Characterization & Lower Bounds for Branching Program Size

Characterization &Lower Bounds for Branching Program Size

using Projective Dimension

Krishnamoorthy Dinesh1 Sajin Koroth1 Jayalal Sarma1

1Department of Computer Science and EngineeringIIT Madras

FSTTCS 2016

1 / 18

Deterministic Branching Programs

Branching programs as a computational model.

x1 x2 x3 x4

x2 x3 x4

R

A

0 0 0 0

0 0 0

1 1 1 1

1 1

1

1

PARITY4 = x1 ⊕ x2 ⊕ x3 ⊕ x4

Size of a branching program computing f – bpsize(f )

I s(n) space Turing machine → 2O(s(n)) size branching program

I Best known lower bound : Nechiporuk (1980) –

∃fn bpsize(fn) = Ω

(n2

(log n)2

)

2 / 18



x1 x2 x3 x4

x2 x3 x4

R

A

0 0 0 0

0 0 0

1 1 1 1

1 1

1

1

PARITY4 = x1 ⊕ x2 ⊕ x3 ⊕ x4





(n2

(log n)2

)

2 / 18



x1 x2 x3 x4

x2 x3 x4

R

A

0 0 0 0

0 0 0

1 1 1 1

1 1

1

1

PARITY4 = x1 ⊕ x2 ⊕ x3 ⊕ x4





(n2

(log n)2

)

2 / 18



x1 x2 x3 x4

x2 x3 x4

R

A

0 0 0 0

0 0 0

1 1 1 1

1 1

1

1

PARITY4 = x1 ⊕ x2 ⊕ x3 ⊕ x4





(n2

(log n)2

)

2 / 18

Projective Dimension

I Graph G (U,V ,E ). Assign subspaces from Fd to vertices sothat ∀(x , y)

(x , y) ∈ E ⇐⇒ φ(x) ∩ φ(y) 6= 0

I Smallest such d : pdF(G ).

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(x , y) ∈ E ⇐⇒ φ(x) ∩ φ(y) 6= 0


G3 / 18



(x , y) ∈ E ⇐⇒ φ(x) ∩ φ(y) 6= 0


G

Spane1

Spane1

Spane1

Spane1

Spane2

Spane2

Spane2

Spane2

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(x , y) ∈ E ⇐⇒ φ(x) ∩ φ(y) 6= 0

I Smallest such d : pdF(G ). pdF(G ) = 2

00 00

01 01

10 10

11 11

x1x2 x3x4

G 3 / 18

Connecting Projective Dimension to Branching Programs(Pudlak and Rodl (1992))

Theorem Over any F, pdF(Gf ) ≤ bpsize(f )Demonstration.

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Theorem Over any F, pdF(Gf ) ≤ bpsize(f )

Demonstration.

4 / 18



4 / 18



00

01

10

11

00

01

10

11

f(x1, x2, x3, x4) = x1 ⊕ x2 ⊕ x3 ⊕ x4

Gf

1. Attach A node to start and assign variable.

2. Assign standard basis vectors to each vertex, difference to each edge.

3. Take span of vectors assigned on the edges reading a variable.

x1 x2 x3 x4

x2 x3 x4

R

A

0 0 0 0

0 0 0

1 1 1 1

1 1

1

1

Branching program computing f = PARITY4

x1x2 x3x4

4 / 18



00

01

10

11

00

01

10

11

f(x1, x2, x3, x4) = x1 ⊕ x2 ⊕ x3 ⊕ x4

Gf

x1 x2 x3 x4

x2 x3 x4

R

A

0 0 0 0

0 0 0

1 1 1 1

1 1

1

1


x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11




x1x2 x3x4

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00

01

10

11

00

01

10

11

f(x1, x2, x3, x4) = x1 ⊕ x2 ⊕ x3 ⊕ x4

Gf

x1 x2 x3 x4

x2 x3 x4

R

A

0 0 0 0

0 0 0

1 1 1 1

1 1

1

1


x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1




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00

01

10

11

00

01

10

11

x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9

f(x1, x2, x3, x4) = x1 ⊕ x2 ⊕ x3 ⊕ x4

Gf

Modified graph giving subspace assignment for Gf




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00

01

10

11

00

01

10

11

f(x1, x2, x3, x4) = x1 ⊕ x2 ⊕ x3 ⊕ x4

Gf

e2 − e3,e3 − e4, e7 − e8

x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9Modified graph giving subspace assignment for Gf




