−1 [ Rτ Sτ ✓ Rτ

Open problems

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n], I (R1, . . . , Rτ) partition [n], and (S1, . . . , Sτ, T) partition [n], I |Si| = [n]/τ for all i 2 [τ], I |T| = k, 8i : 1  i < τ |Ri| = |Si| and |Rτ| = |Sτ| + k.

M[(R1, . . . , Rτ), (S1, . . . , Sτ, T)] = 1 if 8 > > > > > > > < > > > > > > > : T ✓ R1 S1 ✓ R1 [ R2 S2 ✓ R2 [ R3 . . . Sτ−1 ✓ Rτ−1 [ Rτ Sτ ✓ Rτ Prove that the above matrix has rank min{#cols, #rows}(1 o(1))

Open problems

Rows labelled by a tuple of sets (R1, . . . , Rτ) Columns labelled by a tuple of sets (S1, . . . , Sτ, T), where

I Ri, Si, T ✓ [n], I (R1, . . . , Rτ) partition [n], and (S1, . . . , Sτ, T) partition [n], I |Si| = [n]/τ for all i 2 [τ], I |T| = k, 8i : 1  i < τ |Ri| = |Si| and |Rτ| = |Sτ| + k.

M[(R1, . . . , Rτ), (S1, . . . , Sτ, T)] = 1 if 8 > > > > > > > < > > > > > > > : T ✓ R1 S1 ✓ R1 [ R2 S2 ✓ R2 [ R3 . . . Sτ−1 ✓ Rτ−1 [ Rτ Sτ ✓ Rτ Prove that the above matrix has rank min{#cols, #rows}(1 o(1))

Thank You!