Communication vs. Partition Numbersmgoos/tcs_slides.pdf · Chapter 1 ... Communication Complexity...

Chapter 1 . . .

Communication Complexity———– vs. ———–

Partition Numbers(Based on 2 papers to appear at FOCS)

Mika Goos, Toniann Pitassi, and Thomas WatsonUniversity of Toronto

Goos, Pitassi, Watson (Univ. of Toronto) Communication vs. Partition Numbers 30th September 2015 1 / 22

Chapter 1 . . .

Communication Complexity———– vs. ———–

Partition Numbers(Based on 2 papers to appear at FOCS)

Mika Goos, Toniann Pitassi, and Thomas WatsonUniversity of Toronto

Communication complexity [Yao, ’79]

Alicex ∈ 0, 1n

Boby ∈ 0, 1n

Compute: F(x, y) ∈ 0, 1

Deterministic protocols

0 0 0 0

0 F : X ×Y → 0, 1

0 0 0 0

0 1 0 1

0 0 0 0

0 1 0 1 0 1 0 1

A A A A

0 0 0 0

0 1 0 1 0 1 0 1

A A A A

BBBBBBBB

A A A A

BBBBBBBB

0 0 0 0

0 1 0 1 0 1 0 1

1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1

A A A A

BBBBBBBB

A A A A

BBBBBBBB

0 0 0 0

0 1 0 1 0 1 0 1

1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1

A A A A

BBBBBBBB

A A A A

BBBBBBBB

Pcc(F) := Deterministic communication complexity of F

Partition number χ(F) := Least number of monochromaticrectangles required to partition the communication matrix

0 0 0 0

Basic fact:

Pcc(F) ≥ log χ(F)

[Kushilevitz–Nisan]:

Pcc(F) ≤ O(log χ(F)) ?

0 0 0 0

Basic fact:

Pcc(F) ≥ log χ(F)

[Kushilevitz–Nisan]:

Pcc(F) ≤ O(log χ(F)) ?

Our resultsI Theorem 1: ∃F : Pcc(F) ≥ Ω(log1.5 χ(F))

I Theorem 2: ∃F : Pcc(F) ≥ Ω(log2 χ1(F))

I Corollary: ∃F : Pcc(F) ≥ Ω(log2 rank(F))

I Theorem 3: ∃F : coNPcc(F) ≥ Ω(log1.128 χ1(F))

Co-nondeterministiccommunication complexity

I Previously: [Aho–Ullman–Yannakakis, STOC’83]:

∀F : Pcc(F) ≤ O(log2 χ(F))

[Kushilevitz–Linial–Ostrovsky, STOC’96]:∃F : Pcc(F) ≥ 2 · log χ(F)

One-sided partition numbers:

χ(F) = χ1(F) + χ0(F), χi(F) := least number of rectanglesneeded to partition F−1(i)

One-sided partition numbers:

χ(F) = χ1(F) + χ0(F), χi(F) := least number of rectanglesneeded to partition F−1(i)

I Previously: Clique vs. Independent Set [Yannakakis, STOC’88]:

∀F : Pcc(F) ≤ O(log2 χ1(F))

Observation: χ1(F) ≥ rank(F)

I Previously: Log-rank conjecture [Lovasz–Saks, FOCS’88]:

∀F : Pcc(F) ≤ logO(1) rank(F)

[Kushilevitz–Nisan–Wigderson, FOCS’94]:

∃F : Pcc(F) ≥ Ω(log1.63 rank(F))

I Theorem 3: ∃F : coNPcc(F) ≥ Ω(log1.128 χ1(F))=

Nondeterministic protocols

Algorithmic definition:1 Players guess a proof string p ∈ 0, 1C

2 Alice accepts depending on (x, p)3 Bob accepts depending on (y, p)4 (x, y) is accepted iff both players accept

NPcc(F) := Least C for which there is an abovetype protocol accepting F−1(1)