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00

01

10

11

00

01

10

11

f(x1, x2, x3, x4) = x1 ⊕ x2 ⊕ x3 ⊕ x4

Gf

e2 − e3,e3 − e4, e7 − e8

x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6





e1 − e2, e5 − e1, e9 − e6e4 − e5, e8 − e9,

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00

01

10

11

00

01

10

11

f(x1, x2, x3, x4) = x1 ⊕ x2 ⊕ x3 ⊕ x4

Gf

e1 − e2, e5 − e1, e9 − e6e4 − e5, e8 − e9,

e2 − e3,e3 − e4, e7 − e8

x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6





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Known Bounds on pdF

I (Existential) N vertex bipartite G such that

pdF(G ) Field Result

Ω(√

N)

Finite Pudlak and Rodl, 1992

Ω(√

Nlog N

)Infinite Babai et.al, 2002

I (Explicit LB, Pudlak and Rodl, 1992) G = Complement of Nperfect matchings.

pdR(G ) = Ω(logN)

There exists f with pdF(f ) = O(n) while bpsize(f ) = 2Ω(n).

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Known Bounds on pdF



Ω(√

N)


Ω(√

Nlog N



pdR(G ) = Ω(logN)


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Known Bounds on pdF



Ω(√

N)


Ω(√

Nlog N



pdR(G ) = Ω(logN)


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Known Bounds on pdF



Ω(√

N)


Ω(√

Nlog N



pdR(G ) = Ω(logN)


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Starting Thought

I Dimension of Intersecting spaces is 1 !

Projective Dimension with intersection dimension 1 (updF(G ))

Graph G (U,V ,E ). Assign subspaces from Fd to vertices so that

(x , y) ∈ E =⇒ dim(φ(x) ∩ φ(y)) = 1

(x , y) 6∈ E =⇒ dim(φ(x) ∩ φ(y)) = 0

Smallest such d : updF(f ).

Question : Is there a gap between pd and upd ?

6 / 18

Starting Thought




(x , y) ∈ E =⇒ dim(φ(x) ∩ φ(y)) = 1

(x , y) 6∈ E =⇒ dim(φ(x) ∩ φ(y)) = 0



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Starting Thought




(x , y) ∈ E =⇒ dim(φ(x) ∩ φ(y)) = 1

(x , y) 6∈ E =⇒ dim(φ(x) ∩ φ(y)) = 0



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Result : Gap Between pd and upd

TheoremFor f : 0, 1n × 0, 1n → 0, 1, |F| = q, Mf be bipartite

adjacency matrix of Gf rankR(Mf ) = qO(updF(f ))

I Proof idea : All (x , y) such that intersection space is the samenon-zero v forms a rectangle.

I Cannot give a super linear lower bound.

Remark Above result has been generalised to give dimensionlower bounds for family of intersecting subspaces.

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Denote SId : Fd×d2 × Fd×d

2 → 0, 1,

SId(A,B) = 1 iff rowspan(A) ∩ rowspan(B) 6= 0

rankR(MSId ) = 2Ω(d2)

I Follows from (Frankl and Wilson 1986) and (Lv and Wang 2012)

updF(SId) = Ω(d2) pdF(SId) ≤ d

I upd lower bound is still linear in input size !

Lemma (bad news)

∃ f such that updF(f ) = O(n), bpsize(f ) = 2Ω(n)

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2 → 0, 1,






Lemma (bad news)


8 / 18



2 → 0, 1,






Lemma (bad news)


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2 → 0, 1,




updF(SId) = Ω(d2)

pdF(SId) ≤ d


Lemma (bad news)


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2 → 0, 1,






Lemma (bad news)


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2 → 0, 1,






Lemma (bad news)


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2 → 0, 1,






Lemma (bad news)


8 / 18

More Observations on Pudlak & Rodl assignment

00

11

e2 − e3,e3 − e4, e7 − e8

x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9

x1x2

U01 U0

2

Spane2 − e3 Spane3 − e4, e7 − e8

L = Uai | a ∈ 0, 1, i ∈ [n] ,R = V a

i | a ∈ 0, 1, i ∈ [n]

I φ(x) = spani∈[n] Uxii , (subspaces over F2)

I span(i,a)∈S1Ua

i ∩ span(j,b)∈S2Ub

j = 0 for disjoint S1,S2

I Difference of standard basis vectors as basis

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00

11

e2 − e3,e3 − e4, e7 − e8

x1x2

U01 U0

2


x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9

L = Uai | a ∈ 0, 1, i ∈ [n] ,R = V a

i | a ∈ 0, 1, i ∈ [n]