Combinatorial definition:NPcc(F) := log Cov1(F)Cov1(F) := Least number of monochromatic

rectangles needed to cover F−1(1)

Unambiguity: At most one accepting proof

0 0 0 0

NPcc= log Cov1(F)

UPcc= log χ1(F)

2UPcc= log χ(F)

NPcc= log Cov1(F)

UPcc= log χ1(F)

2UPcc= log χ(F)

UP = Unambiguous NP

NPcc= log Cov1(F)

UPcc= log χ1(F)

2UPcc= log χ(F)

Shorthand: 2UP = UP ∩ coUP

[Yannakakis, STOC’88]: Pcc ≤ (UPcc)2

A rectangle is good for Alice (Bob) if at most half ofthe other rectangles intersect it in rows (columns)

y[Yannakakis, STOC’88]: Pcc ≤ (UPcc)2

Players announce a name of a good-for-themrectangle that intersects their row/column

UPcc = k UPcc = k− 1

y[Yannakakis, STOC’88]: Pcc ≤ (UPcc)2

UPcc = k UPcc = k− 1

I Theorem 2: ∃F : Pcc(F) ≥ Ω(UPcc(F)2)

Our resultsI Theorem 1: ∃F : Pcc(F) ≥ Ω(2UPcc(F)1.5)

I Theorem 3: ∃F : coNPcc(F) ≥ Ω(UPcc(F)1.128)

2UP UP

Clique vs. Independent Set

[Yannakakis, STOC’88]:

Clique vs. Independent Set is complete for UPcc

CISG on a graph G = (V , E):

Alice holds a clique C ⊆ VBob holds an independent set I ⊆ VOutput |C ∩ I| ∈ 0, 1

Note: UPcc(CISG) ≤ log |V|

UPcc reduces to CIS [Yannakakis, STOC’88]

Fix a partition of F−1(1). Construct G = (V, E) whereV = rectangles and u, v ∈ E iff u and v share a row

F reduces to CISG: Alice (Bob) maps her row (column)to the set of rectangles intersecting it

UPcc reduces to CIS [Yannakakis, STOC’88]

Fix a partition of F−1(1). Construct G = (V, E) whereV = rectangles and u, v ∈ E iff u and v share a row

Summary: UPcc(F) = UPcc(CISG) = log |V|

More on CIS

Yannakakis’s motivation:Size of LPs for the vertex packing polytope of GBreakthrough: [Fiorini et al., STOC’12]

For an n-node graph:UPcc(CISG) = log n∀G :

Pcc(CISG) ≤ O(log2 n)∀G :coNPcc(CISG) ≤ O(log2 n)∀G :

Yannakakis’s question:

coNPcc(CISG) ≤ O(log n) ?∀G :

Alon–Saks–Seymour conjecture:

chr(G) ≤ bp(G) + 1 ?∀G :

More on CIS

chr(G) ≤ bp(G) + 1 ?∀G :

chr(G) := Chromatic number of Gbp(G) := Least number of bicliques needed to partition E(G)

More on CIS

chr(G) ≤ bp(G) + 1 ?∀G :

[Huang–Sudakov, 2010]: ∃G : chr(G) ≥ bp(G)6/5

[Shigeta–Amano, 2014]: ∃G : chr(G) ≥ bp(G)2

More on CIS

Polynomial Alon–Saks–Seymour conjecture:

chr(G) ≤ poly( bp(G)) ?∀G :

More on CIS

[Alon–Haviv]

=⇒ [Bousquet et al.]

More on CIS

[Alon–Haviv]

=⇒ [Bousquet et al.]