I span(i,a)∈S1Ua




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00

11

x1x2


x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9

U01 + U0

2

U01 U0

2

L = Uai | a ∈ 0, 1, i ∈ [n] ,R = V a

i | a ∈ 0, 1, i ∈ [n]


I span(i,a)∈S1Ua




9 / 18


00

11

x1x2


x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9

U01 + U0

2

U01 U0

2 U11 U1

2


e2 − e7,e3 − e8, e7 − e4

L = Uai | a ∈ 0, 1, i ∈ [n] ,R = V a

i | a ∈ 0, 1, i ∈ [n]


I span(i,a)∈S1Ua




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00

11

x1x2


x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9

U01 + U0

2

U01 U0

2 U11 U1

2


U11 + U1

2

L = Uai | a ∈ 0, 1, i ∈ [n] ,R = V a

i | a ∈ 0, 1, i ∈ [n]


I span(i,a)∈S1Ua




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00

11

x1x2


x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9

U01 + U0

2

U01 U0

2 U11 U1

2


U11 + U1

2

L = Uai | a ∈ 0, 1, i ∈ [n] ,R = V a

i | a ∈ 0, 1, i ∈ [n]


I span(i,a)∈S1Ua




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00

11

x1x2


x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9

U01 + U0

2

U01 U0

2 U11 U1

2


U11 + U1

2

L = Uai | a ∈ 0, 1, i ∈ [n] ,R = V a

i | a ∈ 0, 1, i ∈ [n]


I span(i,a)∈S1Ua




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00

11

x1x2


x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9

U01 + U0

2

U01 U0

2 U11 U1

2


U11 + U1

2

L = Uai | a ∈ 0, 1, i ∈ [n] ,R = V a

i | a ∈ 0, 1, i ∈ [n]


I span(i,a)∈S1Ua




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00

11

x1x2


x1 x2 x3 x4

x2 x3 x4

R0 0 0 0

0 0 0

1 1 1 11 1

11

x30

1e1 e2 e3 e4 e5 e6

e7 e8 e9

U01 + U0

2

U01 U0

2 U11 U1

2


U11 + U1

2

L = Uai | a ∈ 0, 1, i ∈ [n] ,R = V a

i | a ∈ 0, 1, i ∈ [n]


I span(i,a)∈S1Ua




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Bitwise Decomposable Projective Dimension

Call such assignments as Bitwise Decomposable ProjectiveDimension (bitpdim)

bitpdim(f ) = O(bpsize(f )) bpsize(f ) ≤ O(bitpdim(f ))5

Nechiporuk’s methods does not give non-trivial lower bound onbitpdim.

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bitpdim(f ) = O(bpsize(f ))

bpsize(f ) ≤ O(bitpdim(f ))5


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10 / 18





10 / 18

Main Result : Gap Between the Measures

TheoremFor the function SId : Fd×d

2 × Fd×d2 → 0, 1,


bitpdim(SId) = Ω

(d3

log d

)updF(SId) = Ω(d2)

pdF(SId) = d

11 / 18

Main Result : Gap Between the Measures

TheoremFor the function SId : Fd×d

2 × Fd×d2 → 0, 1,


bitpdim(SId) = Ω

(d3

log d

)updF(SId) = Ω(d2)

pdF(SId) = d

11 / 18

Lower Bounds for bitpdim

TheoremFor f : 0, 1n × 0, 1n → 0, 1,

I Partition first n variables into non-empty T1,T2, . . . ,Tm,I ci (f ) = number of sub-functions of f restricted to Ti ,

then

bitpdim(f ) ≥m∑i=1

log ci (f )

log(log ci (f ))

Corollary : bitpdim(SId) = Ω(

d3

log d

)Our proof is a linear algebraic adapation of Nechiporuk’s method.

12 / 18


TheoremFor f : 0, 1n × 0, 1n → 0, 1,

I Partition first n variables into non-empty T1,T2, . . . ,Tm,

I ci (f ) = number of sub-functions of f restricted to Ti ,

then


log ci (f )

log(log ci (f ))


d3

log d


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TheoremFor f : 0, 1n × 0, 1n → 0, 1,


then


log ci (f )

log(log ci (f ))


d3

log d


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TheoremFor f : 0, 1n × 0, 1n → 0, 1,


then


log ci (f )

log(log ci (f ))


d3

log d


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TheoremFor f : 0, 1n × 0, 1n → 0, 1,


then


log ci (f )

log(log ci (f ))


d3

log d

)

Our proof is a linear algebraic adapation of Nechiporuk’s method.