More on CIS

[Alon–Haviv]

=⇒ [Bousquet et al.]I Theorem 3: ∃F : coNPcc(F) ≥ Ω(UPcc(F)1.128)

Communication:I Theorem 1: ∃F : Pcc(F) ≥ Ω(2UPcc(F)1.5)

2-step strategy:Prove analogous query separationsApply a communication↔querysimulation theorem

Decision tree:I Theorem 1: ∃ f : Pdt( f ) ≥ Ω(2UPdt( f )1.5)

I Theorem 2: ∃ f : Pdt( f ) ≥ Ω(UPdt( f )2)

I Theorem 3: ∃ f : coNPdt( f ) ≥ Ω(UPdt( f )1.128)

2-step strategy:Prove analogous query separationsApply a communication↔querysimulation theorem

Step 1: Query separations

Decision tree models

f : 0, 1n → 0, 1

Pdt( f ) = Deterministic query complexity

NPdt( f ) = Nondeterministic query complexity= 1-certificate complexity= DNF width

UPdt( f ) = Unambiguous query complexity= Unambiguous DNF width

Quadratic P-vs-UP gap

0 1 1 1 1 1

1 0 0 1 1 1

0 1 0 1 0 1

1 0 1 1 1 0

1 1 1 1 1 0

1 1 0 1 1 1

Warm-up example f :M is k× k matrix with entries in 0, 1f (M) = 1 ⇐⇒ M contains a unique all-1 column

NPdt = 2k− 1

Pdt = k2

NPdt = 2k− 1

Pdt = k2

NPdt = 2k− 1

Pdt = k2

NPdt = 2k− 1

Pdt = k2

1 1 1 1 1 1

1 1 1 1 1

NPdt = 2k− 1

Pdt = k2

1 1 1 1 1 1

1 1 1 1 1

NPdt = 2k− 1

Pdt = k2

1 1 1 1 1 1

0 1 1 1 0 1

1 0 1 1 1 0

1 1 1 1 1

1 1 0 1 1 1?

NPdt = 2k− 1

Pdt = k2

Actual gap example f :M is k× k matrix with entries in 0, 1×([k]× [k]∪ ⊥)f (M) = 1 ⇐⇒ M contains a unique all-1 columnthat has a linked list through 0’s in other columns

UPdt ≈ 2k− 1

Pdt ≈ k2

1 1 1 1

1 1 1 1 1

UPdt ≈ 2k− 1

Pdt ≈ k2

1 1 1 1

1 1 1 1 1

UPdt ≈ 2k− 1

Pdt ≈ k2

1 1 1 1 1

1 1 1 1

1 1 1 1 1

UPdt ≈ 2k− 1

Pdt ≈ k2

1 1 1 1 1

1 1 1 1

1 1 1 1 1

UPdt ≈ 2k− 1

Pdt ≈ k2

1 1 1 1 1 1

1 1 1 1

1 0 1 1

1 1 1 1 1 1

1 1 1 1 1?

UPdt ≈ 2k− 1

Pdt ≈ k2

1 1 1 1 1 1

1 1 1 1

1 0 1 1

1 1 1 1 1 1

1 1 0 1 1 1

UPdt ≈ 2k− 1

Pdt ≈ k2

1 1 1 1 1 1

1 1 1 1 1

0 1 1 1 0 1

1 0 1 1 1 0

1 1 1 1 1 1

1 1 0 1 1 1

UPdt ≈ 2k− 1

Pdt ≈ k2

1 1 1 1 1 1

1 1 1 1 1

0 1 1 1 0 1

1 0 1 1 1 0

1 1 1 1 1 1

1 1 0 1 1 1

UPdt ≈ 2k− 1

Pdt ≈ k2

Other separations inspired by our function

[Ambainis–Balodis–Belovs–Lee–Santha–Smotrovs](also [Mukhopadhyay–Sanyal]):

Pdt( f ) ≥ ZPPdt( f )2

Counterexample to Saks–Wigderson’86!