12 / 18


TheoremFor f : 0, 1n × 0, 1n → 0, 1,


then


log ci (f )

log(log ci (f ))


d3

log d


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Lower Bounds for bitpdim : Proof I

I # bitpdim assignments is lower bounded by # ofsubfunctions.

Count subfunction separately.

I Challenge : upper bound # of bitpdim assignments possiblefor a given restriction.

ρ =

∗ ∗ ∗ . . .1 0 0 . . .0 0 0 . . ....

......

. . .

,1 1 0 . . .0 1 1 . . .0 1 1 . . ....

......

. . .

x y

T

I Let ρ be restriction s.t. fρ 6≡ const. ∀ ρ bitpdim(fρ) ≤ di .

I bitpdim(f ) =∑m

i=1 diI Aim : Obtain a bitpdim assignment for fρ of small dimension.

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I # bitpdim assignments is lower bounded by # ofsubfunctions. Count subfunction separately.


ρ =

∗ ∗ ∗ . . .1 0 0 . . .0 0 0 . . ....

......

. . .

,1 1 0 . . .0 1 1 . . .0 1 1 . . ....

......

. . .

x y

T


I bitpdim(f ) =∑m


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ρ =

∗ ∗ ∗ . . .1 0 0 . . .0 0 0 . . ....

......

. . .

,1 1 0 . . .0 1 1 . . .0 1 1 . . ....

......

. . .

x y

T


I bitpdim(f ) =∑m


13 / 18




ρ =

∗ ∗ ∗ . . .1 0 0 . . .0 0 0 . . ....

......

. . .

,1 1 0 . . .0 1 1 . . .0 1 1 . . ....

......

. . .

x y

T


I bitpdim(f ) =∑m


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ρ =

∗ ∗ ∗ . . .1 0 0 . . .0 0 0 . . ....

......

. . .

,1 1 0 . . .0 1 1 . . .0 1 1 . . ....

......

. . .

x y

T


I bitpdim(f ) =∑m


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ρ =

∗ ∗ ∗ . . .1 0 0 . . .0 0 0 . . ....

......

. . .

,1 1 0 . . .0 1 1 . . .0 1 1 . . ....

......

. . .

x y

T


I bitpdim(f ) =∑m

i=1 di

I Aim : Obtain a bitpdim assignment for fρ of small dimension.

13 / 18




ρ =

∗ ∗ ∗ . . .1 0 0 . . .0 0 0 . . ....

......

. . .

,1 1 0 . . .0 1 1 . . .0 1 1 . . ....

......

. . .

x y

T


I bitpdim(f ) =∑m


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Lower Bounds for bitpdim : Proof II

ρ =

∗ ∗ ∗ . . .1 0 0 . . .0 0 0 . . ....

......

. . .

,1 1 0 . . .0 1 1 . . .0 1 1 . . ....

......

. . .

x y

T

L+ Za1

L+ Za2

a1

a2

fρ

R

V

I Obs. L ∩ R = 0. Z = Spanx∈V Z x.I We show that there is a bitpdim assignment for fρ from Z .

I Claim :∏

Z (R) = Spanx∈V ∏

Z x (R)I For two restrictions ρ, ρ′, if

∏Z (Rρ) =

∏Z (Rρ′), then bitpdim

assignments are the same.

I Suffice to count different∏

Z (R)

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ρ =

∗ ∗ ∗ . . .1 0 0 . . .0 0 0 . . ....

......

. . .

,1 1 0 . . .0 1 1 . . .0 1 1 . . ....

......

. . .

x y

T

L+ Za1

L+ Za2

a1

a2

fρ

R

V

I Obs. L ∩ R = 0.

Z = Spanx∈V Z x.I We show that there is a bitpdim assignment for fρ from Z .

I Claim :∏



∏Z (Rρ) =




Z (R)

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L+ Za1

L+ Za2

a1

a2

fρ

R

V

I Obs. L ∩ R = 0. Z = Spanx∈V Z x.

I We show that there is a bitpdim assignment for fρ from Z .

I Claim :∏



∏Z (Rρ) =




Z (R)

14 / 18


L+ Za1

L+ Za2

a1

a2

fρ

R

V


I Claim :∏



∏Z (Rρ) =




Z (R)

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Za1

Za2

a1

a2

fρ

∏ZR

V

Z


I Claim :∏


Z x (R)

I For two restrictions ρ, ρ′, if∏

Z (Rρ) =∏

Z (Rρ′), then bitpdimassignments are the same.


Z (R)

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Za1

Za2

a1

a2

fρ

∏ZR

V

Z


I Claim :∏



∏Z (Rρ) =




Z (R)

14 / 18


Za1

Za2

a1

a2

fρ

∏ZR

V

Z


I Claim :∏



∏Z (Rρ) =




Z (R)

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Lower Bounds for bitpdim : Proof III

LemmaLet S = eu − ev | eu − ev ∈ Z. There ∃ linearly independentS ′ ⊆ S such that ∏

Z (R) = SpanS ′

I Suffices to count # of S ′ of size at most di which is2O(di log di ). This is at least ci (f ).

I Hence di = Ω(

log ci (f )log(log ci (f ))

).

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Z (R) = SpanS ′

I Suffices to count # of S ′ of size at most di which is2O(di log di ).

This is at least ci (f ).

I Hence di = Ω(


).

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Z (R) = SpanS ′


I Hence di = Ω(


).

15 / 18



Z (R) = SpanS ′


I Hence di = Ω(


).

15 / 18

Strongly Restricted Variants of Projective Dimension

I Standard Projective Dimension (spdF(f )) All subspace

assignment must be spanned by standard basis vectors

I Standard Projective Dimension with Intersection Dimension 1(uspdF(f ))

Related to Bipartite Cover number (bc(G )) and Bipartite Partitionnumber (bp(G ))

bc(Gf ) = spdF(f ) bp(Gf ) = uspdF(f )

No known computational model corresponding to spd(f ) anduspd(f )

16 / 18








16 / 18








16 / 18








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bc(Gf ) = spdF(f )

bp(Gf ) = uspdF(f )


16 / 18








16 / 18








16 / 18

Summary

I New parameters upd and bitpdim motivated by a Theorem ofPudlak and Rodl.

I ∃c > 0, ∀ f

Ω(bitpdim(f )) ≤ bpsize(f ) ≤ O(bitpdim(f )c)

I For any finite F, ∃ explicit graphs H on N = 2n vertices suchthat

pdF(H) = O(√n) updF(H) = Ω(n),

bitpdim(H) = Ω

(n1.5

log n

)I Strongly restricted variants – spd(f ), uspd(f ).

Coincides with graph parameters bc(Gf ) and bp(Gf ).

17 / 18

Summary


I ∃c > 0, ∀ f




bitpdim(H) = Ω

(n1.5

log n



17 / 18

Summary


I ∃c > 0, ∀ f




bitpdim(H) = Ω

(n1.5

log n

)

I Strongly restricted variants – spd(f ), uspd(f ).Coincides with graph parameters bc(Gf ) and bp(Gf ).

17 / 18

Summary


I ∃c > 0, ∀ f




bitpdim(H) = Ω

(n1.5

log n



17 / 18

Summary


I ∃c > 0, ∀ f




bitpdim(H) = Ω

(n1.5

log n



17 / 18

Open Problems

I Improve updF(f ) lower bound for SId . New techniques toshow bitpdim lower bound ?

I Can any of these results be generalized to non-deterministicBranching programs ?

pd(f)

upd(f) 2D(f)

bitpdim(f) bpsize(f)

bp(Gf )

uspd(f)

bc(Gf )

spd(f)

bitpdim(f)c

D(f) – Deterministic Communication Complexity of fbc(G) – Bipartite Cover number of Gbp(G) – Bipartite Partition number of G

Questions ?

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Open ProblemsI Improve updF(f ) lower bound for SId .

New techniques toshow bitpdim lower bound ?


pd(f)

upd(f) 2D(f)


bp(Gf )

uspd(f)

bc(Gf )

spd(f)

bitpdim(f)c


Questions ?

18 / 18

Open ProblemsI Improve updF(f ) lower bound for SId . New techniques to

show bitpdim lower bound ?


pd(f)

upd(f) 2D(f)


bp(Gf )

uspd(f)

bc(Gf )

spd(f)

bitpdim(f)c


Questions ?

18 / 18


show bitpdim lower bound ?I Can any of these results be generalized to non-deterministic

Branching programs ?

pd(f)

upd(f) 2D(f)


bp(Gf )

uspd(f)

bc(Gf )

spd(f)

bitpdim(f)c


Questions ?

18 / 18




pd(f)

upd(f) 2D(f)


bp(Gf )

uspd(f)

bc(Gf )

spd(f)

bitpdim(f)c


Questions ?

18 / 18




pd(f)

upd(f) 2D(f)


bp(Gf )

uspd(f)

bc(Gf )

spd(f)

bitpdim(f)c


Questions ?18 / 18

Documents

Characterization & Lower Bounds for Branching Program Size