Pdt( f ) ≥ BQPdt( f )4

ZPPdt( f ) ≥ RPdt( f )2

[Ben-David]:

BPPdt( f ) ≥ BQPdt( f )2.5

Other query separations

I Theorem 2:

UPdt( f ) = kPdt( f ) = k2 (Previous slide)

I Theorem 1:

2UPdt(AND f k) = k2

Pdt(AND f k) = k3

Power 1.5 gap—cf. log3 4 ≈ 1.26 from [Savicky’03 / Belovs’06]

=⇒ Involves a delicate recursive construction

Step 2: Simulation theorems

Composed functions f gn

z1 z2 z3 z4 z5 g g g g g

x1 y1 x2 y2 x3 y3 x4 y4 x5 y5

Compose with gn

Examples: • Set-disjointness: OR ANDn

• Inner-product: XOR ANDn

• Equality: AND ¬XORn

Simulation Theorem Template:

Simulate cost-C protocol for f gn in model Mcc

using height-C decision tree for f in model Mdt

i.e., Mcc( f gn) ≈ Mdt( f gn)

x1 y1 x2 y2 x3 y3 x4 y4 x5 y5

Compose with gn

In general: g : 0, 1b × 0, 1b → 0, 1 is a small gadget

Alice holds x ∈ (0, 1b)n

Bob holds y ∈ (0, 1b)n

Inputs x and y encode z := gn(x, y)

x1 y1 x2 y2 x3 y3 x4 y4 x5 y5

Compose with gn

OR OR OR OR OR

z1 z2 z3 z4 z5

x1 y1 x2 y2 x3 y3 x4 y4 x5 y5

Bad example:Gadget must be

chosen carefully!

Known simulation theorems

Model Gadget Reference

P g(x, y) := yxwhere |y| = nΘ(1) [Raz–McKenzie, FOCS’97]

NP g(x, y) := 〈x, y〉 mod 2where |x|, |y| = Θ(log n) [GLMZW, STOC’15]

PP Constant-size g [Sherstov, STOC’08],[Shi–Zhu, QIC’09]

Simulation for P (Our formulation):

Pcc( f gn) = Pdt( f ) ·Θ(log n)

Known simulation theorems

Model Gadget Reference

P g(x, y) := yxwhere |y| = nΘ(1) [Raz–McKenzie, FOCS’97]

NP g(x, y) := 〈x, y〉 mod 2where |x|, |y| = Θ(log n) [GLMZW, STOC’15]

PP Constant-size g [Sherstov, STOC’08],[Shi–Zhu, QIC’09]

SBPWAPP PostBPP PPcorruptionsmooth rectangle

approx rank+

ext. discrepancy discrepancy

F = f gn

Decision tree:I Theorem 1: ∃ f : Pdt( f ) ≥ Ω(2UPdt( f )1.5)

I Theorem 2: ∃ f : Pdt( f ) ≥ Ω(UPdt( f )2)

F = f gn

Future directions

In progress: • Find an F with BPPcc(F) 2UPcc(F)Solved for query complexity:[Kothari–Racicot-Desloges–Santha, RANDOM’15]

Open problems: • Simulation theorem for BPP• Improve gadget size down to b = O(1)

(Gives new proof of Ω(n) bound for disjointness)

Big challenges: • Log-rank conjecture• Lower bounds against PHcc (or AMcc)

Cheers!

Future directions

In progress: • Find an F with BPPcc(F) 2UPcc(F)Solved for query complexity:[Kothari–Racicot-Desloges–Santha, RANDOM’15]

Open problems: • Simulation theorem for BPP• Improve gadget size down to b = O(1)

(Gives new proof of Ω(n) bound for disjointness)

Big challenges: • Log-rank conjecture• Lower bounds against PHcc (or AMcc)

Cheers!Goos, Pitassi, Watson (Univ. of Toronto) Communication vs. Partition Numbers 30th September 2015 22 / 22

Communication vs. Partition Numbersmgoos/tcs_slides.pdf · Chapter 1 ... Communication Complexity...

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